Mathematisches Institut der LMUsawant/Files/Anand_Thesis.pdf · Declaration This thesis is a presentation of my original research work. Wherever contributions of others are involved,

A1-connected components of schemes

A Thesis

Submitted to theTata Institute of Fundamental Research, Mumbai

for the degree of Doctor of Philosophy

in Mathematics

by

Anand Sawant

School of MathematicsTata Institute of Fundamental Research

Mumbai

December 2014

Declaration

This thesis is a presentation of my original research work. Wherever contributions ofothers are involved, every effort is made to indicate this clearly, with due reference tothe literature, and acknowledgement of collaborative research and discussions.

The work was done under the guidance of Professor V. Srinivas, at the Tata Institute ofFundamental Research, Mumbai.

Anand Sawant

In my capacity as supervisor of the candidate’s thesis, I certify that the above statementsare true to the best of my knowledge.

Professor V. SrinivasDate:

Contents

Organization of the thesis 1

1 A brief introduction to A1-homotopy theory 31.1 Simplicial sheaves of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The A1-homotopy category . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 The Morel-Voevodsky Sing˚ construction . . . . . . . . . . . . . . . . . . 6

2 Morel’s conjecture on A1-connected components 72.1 The sheaf of A1-connected components . . . . . . . . . . . . . . . . . . . 72.2 A conjectural description of πA1

0 . . . . . . . . . . . . . . . . . . . . . . . 102.3 The formalism of A1-ghost homotopies . . . . . . . . . . . . . . . . . . . 132.4 Almost proper sheaves and a refinement of a result of Asok-Morel . . . . 18

3 Counterexamples to conjectures of Asok-Morel 233.1 A smooth proper scheme whose Sing˚ is not A1-local . . . . . . . . . . . 233.2 Example showing that πA1

0 need not be birational . . . . . . . . . . . . . 29

4 Relationship of A1-connectedness with R-equivalence 314.1 R-equivalence and near-rationality . . . . . . . . . . . . . . . . . . . . . . 314.2 R-equivalence in anisotropic groups . . . . . . . . . . . . . . . . . . . . . 334.3 A1-connectedness of R-trivial smooth proper varieties . . . . . . . . . . . 38

5 A1-connected components of smooth proper surfaces 435.1 The Morel and Asok-Morel conjectures for non-uniruled surfaces . . . . . 435.2 The case of ruled surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 46

A Model categories 61

B Nisnevich topology 65

Bibliography 69

Organization of the thesis

The thesis consists of five chapters and two appendices; we give a brief outline of thecontents here.

Chapter 1 gives a brief introduction to A1-homotopy theory. We introduce simplicialsheaves of sets on the big Nisnevich site of smooth schemes over a field k, which are theobjects of the A1-homotopy category and discuss some important constructions.

Chapter 2 introduces the sheaf of A1-connected components of a simplicial sheaf,which is the main object of interest of this thesis. We also discuss a conjecture ofMorel on the A1-invariance of this sheaf. We obtain a conjectural description of the A1-connected components sheaf in Section 2.2 and obtain equivalent conditions for Morel’sconjecture to hold. In Section 2.4, we obtain a refinement of a result of Asok-Morel, whichexplicitly describes the sections of the A1-connected components of a proper scheme overfields.

Chapter 3 is devoted to counterexamples to the conjectures of Asok-Morel aboutagreement of the A1-connected components sheaf of a smooth scheme with the sheaf ofits A1-chain connected components and about birational invariance of the A1-connectedcomponents sheaf of smooth proper schemes.

Chapter 4 studies the connection between R-equivalence of points of a scheme andits A1-connected components. In Section 4.2, we show that the sections of A1-connectedcomponents of a semisimple, absolutely almost simple, simply connected anisotropicalgebraic group over a field agree with the R-equivalence classes of points of the group.In Section 4.3, we show that R-trivial smooth proper varieties over infinite fields areA1-connected. We also obtain examples of rational smooth affine varieties for which theA1-connected components fail to agree with the A1-chain connected components.

Chapter 5 explores the A1-connected components of surfaces over an algebraicallyclosed field. Using a method suggested by results of Chapter 2, we show in Section 5.1that Morel’s and Asok-Morel’s conjectures hold for a proper non-uniruled surface over analgebraically closed field. In Section 5.2, we explicitly obtain the conjectural descriptionof A1-connected components of a blow-up of a P1-bundle over a curve at a single closedpoint and show that these are also counterexamples to Asok-Morel’s conjecture aboutbirationality of the A1-connected components sheaf.

Appendix A gives some basic definitions and terminology regarding model categories,and Appendix B describes Nisnevich topology and some of its basic properties.

Chapter 1

A brief introduction to A1-homotopytheory

1.1 Simplicial sheaves of sets

We will always work over a fixed base field k. Let Sm{k denote the category of smooth(separated, finite type) schemes over k with the Nisnevich topology. We refer the readerto Appendix B for definition and basic properties of the Nisnevich topology. In thissection, we shall introduce simplicial sheaves of sets on Sm{k for the Nisnevich topology,which form the objects of the A1-homotopy category.

Let ∆ denote the category of simplices, whose objects are finite ordered sets

rns :“ t0 ă 1 ă ¨ ¨ ¨ ă nu,

for non-negative integers n and morphisms are nondecreasing functions. A simplicialobject in a category C is a functor X : ∆op Ñ C. We usually write Xn for X prnsq.

Let ShvpSm{kq denote the category of sheaves of sets on Sm{k for the Nisnevichtopology. We will continue to denote the sheaf represented by a scheme X by the sameletter. Let ∆oppShvpSm{kqq denote the category of simplicial objects in ShvpSm{kq.Thus, an object X of ∆oppShvpSm{kqq is determined by a collection of sheaves of setsXn, for each non-negative integer n together with face and degeneracy morphisms

dni : Xn Ñ Xn´1 n ą 0, i “ 0, . . . , n;

sni : Xn Ñ Xn`1 n ě 0, i “ 0, . . . , n

satisfying the usual identities of [29, p. 1]. We let ∆oppPShpSm{kqq denote the categoryof simplicial presheaves of sets on Sm{k.

To any simplicial set X, we assign the corresponding constant simplicial sheaf in∆oppShvpSm{kqq, which we will denote by the same letter. Similarly, to any sheaf onSm{k, one can associate the corresponding constant simplicial sheaf, which will also bedenoted by the same letter.

4 Chapter 1. A brief introduction to A1-homotopy theory

A simplicial sheaf X is said to be of simplicial dimension ď n if the natural mapXn ˆ ∆n Ñ X is an epimorphism, that is, Xn ˆ ∆npmq Ñ Xm is an epimorphism ofsheaves for each m. We will identify sheaves of sets on Sm{k with simplicial sheaves ofsimplicial dimension 0.

1.2 The A1-homotopy category

In this section, we define the A1-homotopy category of schemes over k, denoted by Hpkq.Following Quillen [38], we know that an ideal setting for performing homotopy theoryis that of model categories. The definition and basic properties of model categories aregathered in Appendix A. An excellent reference for simplicial homotopy theory is [17].The underlying category of Hpkq is ∆oppShvpSm{kqq and its elements are sometimesreferred to as spaces. One has to put a suitable structure of a model category on∆oppShvpSm{kqq. This was first done by Morel-Voevodsky in two steps, which webriefly describe below.

The first step defines the injective Nisnevich local model structure (see [35, Definition1.2, Theorem 1.4]).

Definition 1.2.1. Let f : X Ñ Y be a morphism of simplicial sheaves of sets on Sm{k.

• f is said to be a simplicial weak equivalence if it induces a weak equivalence ofsimplicial sets on all stalks in the Nisnevich topology.

• f is called a cofibration if it is section-wise injective.

• f is called a fibration if given a commutative diagram in ∆oppShvpSm{kqq withsolid arrows

A //

i��

Xf��

B //

??

Y

in which i is a trivial cofibration, the dotted arrow always exists (this is called theright-lifting property with respect to trivial cofibrations).

Theorem 1.2.2. The notions of weak equivalences, cofibrations and fibrations as inDefinition 1.2.1 give ∆oppShvpSm{kqq the structure of a model category. Moreover, thismodel structure is closed and proper.

See [35, §1], [23, 2.17] and [24] for more details and proofs of results.

Definition 1.2.3. The homotopy category associated with the injective Nisnevich localmodel structure on ∆oppShvpSm{kqq is called the simplicial homotopy category and isdenoted by Hspkq.

1.2. The A1-homotopy category 5

Remark 1.2.4. The injective Nisnevich local model structure was extended to simpli-cial presheaves by Jardine (see [25, Appendix B]), where weak equivalences are local(stalkwise) weak equivalences and cofibrations are monomorphisms. He also showedthat this makes ∆oppPShpSm{kqq a proper, simplicial and cofibrantly generated modelcategory. See [13, Section 5.2 of Voevodsky’s lectures] for a detailed proof.

There is an adjunction

∆oppPShpSm{kqq

aNis

Õi

∆oppShvpSm{kqq,

where i denotes the inclusion and aNis denotes the Nisnevich sheafification functor.Moreover, the above adjunction is a Quillen pair.

Definition 1.2.5. A simplicial sheaf X P ∆oppShvpSm{kqq is said to be A1-local if forany simplicial sheaf Y P ∆oppShvpSm{kqq, the projection map Y ˆ A1 Ñ Y induces abijection

HomHspkqpY ,X q Ñ HomHspkqpY ˆ A1,X q.

Definition 1.2.6. The left Bousfield localization of the injective Nisnevich local modelstructure on ∆oppShvpSm{kqq with respect to the collection of all the projection mapsX ˆ A1 Ñ X is called the A1-model structure. The associated homotopy category iscalled the A1-homotopy category and is denoted by Hpkq.

The weak equivalences, cofibrations and fibrations in this model structure are calledA1-weak equivalences, A1-cofibrations and A1-fibrations, respectively. The followingcharacterization of A1-fibrant objects is often useful.

Lemma 1.2.7. ([35, §2, Proposition 3.19]) A simplicial sheaf on Sm{k is A1-fibrant ifand only if it is simplicially fibrant (that is, fibrant in the injective Nisnevich local modelstructure) and A1-local.

Remark 1.2.8. Since sheaves have simplicial dimension 0, they are always simpliciallyfibrant. Consequently, a sheaf F is A1-fibrant if and only if it is A1-local. This happens ifand only if F is A1-invariant, that is, for every U P Sm{k the projection map UÂ1 Ñ Uinduces a bijection FpUq Ñ FpU ˆ A1q.

Definition 1.2.9. A scheme X P Sm{k is said to be A1-rigid, if for every U P Sm{k,the map

XpUq ÝÑ XpU ˆ A1q

induced by the projection map U ˆ A1 Ñ U is a bijection.

Remark 1.2.10. By Remark 1.2.8, a smooth k-scheme X is A1-local if and only if it isA1-rigid.

6 Chapter 1. A brief introduction to A1-homotopy theory

1.3 The Morel-Voevodsky Sing˚ construction

We now recall the construction of the functor Sing˚ as given in [35, p. 87]. Let ∆‚

denote the cosimplicial scheme

∆n “ Spec

ˆ

krx0, ..., xns

př

i xi “ 1q

˙

,

with usual coface and codegeneracy maps analogous to those on topological simplices.

Definition 1.3.1. For any simplicial presheaf X , define Sing˚pX q to be the diagonal ofthe bisimplicial presheaf Homp∆‚,X q. Thus, for every U P Sm{k, we have

Sing˚pX qnpUq “ HomShvpSm{kqpU ˆ∆n,Xnq.

Note that if X is a simplicial sheaf of sets on Sm{k, then so is Sing˚pX q. There isa functorial morphism X Ñ Sing˚pX q induced by XnpUq Ñ XnpU ˆ ∆nq, which is anA1-weak equivalence. The functor Sing˚ takes A1-fibrant objects to A1-fibrant objectsand takes A1-homotopic maps to simplicially homotopic maps.

The following lemma explicitly describes (see [35, p.107] for a proof) an A1-fibrantreplacement functor ∆oppShvpSm{kqq Ñ ∆oppShvpSm{kqq.

Lemma 1.3.2. Choose a fibrant replacement functor Ex for the injective Nisnevich localmodel structure on ∆oppShvpSm{kqq. Then the functor

LA1 : ∆oppShvpSm{kqq Ñ ∆op

pShvpSm{kqq

defined by settingLA1 :“ Ex ˝ pEx ˝ Sing˚q

N˝ Ex

is such that for any space X , the object LA1pX q is an A1-fibrant object. Moreover, thecanonical morphism X Ñ LA1pX q is a monomorphism and an A1-weak equivalence.

Chapter 2

Morel’s conjecture on A1-connectedcomponents

In this chapter, we study the sheaf of A1-connected components of a simplicial sheaf anddiscuss a conjecture of Morel on the A1-invariance of this sheaf. We begin with somebasic definitions in Section 2.1 and obtain a conjectural description of the A1-connectedcomponents sheaf in Section 2.2. We also obtain equivalent conditions for Morel’s conjec-ture to hold. These results allow us to bring in geometric methods to study A1-connectedcomponents of schemes (sometimes at the cost of heavy book-keeping). We introducethe device called A1-ghost homotopies to facilitate this in Section 2.3. In Section 2.4, weobtain a refinement of a result of Asok-Morel, which explicitly describes the sections ofthe A1-connected components sheaf of a proper scheme over field extensions of the basefield.

2.1 The sheaf of A1-connected components

We will always work over a fixed base field k. Let Sm{k denote the category of smooth fi-nite type schemes over k with the Nisnevich topology. As in Chapter 1, ∆oppPShpSm{kqqand ∆oppShvpSm{kqq will denote the categories of simplicial presheaves and sheaves ofsets on Sm{k, respectively. All presheaves or sheaves will also be considered as constantsimplicial objects.

Definition 2.1.1. For any simplicial presheaf (or a sheaf) X on Sm{k, we define πs0pX qto be the Nisnevich sheafification of the presheaf of sets π0pX q given by

π0pX qpUq :“ HomHspkqpU,X q.

Notation 2.1.2. If we have two sets R, S with maps f, g : R Ñ S, we denote byS

gRf

,

the quotient of S by the equivalence relation generated by declaring fptq „ gptq for all

8 Chapter 2. Morel’s conjecture on A1-connected components

t P R. In this notation, for any simplicial presheaf X ,

π0pX qpUq “X0pUq

d0X1pUqd1,

where d0, d1 : X1pUq Ñ X0pUq are the face maps in the simplicial set X‚pUq.

Definition 2.1.3. The sheaf of A1-connected components of a simplicial sheaf X P

∆oppShvpSm{kqq is defined to be

πA1

0 pX q :“ πs0pLA1pX qq,

where LA1 denotes the A1-fibrant replacement functor defined in Chapter 1.

In other words, for X P ∆oppShvpSm{kqq, πA1

0 pX q is the sheafification of the presheaf

U P Sm{k ÞÑ HomHpkqpU,X q.

Definition 2.1.4. A presheaf F of sets on Sm{k is said to be A1-invariant if for everyU , the map FpUq Ñ FpU ˆ A1q induced by the projection U ˆ A1 Ñ U , is a bijection.

Since LA1pX q is A1-local for any simplicial presheaf X , it is easy to see that thepresheaf

U ÞÑ π0pLA1pX qpUqqis A1-invariant. A conjecture of Morel (see [32, 1.12]) asserts that its Nisnevich sheafifi-cation πA1

0 pX q remains A1-invariant.

Conjecture 2.1.5 (Morel). For any simplicial sheaf X , πA1

0 pX q is A1-invariant.

The following example shows that sheafification in Nisnevich topology can destroyA1-invariance of a presheaf, in general.

Example 2.1.6. Consider the presheaf F on Sm{k whose sections FpUq are defined tobe the set of k-morphisms from U Ñ A1

k which factor through a proper open subset ofA1k. One can show that F is A1-invariant, but its Nisnevich sheafification, which is the

sheaf represented by A1k, is clearly not A1-invariant.

In [9], Morel’s conjecture was proved in the special case when X is an H-groupor a homogeneous space for an H-group. As observed in [9], such simplicial sheavesX have a special property – for U Ă Y , an inclusion of a dense open subset, the mapπs0pX qpY q Ñ πs0pX qpUq is injective. Unfortunately, this fails for general simplicial sheavesX , as shown by the following example.

Example 2.1.7. Let U be a non-empty proper subset of an abelian variety A over k. LetX be the non-separated scheme obtained by gluing two copies of A along U . Let X con-tinue to denote the corresponding constant simplicial sheaf in ∆oppShvpSm{kqq. Clearly,XpAq Ñ XpUq is not injective. However, one can verify that the sheaf represented byX is A1-invariant and hence, X “ πA1

0 pXq.

2.1. The sheaf of A1-connected components 9

Remark 2.1.8. For any simplicial sheaf X , it is easy to see from the definitions thatthe sheaf πA1

0 pX q has the following universal property: any morphism X Ñ F , where Fis an A1-invariant sheaf of sets, factors uniquely through the sheaf πA1

0 pX q. To see this,first note that the map X Ñ F factors through LA1pX q since we have a commutativediagram

X

��

// F

��LA1pX q // LA1pFq

and since the map F Ñ LA1pFq is an isomorphism, F being A1-invariant. The mapLA1pX q Ñ F factors through πA1

0 pX q “ πs0pLA1pX qq since its target is of simplicialdimension 0.

Remark 2.1.9. Observe that a retract of an A1-invariant sheaf is always A1-invariant.Indeed, if a morphism ι : F Ñ G of sheaves of sets for the Nisnevich topology on Sm{k,where G is A1-invariant, admits a retract r : G Ñ F , then we have a commutativediagram for each U P Sm{k:

FpUq //

id

**

p˚

��

GpUq r˚ //

p˚

��

FpUqp˚

��FpU ˆ A1q //

id

44GpU ˆ A1q

r˚ // FpU ˆ A1q

in which the vertical maps are induced by the projection map p : U ˆ A1 Ñ U . Sinceboth the horizontal maps induced by r are surjective and the middle vertical map isa bijection, it follows that p˚ : FpUq Ñ FpU ˆ A1q is surjective. This map is alreadyinjective, since P : U ˆ A1 Ñ U has a section. Consequently, F is A1-invariant. Thisfact will be used later (Lemma 2.2.4).

Definition 2.1.10. Let F be a Nisnevich sheaf of sets on Sm{k, considered as a constantsheaf in 4opShvpSm{kq. Define SpFq to be the sheaf associated to the presheaf Spregiven by

SprepUq :“FpUq

σ0FpU ˆ A1qσ1,

for U P Sm{k, where σ0, σ1 : FpU ˆ A1q Ñ FpUq are the maps induced by the 0- and1-sections U Ñ U ˆ A1, respectively. Note that even if F is a sheaf, SprepFq need notbe a sheaf.

For any sheaf F , there exists a canonical epimorphism F Ñ SpFq. This gives acanonical epimorphism

F Ñ limÝÑn

SnpFq.


Remark 2.1.11. It is clear from the definition of Sing˚pFq that

π0pSing˚pFqqpUq “FpUq

σ0FpU ˆ A1qσ1.

Thus, SpFq “ πs0pSing˚pFqq, for any sheaf F .

Remark 2.1.12. Let X be a smooth scheme over k and view it as a Nisnevich sheafof sets over Sm{k. The sheaf SpXq defined above (Definition 2.1.10) is none other thanthe sheaf πch0 pXq of A1-chain connected components of X introduced by Asok-Morel [1,Definition 2.2.4]. We use S for πch0 in this paper only for typographical reasons.

For smooth schemes over a field k, Asok-Morel have conjectured the following (see[1, Conjecture 2.2.8]):

Conjecture 2.1.13. For any smooth scheme X over a field k, the natural epimorphismSpXq Ñ πA1

0 pXq is an isomorphism.

Asok-Morel also mention that this would be true if one proves that Sing˚pXq is A1-local, for any smooth scheme X over k (see [1, Remark 2.2.9]). However, in Chapter 3,we give examples of schemes for which this property and Conjecture 2.1.13 fail to hold.

2.2 A conjectural description of πA1

0

In this section, we use the sheaf of A1-chain connected components of a (Nisnevich) sheafF on Sm{k to construct a sheaf LpFq that is closely related to πA1

0 pFq. We shall thenshow that these two sheaves are isomorphic, provided πA1

0 pFq is A1-invariant.

Theorem 2.2.1. For any sheaf of sets F on Sm{k, the sheaf

LpFq :“ limÝÑn

SnpFq

is A1-invariant.

Proof. It is enough to show that for every smooth k-scheme U , the map

LpFqpUq Ñ LpFqpU ˆ A1q

is surjective. (It is already injective since the projection map U ˆ A1 Ñ U admits asection.) Let t P LpFqpU ˆ A1q. Since the sheaf LpFq is a filtered colimit, we have foreach U P Sm{k:

LpFqpUq “ limÝÑnPNSnpFqpUq.

2.2. A conjectural description of πA1

0 11

Thus, t is represented by an element tn of SnpFqpU Â1q for some n. We will show thatt is contained in the image of LpFqpUq by showing that the image of tn in Sn`1pU Â1q

is contained in the image of Sn`1pFqpUq. Let m denote the multiplication map

U ˆ A1ˆ A1

Ñ U ˆ A1 ; pu, x, yq ÞÑ pu, xyq.

Consider the elementm˚ptnq P SnpFqpU ˆ A1

ˆ A1q.

Thenσ˚1 ˝m

˚ptnq “ tn, and σ˚0 ˝m

˚ptnq “ p˚ ˝ s˚0ptnq,

where U Â1 pÑ U is the projection, U

s0ÝÑ U Â1 is the zero section and σi : U Â1 Ñ

U Â1Â1 is the i-section of the projection pU Â1qÂ1 Ñ U Â1. Thus, the imageof tn in Sn`1pFqpU ˆ A1q is contained in the image of the map

Sn`1pFqpUq Ñ Sn`1

pFqpU ˆ A1q.

This proves the result.

Remark 2.2.2. In view of Theorem 2.2.1, the canonical map F Ñ LpFq uniquelyfactors through the canonical map F Ñ πA1

0 pFq.

Lemma 2.2.3. LpFq satisfies the following universal property: any map from a sheafF to an A1-invariant sheaf uniquely factors through the canonical map F Ñ LpFq.

Proof. Let G be an A1-invariant sheaf of sets on Sm{k. Let U be a smooth scheme overk and let f, g P FpUq be such that they map to the same element in SpFqpUq underthe canonical epimorphism F Ñ SpFq. Then there exists a Nisnevich cover V Ñ Usuch that f |V and g|V are A1-chain homotopic. Since G is A1-invariant, they map to thesame element in GpV q and consequently, in GpUq. Therefore, the map F Ñ G uniquelyfactors through the map F Ñ SpFq. Continuing in the same way, we see that F Ñ Guniquely factors through F Ñ LpFq.

Recall that πA1

0 also satisfies the same universal property (Remark 2.1.8) but is notknown to be A1-invariant in general.

This is related to Morel’s conjecture on the A1-invariance of A1-connected compo-nents sheaf πA1

0 by the following observations, which give equivalent characterizations ofMorel’s conjecture for simplicial sheaves.

Lemma 2.2.4. Let F be a Nisnevich sheaf of sets on Sm{k, considered as an elementof 4opShvpSm{kq. The following are equivalent:p1q The sheaf πA1

0 pFq is A1-invariant.p2q The canonical map

πA1

0 pFq Ñ LpFqadmits a retract.


Proof. p1q ñ p2q is straightforward by Lemma 2.2.3 and the universal property of πA1

0

(Remark 2.1.8). p2q ñ p1q follows since LpFq is A1-invariant (Theorem 2.2.1) and sincea retract of an A1-invariant sheaf is A1-invariant.

Corollary 2.2.5. Let F be a sheaf of sets on Sm{k such that πA1

0 pFq is A1-invariant.Then the canonical map

πA1

0 pFq Ñ LpFq

is an isomorphism.

Moreover, these results immediately give a description of the A1-connected compo-nents sheaf πA1

0 pX q for a simplicial sheaf of sets X on Sm{k, provided Morel’s conjecture(Conjecture 2.1.5) is true.

Proposition 2.2.6. The following are equivalent:p1q The sheaf πA1

0 pX q is A1-invariant, for all spaces X P 4opShvpSm{kq.p2q The canonical map πA1

0 pX q Ñ πA1

0 pπs0pX qq is an isomorphism, for all spaces X P

4opShvpSm{kq.

Proof. p1q ñ p2q : Consider the following commutative diagram with the natural maps(all of which are epimorphisms)

X // πs0pX q //

��

πA1

0 pX q

xx

πA1

0 pπs0pX qq

Since πA1

0 pX q is A1-invariant by assumption, the map πs0pX q Ñ πA1

0 pX q has to uniquelyfactor through πA1

0 pπs0pX qq, by the universal property of πA1

0 pπs0pX qq. This gives an

inverse to the map πA1

0 pX q Ñ πA1

0 pπs0pX qq, by uniqueness.

p2q ñ p1q : Let X be a space and let F :“ πs0pX q be the sheaf of (simplicial) connectedcomponents of X . We are given that πA1

0 pX q Ñ πA1

0 pFq is an isomorphism. We willprove that πA1

0 pFq is A1-invariant. We have a natural map

SpFq “ πs0pSing˚pFqq Ñ πA1

0 pSing˚pFqq » πA1

0 pFq.

By Remark 2.1.11, we have

S2pFq “ Spπs0pSing˚pFqqq “ πs0pSing˚π

s0pSing˚pFqqq,

whence we have a map S2pFq Ñ πA1

0 pSing˚πs0pSing˚pFqqq. But hypothesis p2q implies

that

πA1

0 pSing˚πs0pSing˚pFqqq » πA1

0 pπs0pSing˚pFqqq » πA1

0 pSing˚pFqq » πA1

0 pFq.

2.3. The formalism of A1-ghost homotopies 13

The composition of these maps gives a natural map S2pFq Ñ πA1

0 pFq, which makes thefollowing diagram commute, where the maps are the ones defined above.

SpFq

$$

// S2pFq

zz

πA1

0 pFq

Continuing in this way, we obtain a compatible family of maps SipFq Ñ πA1

0 pFq, foreach i, giving a retract LpFq Ñ πA1

0 pFq. Lemma 2.2.4 now implies that πA1

0 pFq isA1-invariant, proving p1q.

2.3 The formalism of A1-ghost homotopies

Let F be any sheaf of sets on Sm{k. In order to address the question of A1-invarianceof SpFq, we need to obtain an explicit description for the elements of the set SpFqpUqwhere U is a smooth scheme over k. We will then specialize to the case when F isrepresented by a proper scheme X over k.

As before, for any scheme U over k, we let σ0 and σ1 denote the morphisms U Ñ

U ˆ A1 given by u ÞÑ pu, 0q and u ÞÑ pu, 1q, respectively.

Definition 2.3.1. Let F be a sheaf of sets over Sm{k and let U be a smooth schemeover k.

(1) An A1-homotopy of U in F is a morphism an element h of FpU ˆ A1kq. We say

that t1, t2 P FpUq are A1-homotopic if there exists an A1-homotopy h P FpU Â1q

such that σ˚0 phq “ t1 and σ˚1 phq “ t2q.

(2) An A1-chain homotopy of U in F is a finite sequence h “ ph1, . . . , hnq where eachhi is an A1-homotopy of U in F such that σ˚1 phiq “ σ˚0 phi`1q for 1 ď i ď n´ 1. Wesay that t1, t2 P FpUq are A1-chain homotopic if there exists an A1-chain homotopyh “ ph1, . . . , hnq such that σ˚0 ph1q “ t1 and σ˚1 phnq “ t2.

Clearly, if t1 and t2 are A1-chain homotopic, they map to the same element ofSpFqpUq. The converse is partially true - if t1, t2 P FpUq are such that they mapto the same element in SpFqpUq, then there exists a Nisnevich cover V Ñ U such thatt1|V and t2|V are A1-chain homotopic.

Now we explicitly describe the sections of SpFq. Let t P SpFqpUq. Since F Ñ SpFqis an epimorphism of sheaves, there exists a finite Nisnevich cover V Ñ U such that t|Vcan be lifted to s P FpV q. Let pr1, pr2 : V Û V Ñ V be the two projections. Then,since the two elements pr˚i psq map to the same element in SpFqpV Û V q, there existsa finite Nisnevich cover W Ñ V Û V such that pr˚1 psq|W and pr˚2 psq|W are A1-chainhomotopic. Conversely, given a finite cover V Ñ U and an element s of FpV q such that


pr˚1 psq and pr˚2 psq become A1-chain homotopic after restricting to a Nisnevich cover ofV Û V , we can obtain a unique element t of SpFqpUq.

Applying the same argument to A1U , we are led to define the notion of an A1-ghost

homotopy.

Definition 2.3.2. Let F be a sheaf of sets and let U be a smooth scheme over k. AnA1-ghost homotopy of U in F consists of the data

H :“ pV Ñ A1U ,W Ñ V Â1

UV, h, hW q

which is defined as follows:

(1) V Ñ A1U is a Nisnevich cover of A1

U .

(2) W Ñ V Â1UV is a Nisnevich cover of V Â1

UV .

(3) h is a morphism V Ñ F .

(4) hW is an A1-chain homotopy connecting the two morphisms W Ñ V Â1UV

priÑ

V Ñ F where pr1 and pr2 are the projections V Â1UV Ñ V . (Thus hW is a finite

sequence phW1 , . . .q of A1-homotopies satisfying the appropriate conditions as givenin Definition 2.3.1.)

Let t1, t2 P FpUq. We say that H connects t1 and t2 (which are then said to be A1-ghosthomotopic) if the morphisms σ0, σ1 : U Ñ A1

U admit lifts rσ0 : U Ñ V and rσ1 : U Ñ Vsuch that h ˝ rσ0 “ t1 and h ˝ rσ1 “ t2.

By the discussion above, a homotopy in SpFq gives rise (non-uniquely) to a ghosthomotopy in F . On the other hand, a ghost homotopy in F gives rise to a uniquehomotopy in SpFq. Also, if t1, t2 are A1-ghost homotopic elements of FpUq, then itis clear that their images in SpFqpUq are A1-homotopic. The converse is partially true- if rt1, rt2 P SpFqpUq are A1-homotopic, then they have preimages t1, t2 P FpUq whichbecome A1-ghost homotopic over some Nisnevich cover of U . In general, if H is anA1-ghost homotopy of U , it is possible that there do not exist two elements t1, t2 P FpUqwhich are connected by H. This is because the lifts rσ0 and rσ1 may not exist until wepass to a Nisnevich cover of U . However, if U is a smooth Henselian local scheme overk, then the morphisms rσ0 and rσ1 do exist and thus there exist preimages of rt1 and rt2which are A1- ghost homotopic.

Notation 2.3.3. Let U be an essentially smooth scheme over k. Then U “ limαXα,where tXαuα is a filtered inverse system of (finite-type, separated) smooth schemes overk in which the transition maps are etale affine morphisms. Let F be a sheaf of sets onSm{k. We set FpUq :“ colimαFpXαq. This is well-defined by the results in [18, 8.14].

We summarize the above discussion as follows:


Lemma 2.3.4. Let F be a sheaf of sets over Sm{k. Then, SpFq “ S2pFq if and onlyif for every smooth Henselian local scheme U , if t1, t2 P FpUq are A1-ghost homotopic,then they are also A1-chain homotopic.

In other words, if U is a smooth Henselian local scheme over k, then S2pFqpUq isprecisely the quotient of the set FpUq by the equivalence relation generated by A1-ghost homotopies. We will now set up the notation for such data in order to representhomotopies in SnpFq. We will only need to do this for the cases n “ 1 and 2 for thepurposes of this thesis. However, we give a general inductive definition for the sake ofcompleteness.

Definition 2.3.5. Let F be a sheaf of sets and let U be an essentially smooth schemeover k. Let n ě 0 be an integer. The notion of an n-ghost homotopy and n-ghost chainhomotopy is defined as follows:

(1) A 0-ghost homotopy is the same as an A1-homotopy as defined in Definition 2.3.1.Similarly, a 0-ghost chain homotopy is the same as an A1-chain homotopy.

(2) Assuming that the notion of m-ghost homotopy and m-ghost chain homotopy hasbeen defined for m ă n, we define an n-ghost homotopy. For t1, t2 P FpUq, ann-ghost homotopy connecting t1, t2 consists of the data:


UV, rσ0, rσ1, h,HW

q

which is defined as follows:

(a) V Ñ A1U is a Nisnevich cover of A1

U .

(b) For i “ 0, 1, rσi : U Ñ V is a lift of σi : U Ñ A1U .

(c) W Ñ V Â1UV is a Nisnevich cover of V Â1

UV .

(d) h is a morphism V Ñ F such that h ˝ rσi “ ti.

(e) HW is an pn´ 1q-ghost chain homotopy connecting the two morphisms W Ñ

V Â1UV

priÑ V Ñ F where pr1 and pr2 are the projections V Â1

UV Ñ V .

With this notation, we will also write Hpiq “ ti for i “ 0, 1.

(3) Suppose the notion of an n-ghost homotopy has been defined. Then for elementst1, t2 P FpUq, an n-ghost chain homotopy connecting t1, t2 is a finite sequenceH “ pH1, . . . ,Hrq where each Hi is an n-ghost homotopy of U in F such thatpHiqp1q “ pHiqp0q for 1 ď i ď r ´ 1, pH1qp0q “ t1 and pHrqp1q “ t2.

Note that with the above definition, a 1-ghost homotopy is nothing but a ghosthomotopy in the sense of Definition 2.3.2.


Lemma 2.3.6. Let F be a sheaf of sets, let U be an essentially smooth scheme overk and let n ě 0 be an integer. Let t1, t2 P FpUq. If t1, t2 are n-ghost homotopic thentheir images in SnpFqpUq are A1-homotopic. Conversely, if the images of t1, t2 are A1-homotopic, there exists a Nisnevich cover p : U 1 Ñ U such that the images of ti ˝ p inSnpFqpUq are n-ghost homotopic.

Proof. First we prove that if t1 and t2 are n-ghost homotopic, then their images in SnpFqare A1-homotopic. We prove this by induction on n. The case n “ 0 is trivial. We firstcheck the case n “ 1. Thus, suppose we have a 1-ghost homotopy


UV, rσ0, rσ1, h, h

Wq

connecting t1 and t2. Then for i “ 1, 2, if pri denotes the projection on the i-th factor,we see that the two morphisms

W Ñ V Â1UV

priÑ V

hÑ F Ñ SpFq

are equal. Since W Ñ V Â1UW is a Nisnevich cover, we see that the morphism

V Ñ F Ñ SpFq descends to a morphism A1U Ñ F , connecting the images of t1 and t2

in SpFq.Now suppose that it is known that for any scheme T , if two morphisms T Ñ F are

pn ´ 1q-ghost homotopic, then their images in Sn´1pFq are A1-homotopic. Suppose weare given n-ghost homotopy


UV, rσ0, rσ1, h, h

Wq

between t1, t2 P FpUq. Then, by the induction hypothesis, on composing with themorphism F Ñ Sn´1pF q, we get a 1-ghost homotopy of U in Sn´1pFq. Since we knowthe result to be true for n “ 1, we see that a 1-ghost homotopy in Sn´1pFq gives rise toan A1-homotopy of U in SnpFq connecting the images of t1 and t2 in SnpFqpUq.

Now we prove the converse, again by induction on n, the case n “ 0 being trivial.Suppose t1, t2 P FpUq are such that their images in SnpFqpUq are connected by a singleA1-homotopy. As F Ñ SnpFq is an epimorphism, there exists a Nisnevich cover V Ñ A1

U

such that the morphism V Ñ A1U Ñ SnpFq can be lifted to a morphism V Ñ F .

By replacing U by some suitable Nisnevich cover U 1 Ñ U , we may assume that themorphisms σi : U Ñ A1

U lift to V . Now, in the notation of Definition 2.3.2, it sufficesto construct the Nisnevich cover W Ñ V Â1

UV and the pn´ 1q-ghost chain homotopy

HW .The two morphisms

V Â1UV

priÑ V

hÑ F Ñ Sn´1

pFq

become equal when composed with the morphism Sn´1pFq Ñ SnpFq. Thus, they mustbe compatible up to A1-homotopy in Sn´1pF q, i.e. there exists a Nisnevich cover W Ñ


V Â1UV such that the two morphisms

W Ñ V Â1UV

priÑ V

hÑ F Ñ Sn´1

pFq

are A1-chain homotopic in Sn´1pFq, i.e. there exists a finite sequence of homotopieshW :“ ph1, . . . , hrq in Sn´1pF q such that σ˚1 phiq “ σ˚0 phi`1q for 1 ď i ď r ´ 1 andsuch that σ˚0 ph1q and σ˚1 phrq are the two morphisms of W . Now, by replacing W bysome suitable Nisnevich cover, we may assume that the for every i, 1 ď i ď r ´ 1, themorphisms σ˚1 phiq lift to F . Now, applying the induction hypothesis, we see that all theselifts are pn ´ 1q-ghost homotopic. Thus, the two morphisms W Ñ F are pn ´ 1q-ghostchain homotopic. This completes our proof.

In order to avoid being flooded by notation in some of the proofs, we need to introducethe notion of the “total space” of an n-ghost homotopy. For a given n-ghost homotopyH of a scheme U in a sheaf F , this is simply the union of all the schemes that show upin its definition. This is a scheme over U and is equipped with a morphism into F whichis simply the coproduct of all the morphisms that come up in the definition of H. Forthe sake of precision, we write down the definition explicitly as follows:

Definition 2.3.7. Let F be a sheaf of sets, let U be a smooth scheme. For any n-ghosthomotopy H of U in F , we will define a scheme SppHq, called the total space of H, andmorphisms fH : SppHq Ñ U and hH : SppHq Ñ F . We do this by induction on n asfollows:

(1) For a 0-ghost homotopy H which is given by a morphism A1U Ñ F , we define

SppHq “ A1U “ A1 ˆ U . The morphism fH : A1

U Ñ U is simply the canonicalprojection on U . The morphism hH : A1

U Ñ F is none other than the morphismdefining the homotopy.

(2) Now suppose this construction has been done for pn´ 1q-ghost homotopies. Let



q

be an n-ghost homotopy where HW “ pH1, . . . ,Hrq is a pn´ 1q-ghost chain homo-topy. Then we define

SppHq “ V > prž

i“1

SppHiqq.

We will define the morphisms fH : SppHq Ñ U and hH : SppHq Ñ F by specifyingtheir restrictions to V and to SppHiq for all i. We define fH|V to be the compositionof the given morphism V Ñ A1

U with the projection A1U Ñ U . The morphism

fH|SppHiq is defined to be the composition

SppHiqfHiÑ W Ñ V Â1

UV

priÑ V Ñ A1

U Ñ U


where pri could be either of the projections V Â1UV Ñ V (the composition clearly

does not depend on this choice). The morphism hH : SppHq Ñ F is defined byhH|V “ h and hH|SppHiq “ hHi

.

Observe that all the data appearing in the definition of an n-ghost homotopy iscaptured in SppHq, fH and hH.

2.4 Almost proper sheaves and a refinement of a

result of Asok-Morel

The aim of this section is to obtain a refinement of the following theorem of Asok-Morel(see [1, Theorem 2.4.3]).

Theorem 2.4.1 (Asok-Morel). Let X be a proper scheme of finite type over a field k.For every finitely generated separable extension K{k, the canonical epimorphism

SpXqpKq Ñ πA1

0 pXqpKq

is a bijection.

We introduce the concept of almost proper sheaves (see Definition 2.4.4 below), whichwe shall find helpful while handling the iterations of the functor S. Since our method isgeometric in nature, it allows us to prove 2.4.1 for all finitely generated field extensionsof the base field k.

Notation 2.4.2. Let Z be a variety over k and let x be a point of Z. Let kpxq “ OZ,x{mx

be the residue field at x. The quotient homomorphism OX,x Ñ kpxq induces a morphismSpecpkpxqq Ñ SpecpOZ,xq Ñ Z which will also be denoted by x : Specpkpxqq Ñ Z.

Let F be a sheaf of sets and let Z be a variety over k. Let s1, s2 P FpZq. We willwrite s1 „0 s2 if for any point x of Z, the morphisms s1 ˝ x, s2 ˝ x : Specpkpxqq Ñ F areidentical.

Remark 2.4.3. In Notation 2.4.2, the extension kpxq{k may have positive transcendencedegree. For instance, this may happen when U is a smooth, irreducible scheme over kand x is the generic point of U . Then Specpkpxqq is viewed as an essentially smoothscheme over k and the set FpSpecpkpxqq is the direct limit

FpSpecpkpxqq :“ limÝÑU 1

FpU 1q

where U 1 varies over all the open subschemes of U . Thus any morphism Specpkpxqq Ñ Fcan be represented by a morphism U 1 Ñ F where U 1 is some open subscheme of U . Inthe following discussion, we will also have to consider morphisms of the form A1

kpxq Ñ F .

Such a morphism can be represented by a morphism A1U 1 Ñ F where U 1 is an open

subscheme of U .

2.4. Almost proper sheaves and a refinement of a result of Asok-Morel 19

Definition 2.4.4. Let F be a sheaf of sets over Sm{k. We say that F is almost properif the following two properties hold:

(AP1) Let U be a smooth, irreducible variety of dimension ď 2 and let s : U Ñ F beany morphism. Then there exists a smooth projective variety U , a birational mapi : U 99K U and a morphism s : U Ñ F which “extends s on points”. By this wemean that if U 1 is the largest open subscheme of U such that i is represented by amorphism i1 : U 1 Ñ U , then we have s ˝ i1 „0 s|U 1 .

(AP2) Let U be a smooth, irreducible curve over k and let U 1 be an open subscheme.Suppose we have two morphisms s1, s2 : U Ñ F such that s1|U 1 “ s2|U 1 . Thens1 „0 s2.

Lemma 2.4.5. Let F be an almost proper sheaf of sets on Sm{k. Let U be a smoothcurve, let x : Specpkq Ñ U be a k-rational point on U and let U 1 be the open subschemeUztxu of U . Let f, g : U Ñ F be such that the morphisms f |U 1 and g|U 1 are A1-chainhomotopic. Then the morphisms f ˝ x, g ˝ x : Specpkq Ñ F are A1-chain homotopic.

Proof. First we prove the result in the case when f |U 1 and g|U 1 are simply A1-homotopic,i.e. there exists a morphism h : U 1Â1 Ñ F such that h|U 1ˆt0u “ f |U 1 and h|U 1ˆt1u “ g|U 1 .

By (AP1), there exists a smooth proper surface X and a proper, birational morphismi : U 1Â1 99K X and a morphism h : X Ñ F “extending h on points”. As in Definition2.4.4, we elaborate on the precise sense in which h extends h. Suppose that W is thelargest open subscheme of U 1 ˆ A1 on which i is defined. Then h ˝ i|W „0 h|W . Notethat U 1 Â1zW is a closed subscheme of codimension 2. Thus, the curves U 1 ˆ t0u andU 1 ˆ t1u both intersect W . The curves U 1 ˆ t0u XW and U 1 ˆ t1u XW can be mappedinto X via i to give morphisms t0, t1 : U Ñ X, where U is a smooth compactification ofU . Clearly, h ˝ t0|U 1 „0 f |U 1 and h ˝ t1|U 1 „0 g|U 1 . This implies that there exists a Zariskiopen subset U2 of U 1 such that h ˝ t0|U2 “ f |U2 and h ˝ t0|U2 “ f |U2 . Thus, by (AP2),h ˝ t0 ˝ x “ f ˝ x and h ˝ t1 ˝ x “ g ˝ x. Thus in order to show that f ˝ x and g ˝ x areA1-chain homotopic, it suffices to show that t0 ˝ x and t1 ˝ x are A1-chain homotopic inX. Since X is birational to UˆP1, by resolution of indeterminacy (see, for example, [20,

Chapter II, 7.17.3]), there exists a smooth projective surface rX and birational proper

morphisms rX Ñ U ˆ P1 and rX Ñ X.

rX

{{ ��U ˆ P1 X

W2 R

cci

>>

Let t0 : U Ñ U ˆ P1 be the map that identifies U with the curve U ˆ tp0 : 1qu andt1 : U Ñ U ˆ P1 be the map that identifies U with the curve U ˆ tp1 : 1qu. It is easy


to see that there exist unique morphisms rt0,rt1 : U Ñ rX such that rt0 is a lift of botht0 and t0 and rt1 is a lift of both t1 and t1. This completes the proof of the result whenf |U 1 and g|U 1 are simply A1-homotopic by [1, Lemma 6.2.9], since rt0 and rt1 are A1-chainhomotopic.

Now, suppose f |U 1 and g|U 1 are A1-chain homotopic. Thus, we have a sequencef0 “ f |U 1 , . . . , fn “ g|U 1 where fi is A1-homotopic to fi`1 for every i. By (AP1), for each

i, there exists a rfi : U Ñ F such that rfi|U 1 “ fi. Thus rfi ˝ x and rfi`1 ˝ x are A1-chain

homotopic for every i. By (AP2), rf0 ˝ x “ f ˝ x and rfn ˝ x “ g ˝ x.

Lemma 2.4.6. Let F be an almost proper sheaf. Then SpFq is also an almost propersheaf.

Proof. We first check condition (AP2). Thus, let U be a smooth curve and let U 1 Ă Ube an open subscheme. We assume that we have two morphisms s1, s2 : U Ñ SpFqsuch that s1|U 1 “ s2|U 1 and prove that s1 „0 s2. Without any loss of generality, we mayassume that UzU 1 consists of a single closed point. By a change of base, if necessary,we may assume that this point is k-rational, that is, it is the image of a morphismx : Specpkq Ñ U . We need to prove that s1 ˝ x “ s2 ˝ x.

The morphism φ : F Ñ SpFq is an epimorphism of sheaves. Thus, there exists asmooth, irreducible curve C and an etale morphism p : C Ñ U such that x can be liftedto a morphism c0 : Specpkq Ñ C, (i.e. p ˝ c0 “ x) and such that the morphisms p ˝ s1

and p ˝ s2 can be lifted to F . In other words, there exist morphisms f, g : C Ñ F suchthat φ ˝ f “ s1 ˝ p and φ ˝ g “ s2 ˝ p. Since s1|U 1 “ s2|U 1 , there exists a Nisnevichcover q : V Ñ U 1 such that q ˝ f and q ˝ g are A1-chain homotopic. Let K denote thefunction field of C. Then the canonical morphism η : SpecpKq Ñ F lifts to V . Thus themorphisms f ˝η and g˝η are A1-chain homotopic. Thus, there exists an open subschemeC 1 Ă C such that if i : C 1 ãÑ C is the inclusion, the morphisms f ˝i and g˝i are A1-chainhomotopic. Applying Lemma 2.4.5, we see that f ˝ c0, g ˝ c0 : Specpkq Ñ F are identical.Thus

s1 ˝ x “ s1 ˝ p ˝ c0 “ φ ˝ f ˝ c0 “ φ ˝ g ˝ c0 “ s2 ˝ p ˝ c0 “ s2 ˝ x

as desired. This shows that the sheaf SpFq satisfies the condition (AP2).Now we check the condition (AP1). First suppose that U is a smooth, irreducible

variety of dimension ď 2 and we have a morphism s : U Ñ SpF q. Since φ : F Ñ SpFqis surjective, there exists a Nisnevich cover p : V Ñ U of such that the morphism s ˝ plifts to F . Thus there exists an open subscheme U 1 Ă U such that the morphism s|U 1lifts to a morphism t : U 1 Ñ F . Applying the condition (AP1) for F , there exists asmooth projective variety U , a birational map i : U 1 99K U and a morphism s : U Ñ F“extending” t. Let U2 be the largest open subscheme of U 1 such that the rational mapi can be represented by a morphism i1 : U2 Ñ U . We know that s ˝ i1 „0 t|U2 .

We claim that φ˝ s is the required morphism. We note that i1 defines a rational mapfrom U to U . Let U3 Ă U be the largest subscheme of U such that this rational map canbe represented by a morphism i2 : U3 Ñ U . We wish to prove that for any point x in

2.4. Almost proper sheaves and a refinement of a result of Asok-Morel 21

U3, we have s ˝x “ φ ˝ s ˝ i2 ˝x. When x P U2, we already know that s ˝ i1 ˝x “ t|U2 ˝x.Composing by φ, we get that φ ˝ s ˝ i1 ˝ x “ φ ˝ t|U2 ˝ x. But φ ˝ t|U2 “ s|U2 . Thus wesee that φ ˝ t|U2 ˝ x “ s ˝ x. Since i2|U2XU3 “ i1|U2XU3 , we obtain the desired equality inthis case. Thus we now assume that x P U3zU2.

Since U2 is a dense open subscheme of U , there exists a smooth curve D over k anda locally closed immersion j : D Ñ U3 such that jpDq contains x and also intersectsU2XU3. (When dimpUq “ 1, D will be an open subscheme of U .) Let z be the genericpoint of D. Then we note that jpzq is a point of U2 X U3 with residue field kpzq. Bywhat we have proved in the previous paragraph, we know that s ˝ jpzq “ φ ˝ s ˝ i2 ˝ jpzq.Thus there exists an open subscheme D1 of D such that j maps D1 into U2 X U3 ands˝ j|D1 “ φ˝s˝ i2 ˝ j|D1 . Since we have already verified (AP2) for SpFq, we can concludethat s ˝ x “ φ ˝ s ˝ i2 ˝ x. This completes the proof of the fact that SpFq is almostproper.

Theorem 2.4.7. Let F be an almost proper sheaf. Then for any finitely generatedseparable field extension K of k, SpFqpKq “ SnpFqpKq for any integer n ě 1.

Proof. By a base change, we may assume that K “ k. Thus we need to prove that if Fis almost proper, we have SpFqpkq “ SnpFqpkq for all n ě 1. In view of Lemma 2.4.6,it suffices to show that SpFqpkq “ S2pFqpkq.

Let φ denote the morphism of sheaves F Ñ SpFq. Let x and y be elements ofSpFqpkq and suppose there exists a morphism h : A1

k Ñ SpFq such that hp0q “ x andhp1q “ y. Then there exists an open subscheme U of A1

k such that h can be lifted to F ,i.e. there exists a morphism h1 : U Ñ F such that φ˝h1 “ h|U . By (AP1), there exists amorphism h : P1

k Ñ F which “extends h1 on points”, i.e. if i : U Ñ P1k is the composition

U ãÑ A1k ãÑ P1

k, for all points x of U , we have h ˝ i ˝ x “ h1 ˝ x. (Here we identify A1k

with the open subscheme of P1k “ ProjpkrT0, T1sq given by T1 ‰ 0.) Applying this to

the generic point of U , we see that there exists an open subscheme U 1 of U such thath ˝ i|U 1 “ h1|U 1 . Since i is just the inclusion, we may write this as h|U 1 “ h1|U 1 .

Consider the morphism rh :“ φ˝h|A1k

: A1k Ñ SpFq. We have the equalities rh|U 1 “ φ˝

h1|U 1 “ h|U 1 . Thus, by (AP2), we see that rh „0 h. But this means that p “ pφ˝hqpp0 : 1qqand q “ pφ˝hqpp1 : 1qq. In other words, p and q are images under φ of two A1-homotopicmorphisms from Specpkq into F . But by the definition of the functor S, this impliesthat p “ q.

Corollary 2.4.8. If F is an almost proper sheaf of sets, the canonical surjection

SpF qpKq Ñ πA1

0 pKq

is a bijection, for any finitely generated separable field extension K{k. In particular, thisholds for any proper scheme X of finite type over k.

Proof. Theorem 2.4.7 and Theorem 2.2.1 together imply that

SpFqpKq Ñ LpFqpKq


is a bijection for all finitely generated field extensions K of k. The result now followsby Remark 2.2.2. To prove the second claim, we observe that every proper scheme overa field k is an almost proper sheaf in the sense of Definition 2.4.4. Indeed, the property(AP1) follows by resolution of indeterminacy and the property (AP2) follows from thevaluative criterion of properness (see [20, Chapter II, Theorem 4.7]).

Chapter 3

Counterexamples to conjectures ofAsok-Morel

A conjecture of Asok-Morel (Conjecture 2.1.13) asserts that the canonical epimorphismSpXq Ñ πA1

0 pXq is an isomorphism for all smooth schemes X over k. They also remarkthat this would hold if Sing˚pXq is A1-local for all X, where Sing˚ denotes the Morel-Voevodsky singular complex construction. Another conjecture of Asok-Morel says thatπA1

0 is a birational invariant of smooth proper schemes over a field. In Section 3.1, weconstruct an example of a smooth proper scheme over C for which Conjecture 2.1.13fails. In Section 3.2, we give an example showing that πA1

0 is not a birational invariantof smooth proper schemes.

3.1 A smooth proper scheme whose Sing˚ is not A1-

local

The aim of this subsection is to construct an example of a smooth, projective variety Xover C such that:

(i) SpXq ‰ S2pXq,

(ii) SpXq Ñ πA1

0 pXq is not a monomorphism, and

(iii) Sing˚pXq is not A1-local.

We will use a well-known characterization of Nisnevich sheaves, which we will recallhere for the sake of convenience.

For any scheme U , an elementary Nisnevich cover of U consists of two morphismsp1 : V1 Ñ U and p2 : V2 Ñ U such that:

(1) p1 is an open immersion.

24 Chapter 3. Counterexamples to conjectures of Asok-Morel

(2) p2 is an etale morphism and its restriction to p´12 pU z p1pV1qq is an isomorphism

onto U z p1pV1q.

Then a presheaf of sets F on Sm{k is a sheaf in Nisnevich topology if and only ifthe morphism

FpUq Ñ FpV1q ˆFpV1ÛV2q FpV2q

is an isomorphism, for all elementary Nisnevich covers tV1, V2u of U . If F is a Nisnevichsheaf, the fact that this morphism is an isomorphism is an immediate consequence ofthe definition of a sheaf. The converse (see [35, §3, Proposition 1.4, p.96] for a proofor Proposition B.7 in Appendix B) is useful since it simplifies the criterion for checkingwhether sections si P FpViq determine a section s P FpUq. Indeed, suppose pr1 andpr2 are the two projections V Û V Ñ V . Then we do not need to check that the twoelements pr˚1 ps1q and pr˚2 ps2q of FpV Û V q are identical. We merely need to check thatthe images of s1 and s2 under the maps FpV1q Ñ FpV1Û V2q and FpV2q Ñ FpV1Û V2q

respectively are identical.Now suppose F is a sheaf of sets. Applying the above criterion to the sheaf SpFq, we

see that when we work with elementary Nisnevich covers, we can construct morphismsA1U Ñ SpFq with only part of the data that is required for an A1-ghost homotopy of U

in F .The key observation that will be repeatedly used while constructing counterexamples

to Asok-Morel’s conjecture is as follows:

Lemma 3.1.1. Let F be a sheaf of sets over Sm{k. Let U be a smooth scheme over kand let f, g : U Ñ F be two distinct morphisms. Suppose that we are given data of theform

ptpi : Vi Ñ A1UuiPt1,2u, tσ0, σ1u, th1, h2u, h

Wq

where:

• The two morphisms tpi : Vi Ñ A1Uui“1,2 constitute an elementary Nisnevich cover.

• For i P t0, 1u, σi is a morphism U Ñ V1

š

V2 such that pp1

š

p2q˝σi : U Ñ UÂ1

is the closed embedding U ˆ tiu ãÑ U ˆ A1,

• For i P t1, 2u, hi denotes a morphism Vi Ñ F such that ph1

š

h2q ˝ σ0 “ f andph1

š

h2q ˝ σ1 “ g.

• Let W :“ V1 Â1UV2 and let pri : W Ñ Vi denote the projection morphisms.

Then hW “ ph1, . . . , hnq is an A1-chain homotopy connecting the two morphismshi ˝ pri : W Ñ F .

Then, f and g map to the same element under the map FpUq Ñ πA1

0 pFqpUq. Moreover,if f and g are not A1-chain homotopic, then Sing˚pFq is not A1-local.

3.1. A smooth proper scheme whose Sing˚ is not A1-local 25

Proof. Choose a simplicial weak equivalence Sing˚pFq Ñ X , where X is simpliciallyfibrant (that is, a simplicially fibrant replacement of Sing˚pFq). We denote by Hi the

composition VihiÑ F Ñ Sing˚pFq Ñ X , for i “ 1, 2. The above data induces a morphism

ψ : A1U Ñ SpFq “ πs0pX q. Let c0 : A1

U Ñ A1U be the morphism A1

U “ UÂ1Cpr1Ñ U

σ0Ñ A1

U .Then, by construction, ψ ˝ c0 ‰ ψ.

The A1-chain homotopy hW connects hi|W for i “ 1, 2. Thus, by the definition ofSing˚pFq, it gives rise to a chain of simplicial homotopies ∆1 Ñ X pW q, which canbe composed (since X is simplicially fibrant) to give a homotopy Hs : ∆1 Ñ X pW qconnecting Hi for i “ 1, 2. Thus we have the following diagram (where ˚ denotes thepoint sheaf)

˚ //

��

X pV2q

��∆1 //

D

<<

X pW q

where the upper horizontal arrow maps ˚ to H2. Since W “ V1 ˆX V2 Ñ V2 is acofibration (being an open immersion), the morphism X pV2q Ñ X pW q of simplicial setsis a fibration (this is a standard fact about model categories; see [21, Proposition 9.3.7]for a proof, for example). Thus the dotted diagonal lift exists in the above diagram.

Thus we see that we can find a morphism H 1 : V2 Ñ X such that:

(1) H 1 and H2 induce the same morphism V2 Ñ πs0pX q (since they are simpliciallyhomotopic in X pV2q), and

(2) H 1|W “ H1|W .

Since X is a sheaf, the following diagram is cartesian:

X pA1Uq

//

��

X pV2q

��X pV1q // X pW q.

Thus, H1 and H 1 can be glued to give an element of X pA1Uq which we can think of as

a morphism rψ : A1U Ñ X . Clearly, this lifts ψ, as can be checked by restriction to the

Nisnevich cover tV1, V2u.

Thus rψ and rψ ˝ c0 define two morphisms A1U Ñ X which are distinct in the simplicial

homotopy category as, by assumption, the induced maps to πs0pX q are distinct. Thisshows that X is not A1-local.

We now claim that if rψ1 denotes the composition A1U

rψÑ X Ñ Sing˚pX q, then rψ1 and

rψ1 ˝ c0 are homotopic. To see this, we first observe that rψ1 is represented by a morphismfrom A1

U to X0. Now define a morphism

A1U ˆ A1

CHÝÑ X0


by precomposing the map A1U

rψÝÑ X with

A1U ˆk A1

C “ U ˆk A1C ˆ A1

Cps,t1,t2qÞÑps,t1t2qÝÝÝÝÝÝÝÝÝÝÑ U ˆ A1

C.

Let s0 : X0 Ñ X1 denote the degeneracy map, then by definition of Sing˚pX q, s0 ˝Hdefines a map from A1

U Ñ Sing˚pX q1 or equivalently a map A1U ˆ ∆1 Ñ Sing˚pX q. It

is straightforward to check that this gives a homotopy between rψ1 and rψ1 ˝ c0. Thisshows that the two distinct elements ψ and ψ ˝ c0 in πs0pX qpA1

Uq go to the same elementin πs0pSing˚pX qq. From the definition of LA1 (see Lemma 1.3.2), we see that the mapπs0pX q Ñ πA1

0 pFq “ πs0pLA1pFqq factors through πs0pSing˚pX qq. Therefore, f and g mapto the same element under the canonical map

πs0pX q Ñ πA1

0 pFq.

Definition 3.1.2. We will refer to the data of the form

ptpi : Vi Ñ A1UuiPt1,2u, tσ0, σ1u, th1, h2u, h

Wq

as in Lemma 3.1.1 as an elementary A1-ghost homotopy.

We now proceed to the construction of the schemes satisfying properties (i)-(iii)stated at the beginning of the section.

Construction 3.1.3. To clarify the idea behind this construction, we first constructa non-proper, singular variety satisfying (i)-(iii). Later (in Construction 3.1.5), we willmodify this example suitably to create an example of a smooth, projective variety overC which will also satisfy (i)-(iii).

(1) Let λi P Czt0u for i “ 1, 2, 3 be distinct. Let E Ă A2C be the affine curve cut out

by the polynomial y2 ´ś

ipx ´ λiq. Let π : E Ñ A1C denote the projection onto

the x-axis.

(2) Let S1 and S2 be the closed subschemes of A3C cut out by the polynomials y2 “

t2ś

ipx´ λiq and y2 “ś

ipx´ λiq respectively. Let f : S2 Ñ S1 be the morphismcorresponding to the homomorphism of coordinate rings given by x ÞÑ x, y ÞÑ ytand t ÞÑ t.

(3) Let α0 be a square-root of ´λ1λ2λ3 so that the point p0, α0q is mapped to the pointx “ 0 under the projection π. Let p0, p1 and q denote the points p0, 0, 0q, p0, α0, 1qand p1, 0, 0q of S1.

(4) Consider the morphism A1C Ñ S2 given by s ÞÑ p0, α0, sq for s P A1

C. Composingthis morphism with f , we obtain a morphism A1

C Ñ S1, the image is a closedsubscheme of S1 which we denote by D.

3.1. A smooth proper scheme whose Sing˚ is not A1-local 27

Now let S Ă A3C be the scheme S1ztp0u. We claim that S satisfies condition (i)

above. We first claim that any morphism h : A1C Ñ S such that p1 lies in the image

of h is constant. To see this, first observe that this is obvious if the image of h iscontained completely inside Dztp0u (since any morphism from A1

C into A1Czt0u is a

constant morphism). Now suppose that the image of h is not contained in Dztp0u. Notethat the morphism f : S2 “ E ˆ A1

C Ñ S1 is an isomorphism outside the locus of t “ 0.Hence, h induces a rational map A1

C 99K E, which can be completed to a morphismP1C Ñ E, where E denotes a projective compactification of E. This has to be a constant

morphism by the Hurwitz formula, since the genus of E is greater than that of P1C. Thus,

there is no non-constant morphism A1C Ñ S having p1 in its image.

Thus we see that p1 and q are not connected by an A1-chain homotopy. However,we claim that they map to the same element in S2pSqpCq.

In order to be able to apply Lemma 3.1.1, we first construct an elementary Nisnevichcover of A1

C. We observe that π is etale except at the points where x “ λi, i “ 1, 2, 3.Now we define V 1 “ A1

Czt0u and V 2 “ π´1pA1Cztλ1, λ2, λ3uq Ă E. The morphism

V 1 Ñ A1C is the inclusion and the morphism V 2 Ñ A1

C is simply π|V 2. As λi ‰ 0 for all

i, we see that this is a Nisnevich cover of A1C. We define W :“ V 2zπ

´1p0q and observethat V 1 Â1

CV 2 – W . In order to give an A1-ghost homotopy in S using this Nisnevich

cover, we will now define morphisms hi : V i Ñ S and a morphism hW : W ˆ A1 Ñ S,which is a homotopy connecting pπ ˝ h1q|W and π ˝ h2|W .

We define h1 : V 1 Ñ S by s ÞÑ ps, 0, 0q for s P A1C and h2 : V 2 Ñ S by pα, βq ÞÑ

pα, β, 1q for pα, βq P V 2. To define hW , we first identify S2 with E ˆ A1 and define hW

to be f |WÂ1 (recall that W Ă V 2 Ă E). It is clear that the the C-valued points p1 andq are mapped to the same element in S2pSqpCq. This shows that the surface S satisfiesthe condition (i) listed above.

We have thus obtained the data´

V 1

ž

V 2 Ñ A1U ,W Ñ V 1 Â1

UV 2,rh1

ž

rh2,rhW¯

,

which satisfies the hypotheses of Lemma 3.1.1, which proves that S satisfies (ii) and (iii).The following lemma is the key tool in creating a smooth example that satisfies (i) -

(iii) stated at the beginning of the section:

Lemma 3.1.4. Let k be a field and let S be an affine scheme over k. Then there existsa closed embedding of S into a smooth scheme T over k such that for any field extensionL{k, if H : A1

L 99K T is a rational map such that the image of H meets S, then Hfactors through S ãÑ T .

Proof. Choose an embedding of S into Ank for some n and suppose that as a subvariety of

Ank , S is given by the ideal xf1, . . . , fry Ă krX1, . . . , Xns. Then the polynomials fi define

a morphism f : An Ñ Ar such that S is the fibre over the origin. Now let g : C Ñ A1k

be an etale morphism such that C is an affine curve, whose compactification C is of


genus ě 1 and the preimage of the origin consists of a single k-valued point c0 of C.Then this gives us an etale morphism gr : Cr Ñ Ar such that the preimage of the originis the point pc0, . . . , c0q. Let T “ An

k ˆf,Ark,g

r Cr. It is clear that T Ñ Ank induces an

isomorphism over S. We claim that T is the desired scheme.Let L{k be a field extension and let H : A1

L 99K T be a rational map such that theimage of H meets the preimage of S in T . Since C is A1-rigid, the composition of Hwith the projection T Ñ Cr factors through pc0, . . . , c0q. It follows from this that Hfactors through the preimage of S in T .

Construction 3.1.5. We now construct a smooth and projective variety X over Csatisfying the conditions (i)-(iii) listed at the beginning of this subsection:

(1) Using Lemma 3.1.4, we construct a scheme T such that S1 Ă T as a closed sub-scheme and such that for any field extension L{C, if H : A1

L Ñ T is a morphismsuch that the image of H meets S1, then H factors through S1.

(2) Let T c Ą T be a smooth, projective compactification of T .

(3) Let C be a smooth, projective curve over C of genus ą 0. Let c0 be a C-valuedpoint of C. Let R “ OhC,c0 and let U “ SpecpRq. Let u be the closed point of U .Let γ : U Ñ C be the obvious morphism.

(4) Let p2 be the point p0, 4α0, 2q P S1. Let X be the blowup of C ˆ T c at the pointspc0, p0q (see Notation 3.1.3, (3)) and pc0, p2q .

(5) For any C valued point p : SpecpCq Ñ T c of T c, let θp be the induced morphismγ ˆ p : U “ U ˆ SpecpCq Ñ C ˆ T c. Note that θp is a lift of γ to C ˆ T c withrespect to the projection C ˆ T c Ñ C.

(6) Let ξp1 and ξq denote the lifts of θp1 and θq to X (which exist and are unique).

Claim: ξp1 and ξq are connected by an elementary A1-ghost homotopy.To prove this, we construct an elementary A1-ghost homotopy connecting θp1 and θq

which lifts to X. This is done by simply taking the product of the morphism γ : U Ñ Cwith the elementary A1-ghost homotopy of SpecpCq in S Ă T c which we constructedabove. In other words, we define Vi “ U ˆV i and W “ U ˆW . We define hi :“ γˆhi :Vi “ U ˆ V i Ñ C ˆ T c and define hW :“ γ ˆ hW : W ˆ A1 “ U ˆ W Ñ C ˆ T c.(Here we have abused notation by viewing hi and hW as morphisms into T c rather thanas morphisms into S Ă T c.) We observe that W – V1 Â1

UV2 and we have constructed

an elementary A1-ghost homotopy H of U in C ˆ T c. Since this elementary A1-ghosthomotopy factors through the complement of the the points pc0, p0q and pc0, p2q, it lifts

to an elementary A1-ghost homotopy rH of U in X and connects ξp1 and ξq.Claim: ξp1 and ξq are not A1-chain homotopic.

For this it will suffices to show that if ξp1 is A1-homotopic to any ξ : U Ñ X, thenξpuq “ ξp1puq. Projecting this A1-homotopy down to C ˆ T c, we see that it suffices to

3.2. Example showing that πA1

0 need not be birational 29

prove that if h is an A1-homotopy of U in C ˆ T c which lifts to X and if h ˝ σ0 “ θp1 ,then h maps the closed fibre A1

C Ă A1U to θp1puq.

Let D be the closure of D (see Construction 3.1.3, (4)) in C ˆ T c. By construction,D is the only rational curve through the point θp1puq. We claim that any A1-homotopyh of U in C ˆ T c such that h ˝ σ0 “ θp1 must map A1

C Ă A1U into D. Indeed, this follows

immediately from the fact that the composition of this A1-homotopy with the projectionpr1 : C ˆ T c Ñ C must be the constant homotopy (since C is A1-rigid). In fact, it iseasy to see that pr1 ˝ h must factor through γ.

Now we claim that the image of h|A1C

does not contain the point p0. Indeed, if it wereto contain this point, since h can be lifted to X, the preimage of p0 under h would bea non-empty, codimension 1 subscheme of A1

U which is contained in A1C Ă A1

U . Thus itwould be equal to A1

C. Since h ˝ σ0puq “ p1 ‰ p0, this is impossible. Thus, we see thatthe image of h|A1

Cdoes not contain p0. By the same argument it does not contain p2.

However, a morphism from A1C into a rational curve which avoids at least two points

on the rational curve must be a constant (since Gm is A1-rigid). Thus h|A1C

is a constant.

This completes our proof of the claim that ξq and ξp1 are not A1-chain homotopic.Thus, we have now proved that X satisfies the property (i) listed above. Lemma 3.1.1implies that it also satisfies property (ii). If Sing˚pXq were A1-local, the morphismSpXq :“ πs0pSing˚pXqq Ñ πA1

0 pXq would be an isomorphism. Thus X also satisfiesproperty (iii).

3.2 Example showing that πA1

0 need not be birational

Asok-Morel in [1, Section 6.2] define the sheaf πbA1

0 pXq of birational A1-connected com-ponents of a smooth proper scheme X over k. The sheaf πbA

1

0 pXq is a birational invariantof smooth proper schemes over k and admits a canonical surjection from πA1

0 pXq. Asok-Morel have conjectured that the natural map πA1

0 pXq Ñ πbA1

0 pXq is an isomorphism,for all proper schemes X of finite type over a field (see [1, Conjecture 6.2.7]). We endthis chapter by noting a counterexample to the birationality of πA1

0 of smooth properschemes over a field k. This disproves [1, Conjecture 6.2.7].

Recall that a scheme X P Sm{k is said to be A1-rigid, if for every U P Sm{k, themap XpUq ÝÑ XpU ˆ A1q induced by the projection map U ˆ A1 Ñ U is a bijection.

Lemma 3.2.1. If X P Sm{k is A1-rigid, then the canonical map X Ñ πA1

0 pXq is anisomorphism of Nisnevich sheaves of sets.

Proof. If X P Sm{k is A1-rigid, then by Remark 1.2.10, it is A1-local. Hence, for anyU P Sm{k the canonical map XpUq Ñ HomHpkqpU,Xq is a bijection. Consequently, the

canonical map X Ñ πA1

0 pXq is an isomorphism of Nisnevich sheaves of sets.

Lemma 3.2.2. Let π : Y Ñ X be a birational proper morphism of schemes over k. Ifπ has etale local sections, then it is an isomorphism.


Proof. Let p : W Ñ X be an etale cover of X such that the pullback π1 : Y ˆXW Ñ Wof π along p admits a section s. Since π is a birational proper morphism, so is π1. Since pis faithfully flat, we are reduced to proving that p1, which is a birational proper morphismthat admits a section, is an isomorphism. We may assume that W is connected.

Y ˆX W //

π1

��

Y

π��

W

s

CC

p // X

Now, Impsq is closed in Y ˆX W . It is also open, since π1 is birational and p is etale;hence, π1 is an isomorphism, and we are done.

Example 3.2.3. Let X be an abelian variety of dimension at least 2. Since X is an A1-rigid scheme, X » πA

1

0 pXq, as noted above. Now, blow up X at a point to get anotherscheme Y . Then the blow-up map Y Ñ X is a birational proper map which induces amap πA1

0 pY q Ñ πA1

0 pXq. If πA1

0 is a birational invariant of smooth proper schemes overk, then πA1

0 pY q Ñ πA1

0 pXq must be an isomorphism.

Y

��

// // πA1

0 pY q

��

X„ // // πA1

0 pXq

However, Y Ñ πA1

0 pY q is an epimorphism, implying that the map Y Ñ X is an epimor-phism (in Nisnevich topology). But then it would admit a section over a Nisnevich coverof X. This in turn, would imply that it is an isomorphism by Lemma 3.2.2, Y Ñ Xbeing a birational proper morphism. This is a contradiction to the statement that πA1

0

is a birational invariant of smooth proper schemes.

Chapter 4

Relationship of A1-connectednesswith R-equivalence

The notion of R-equivalence of rational points on a variety, introduced by Manin [28]in 1970’s, has been extensively studied in the context of algebraic groups, where itprovides a lot of information in the study of their rationality properties. In this chapter,we explore the connection between the notions of R-equivalence in an algebraic groupand the sheaf of its A1-connected components. We begin with a quick review of R-equivalence and various notions of near-rationality in Section 4.1. In Section 4.2, we showhow to identify R-equivalence classes in a semisimple, absolutely almost simple, simplyconnected group over a field k as sections of its sheaf of A1-connected components. Thisis the main result of this chapter. Finally, in Section 4.3, we show that smooth properR-trivial varieties are A1-chain connected (and hence, A1-connected). We also giveexamples of rational affine varieties that are A1-connected but not A1-chain connected.

4.1 R-equivalence and near-rationality

Definition 4.1.1. Let X be an algebraic variety over a field k such that Xpkq isnonempty. Two k-rational points x, y of X are said to be strictly R-equivalent if there isa rational map f : P1

k 99K X defined at 0 and 1 such that fp0q “ x and fp1q “ y. Twopoints x, y P Xpkq are said to be R-equivalent and we write x „ y if there exists a finitesequence x “ x0, x1, . . . , xn “ y of points in Xpkq such that xi is strictly R-equivalentto xi`1, for every i.

Notation 4.1.2. It is easy to see that R-equivalence is an equivalence relation on theset Xpkq. The set of equivalence classes of Xpkq under R-equivalence is denoted byXpkq{R. For a field extension F of k, we set

XpF q{R :“ pX ˆSpec k SpecF qpF q{R.

32 Chapter 4. Relationship of A1-connectedness with R-equivalence

Definition 4.1.3. A variety X over a field k is said to be R-trivial if XpF q{R is asingleton, for every field extension F of k.

We next summarize the basic properties of R-equivalence into a proposition. Forproofs, see [42, Chapter 6].

Proposition 4.1.4. Let X and Y be algebraic varieties over a field k. We have thefollowing:

p1q Any morphism X Ñ Y induces a map of sets Xpkq{RÑ Y pkq{R.

p2q pX ˆk Y q{R » Xpkq{R ˆ Y pkq{R.

p3q Let F {k be a field extension and let Z be a variety over F with Zpkq ‰ H. Let RF {k

denote the Weil restriction from F to k. Then we have RF {kpZqpkq{R “ ZpF q{R.

p4q (Colliot-Thelene-Sansuc [11]) Xpkq{R is a birational invariant of smooth projectivevarieties over k.

Let G be an algebraic group over a field k. The relation of R-equivalence generatesa normal subgroup of Gpkq. Thus, Gpkq{R is a group. We now briefly recollect thesignificant properties of R-equivalence in algebraic groups (see [42, Section 16.2] forproofs).

Proposition 4.1.5. Let G be an algebraic group over a field k. We have the following:

p1q Any two R-equivalent points of Gpkq are strictly R-equivalent.

p2q Let kptq be a purely transcendental extension of k. Then the natural map Gpkq{RÑGpkptqq{R is an isomorphism.

p3q Let U be a nonempty Zariski open subset of G. Then the natural map Upkq{R ÑGpkq{R is a bijection.

The following is a long-standing open question:

Question 4.1.6. Is the group of R-equivalence classes Gpkq{R of a reductive algebraicgroup always abelian?

Remark 4.1.7. This has been proved by Chernousov and Merkurjev (see [15, Theoreme7.7] and [8, 1.2]) in the case when G is a semisimple, simply connected, absolutely almostsimple and of classical type over k. R-equivalence has been extensively studied by theMinsk group of algebraic geometers. For a survey of known results, we refer the readerto [42, Chapter 6] and [15].

We next briefly review the various near-rationality concepts regarding algebraic va-rieties.

4.2. R-equivalence in anisotropic groups 33

Definition 4.1.8. Let X be an algebraic variety over a field k. We say that X is

(1) k-rational if it is birational to Pnk , for some n;

(2) stably k-rational if for some n, X ˆ An is k-rational;

(3) factor k-rational if there exists a variety Y such that X ˆ Y is k-rational;

(4) retract k-rational if there exists a Zariski open set U of X such that the identitymorphism U Ñ U factors through an open subset of some affine space.

(5) k-unirational if there exists a k-rational variety Y and a dominant rational mapY Ñ X.

It is known that each of the properties in the Definition 4.1.8 implies the next one(see [12, Proposition 1.4] for a proof). It is also known that there exist factor k-rationalvarieties that are not stably k-rational and stably k-rational varieties that are not k-rational. Also, it is known that any smooth proper retract k-rational variety is R-trivial(see [1, Theorem 2.3.6] for a proof, for example). It is also known that any algebraicgroup over k is k-unirational ([5, Theorem 18.2]).

For smooth proper varieties, R-equivalence classes are related to the sheaf of A1-connected components in the following way (see [1, Section 2] for a proof).

Lemma 4.1.9. Let k be a field of characteristic 0. Let X be a smooth proper varietyover k. Then for any finitely generated field extension F of k, we have SpXqpF q “πA1

0 pXqpF q “ XpF q{R.

4.2 R-equivalence in anisotropic groups

In this section, we show that if G is an anisotropic, semisimple, absolutely almost simple,simply connected group over a field k, then two elements of G over any field extensionof k are R-equivalent if and only if they are A1-equivalent. As a consequence, we seethat Sing˚pGq cannot be A1-local for such groups. This implies that the A1-connectedcomponents of a semisimple, absolutely almost simple, simply connected algebraic groupof classical type over a field k form a sheaf of abelian groups. For basic definitions andgeneralities regarding algebraic groups, we refer the reader to [26, Chapter VI].

Conventions 4.2.1. We will use the following conventions in this section:

(1) For any scheme X over k and any field extension L{k, XL will denote the pullbackX ˆSpecpkq SpecpLq over L. Similarly, for any morphism f : X Ñ Y betweenschemes over k, we will denote by fL : XL Ñ YL the pullback of f with respect tothe projection YL Ñ Y .


(2) For any smooth scheme U over k and any sheaf F on Sm{k, we will say thatf, g P FpUq are A1-equivalent if they map to the same element of πA1

0 pFqpUq.

The following notion will turn out to be very useful in the proof.

Definition 4.2.2. For an algebraic group G over a field k and a field extension F of k,let GpF q` be the subgroup of GpF q generated by the subsets UpF q where U varies overall F -subgroups of G which are isomorphic to the additive group Ga. The group

W pF,Gq :“ GpF q{GpF q`

is called the Whitehead group of G over F .

Thus, if W pF,Gq is trivial, then GpF q is generated by unipotent elements. Sinceevery unipotent element is A1-chain connected to the identity element of G, it followsthat SpGqpF q “ ˚. Also, if the natural map W pk,Gq Ñ W pF,Gq for a field extension Fof k is an isomorphism, then any element of GpF q is A1-chain connected to an elementlying in the image of the natural map Gpkq Ñ GpF q.

We now state an interpretation of the known results in the isotropic case, which willplay an important role in our proof of the Main Theorem.

Theorem 4.2.3. Let G be an isotropic, semisimple, simply connected, absolutely almostsimple group over an infinite field k. Then there is an isomorphism

πA1

0 pGqpkq » Gpkq{R.

Proof. Note that the canonical quotient map Gpkq Ñ Gpkq{R clearly factors through themap Gpkq Ñ SpGqpkq. By [15, Theoreme 7.2], we identify Gpkq{R with the Whiteheadgroup W pk,Gq. Therefore, any two R-equivalent elements of Gpkq differ by an elementof Gpkq`, which gives an A1-chain homotopy between the two elements. This shows thatSpGqpkq “ Gpkq{R.

A result of Volkel-Wendt [41, Corollary 3.4, Proposition 4.1] and Moser [36, 3.7] saysthat for an isotropic reductive group G, Sing˚pGq is A1-local. Therefore, the canonicalmap SpGq Ñ πA1

0 pGq is an isomorphism.

We next quote a straightforward consequence of [6, 8.2].

Theorem 4.2.4 (Borel-Tits). Let G be a smooth affine group scheme over a perfect fieldk. Then the following are equivalent:p1q G admits no k-subgroup isomorphic to Ga or Gm.p2q G admits a G-equivariant compactification G such that Gpkq “ Gpkq.

Lemma 4.2.5. Let G be an anisotropic group over a perfect field k. Then any rationalmap h : P1

k 99K G is defined at all the k-rational points of P1k.


Proof. By Theorem 4.2.4, there exists a compactification G of G such that Gpkq “ Gpkq.Clearly h can be extended to a morphism h : P1

k Ñ G and the lemma follows.

It is well known that if G is an anisotropic, semisimple group over a local field k,then Gpkq is a compact set [7, 6.14]. The following simple observation can be seen as ananalogue of this, which will be useful in the proof of our main theorem.

Lemma 4.2.6. Let G be an anisotropic group over a perfect field k. Then there are nonon-constant morphisms from A1

k into G and consequently,

SpGqpkq “ Gpkq.

Proof. Again, obtain a compactification G of G such that Gpkq “ Gpkq by applyingTheorem 4.2.4. Any morphism h : A1

k Ñ G can be extended to a morphism h : P1k Ñ G.

By Lemma 4.2.5, the morphism h maps all the k-rational points of P1k into Gpkq. Since h

maps every point of P1k other than 8 into G anyway, we see that h maps P1

k into G whichis an affine scheme. Thus, h is the constant map. This shows that SpGqpkq “ Gpkq.

We recall that according to [9, Theorem 4.18], for any algebraic group G, the sheafπA1

0 pGq is A1-invariant. This allows us to use Lemma 3.1.1 in the following proof.

Theorem 4.2.7. Let G be an anisotropic, semisimple, absolutely almost simple, simplyconnected group over a field k of characteristic 0. Let F be a field extension of k. Thentwo elements of GpF q are R-equivalent if and only if they are A1-equivalent.

Proof. In view of Theorem 4.2.3, observe that it suffices to prove the theorem in thecase F “ k.Proof of the “if” part: By Theorem 4.2.4, there exists a compactification G of G suchthat Gpkq “ Gpkq. If two elements p and q of Gpkq are A1-equivalent, then p and q mapto the same element in πA1

0 pGqpkq. Since G is proper over k, we can apply Theorem [1,Theorem 2.4.3] to conclude that p and q map to the same element in SpGqpkq. Therefore,p and q are A1-chain homotopic k-rational points of G. Since GpkqzGpkq “ H, it followsthat p and q map to the same element in Gpkq{R.Proof of the “only if” part: Let p and q be two elements of Gpkq, which are R-equivalent.Thus, there is a rational map h : P1

k 99K G which is defined on 0 and 1 such that hp0q “ pand hp1q “ q. Choose a compactification G of G such that Gpkq “ Gpkq. The rationalmap h can be uniquely extended to a morphism h : P1

k Ñ G. By Lemma 4.2.5, hmaps all the k-rational points of P1

k into G. Thus, we see that h is undefined only atpoints of A1

k having residue fields that are non-trivial finite extensions of k. We define

V :“ h´1pGq X A1

k which is a Zariski open subscheme of A1k. Let A1

kzV “ tp1, . . . , pnuand let the residue field at pi be Li. We define hV : V Ñ G by hV :“ h|V .

We claim that for each i, GLiis an isotropic group. Indeed, the rational map hLi

:P1Li

99K GLiis not defined at an Li-rational point. Hence, by Lemma 4.2.5, GLi

cannotbe anisotropic.


Since the group GLiis isotropic, we may apply [15, Theoreme 5.8], which says that

W pLi, Gq “ W pLiptq, Gq. Thus any element of GLipLiptqq can be connected by an A1-

chain homotopy to an element in the image of the natural map GLipLiq Ñ GLi

pLiptqq.Applying this to the map phV qLi

: VLiÑ GLi

, we see that there exists some opensubscheme V 1i of VLi

such that the map phV qLi|V 1i can be connected by an A1-chain

homotopy to a constant map taking V 1i to some Li-rational point q1i P GLipLiq.

Choose a preimage p1i of pi under the projection map A1LiÑ A1

k for each i and denoteby Vi the open subscheme of A1

Ligiven by V 1i Ytp

1iu. Let qi be the image of q1i under the

projection GLiÑ G. We define hi : Vi Ñ G to be the constant map taking Vi to the

point qi. Let W :“š

i Vi and let hW : W Ñ G be the mapš

i hi.We define pV : V Ñ A1

k to be the inclusion. For each i, we define pi : Vi Ñ A1k

to be the composition Vi ãÑ A1LiÑ A1

k. Let pW : W Ñ A1k be the map

š

i pi. Sincep´1W pA1

kzV q “ tp11, . . . , p1nu, it is easy to see that tpV , pW u is an elementary Nisnevich

cover of A1k. In order to apply Lemma 3.1.1, we need to show that the morphisms

hV ˝ prV and hW ˝ prW from V Â1kW to G are A1-chain homotopic.

For every 1 ď i ď n, we have V Â1kVi “ V 1i . Thus V Â1

kW “

š

i V1i . The morphism

prV |V 1i is equal to the composition V 1i ãÑ VLiÑ V . Also, the morphism prW |V 1i is equal

to the composition of inclusions V 1i Ă Vi Ă W .For each i, we have the commutative diagrams

V 1i� � // VLi

phV qLi//

��

GLi

��V

hV// G

and

V 1i� � // Vi

cq1i //

hW |Vi !!

GLi

��G

where cq1i is the constant map taking the scheme VLito q1i. By assumption, there exists

an A1-chain homotopy connecting the maps phV qLi|V 1i to the map cq1i |V 1i . On composing

with the projection map GLiÑ G, this gives an A1-chain homotopy connecting the

morphism hV ˝ prV |V 1i to the morphism hW ˝ prW |V 1i . Thus, there exists an A1-chainhomotopy connecting the morphisms hV ˝ prV to the morphism hW ˝ prW .

Thus, we may now apply Lemma 3.1.1 to conclude that p and q map to the sameelement in πA1

0 pGqpkq. This completes the proof of Theorem 4.2.7.

Remark 4.2.8. An unpublished result of Gabber (see [16, Theorem B] or [14, Theorem4]) generalizes Theorem 4.2.4 to fields that are not perfect and to groups that are notnecessarily smooth. This can be used to generalize Theorem 4.2.7 to fields that are notperfect by closely following the proof of Theorem 4.2.7 with a few adjustments. We will


use the same notation as in the proof of Theorem 4.2.7. One of the adjustments neededis in the proof of the “if” part, where one replaces the use of [1, Theorem 2.4.3] with theuse of [2, Theorem 2 in the Introduction] in order to deal with field extensions that arenot separable. Another adjustment needed is in the choice of the Li. In the case whenchar k ą 0, we choose for Li the separable closure of the base field k inside the residuefield of pi. It remains to verify that GLi

is isotropic. This follows from the fact that thebase change of G to the residue field of pi is isotropic and the following basic result (see[43, 7.3], for example): taking character groups gives an anti-equivalence between thecategories of group schemes of multiplicative type and abelian groups with continuousaction of Galpksep{kq. As a consequence, a torus is split by a purely inseparable extensionof the base field if and only if it was split to begin with.

Corollary 4.2.9. Let G be as in Theorem 4.2.7. Then Sing˚pGq cannot be A1-local.

Proof. We simply note that there does exist a pair of distinct R-equivalent elements inGpkq. Indeed, this is an immediate consequence of the fact that G is unirational over k(see [5, Theorem 18.2]). Thus, the map SpGqpkq Ñ πA1

0 pGqpkq is not a bijection. Thisshows that Sing˚pGq cannot be A1-local.

In view of Question 4.1.6 and Remark 4.1.7, it is natural to conjecture the following.

Conjecture 4.2.10. Let G be a semisimple algebraic group over a field k. Then

πA1

0 pGqpF q “ GpF q{R,

for all field extensions F of k.

We end with an open question, posed by Anastasia Stavrova:

Question 4.2.11. Let G be a semisimple algebraic group over a field k. Is πA1

0 pGq asheaf of abelian groups?

Remark 4.2.12. We briefly explain how giving an affirmative answer to Question 4.2.11is equivalent to giving an affirmative answer to the Question 4.1.6 for a semisimplealgebraic group over a field k, if Conjecture 4.2.10 holds. One implication is obvious.For the other, observe that if GpF q{R is an abelian group for any field extension F {k,to answer Question 4.2.11 affirmatively, it suffices to prove that πA1

0 pGqpSpecAq is anabelian group for regular henselian rings A containing k. This follows from [9, Corollary4.17], which implies that πA1

0 pGqpSpecAq injects into πA1

0 pGqpSpecQpAqq, where QpAqdenotes the quotient field of A. This proves the other implication.

In particular, using Remark 4.1.7, this gives an affirmative answer to Question 4.2.11in the case when G is a semisimple, simply connected, absolutely almost simple and ofclassical type over a field k.


4.3 A1-connectedness of R-trivial smooth proper va-

rieties

The aim of this section is to show that a smooth proper R-trivial variety is A1-chainconnected. This in particular implies that Conjectures 2.1.5 and 2.1.13 hold for suchvarieties X, since SpXq and πA1

0 both turn out to be trivial one-point sheaves. We willrequire an analogue of a result of Morel [34, Lemma 6.1.3] in the context of A1-chainconnected components, which we prove first. We will make use of the following lemma:

Lemma 4.3.1. Let V be an irreducible smooth scheme over k and let W ãÑ V be theinclusion of a dense open subscheme. Then πs0pSing˚pV {W qq is trivial.

Proof. We first show that any point x P V has an open neighbourhood U such thatπs0pSing˚pU{W XUqq is trivial. Let Z ãÑ V be the closed immersion of the complementof W , with the reduced induced subscheme structure. Since k is an infinite field, byGabber’s presentation lemma [10, Theorem 3.1.1], x admits an open neighbourhood Uand an etale morphism π : U Ñ A1

V 1 , for some open subscheme V 1 of Ad´1, where d isthe dimension of V at x, such that π induces a closed immersion ZXU ãÑ A1

V 1 satisfyingZ XU “ π´1pπpZ XUqq and such that Z XU Ñ V 1 is a finite morphism. Therefore, wehave an isomorphism of Nisnevich sheaves

U{pU ´ Z X Uq„ÝÑ A1

V 1{pA1V 1 ´ πpZ X Uqq.

Hence, it suffices to check that πs0pSing˚pA1V 1{pA1

V 1 ´ πpZ X Uqqqq is trivial. Now, sinceZ X U Ñ V 1 is a finite morphism, Z X U Ñ P1

V 1 is proper. This closed immersion doesnot intersect the section at infinity s8 : V 1 Ñ P1

V 1 . By Mayer-Vietoris excision (see B.6),we have an isomorphism of Nisnevich sheaves

A1V 1{pA1

V 1 ´ πpZ X Uqq„ÝÑ P1

V 1{pP1V 1 ´ πpZ X Uqq.

Also observe that A1V 1 Ñ P1

V 1{pP1V 1´πpZXUqq is onto and that Sing˚pA1

V 1q » Sing˚pV1q

(since Sing˚ preserves A1-weak equivalences). Thus, the composition

V 1 Ñ A1V 1 Ñ πs0pSing˚pA1

V 1{pA1V 1 ´ πpZ X Uqqqq Ñ πs0pSing˚pP1

V 1{pP1V 1 ´ πpZ X Uqqqq

is surjective for any section V 1 Ñ A1V 1 ; in particular, for the zero section. But, in P1

V 1 , thezero section is A1-homotopic to the section at infinity s8. Since s8pV

1q Ď P1V 1´πpZXUq,

it follows thatV Ñ πs0pSing˚pP1

V 1{pP1V 1 ´ πpZ X Uqqqq

is the trivial morphism, as desired.Since V Ñ πs0pSing˚pV {W qq is an epimorphism, the above discussion, by functorial-

ity, implies that πs0pSing˚pV {W qq is trivial.

4.3. A1-connectedness of R-trivial smooth proper varieties 39

Proposition 4.3.2. Let k be an infinite field. Let X be a simplicial sheaf of sets onSm{k. The following are equivalent:

(1) SpX qpF q “ ˚, for all finitely generated field extensions F {k.

(2) SpX q is trivial as a sheaf.

Proof. p2q ñ p1q is obvious. We give a proof of p1q ñ p2q below, which closely followsMorel ([34, Proof of Lemma 6.1.3]) and [33, Lemma 3.3.7]).

We have to show that for every U P Sm{k, the pointed set SpX qpUq is trivial.It suffices to show that for every morphism U Ñ SpX q, there is a Nisnevich coverV “

š

Vi Ñ U such that the composite V Ñ U Ñ SpX q “ πs0pSing˚X q is trivial.Since Sing˚X Ñ SpX q “ πs0pSing˚X q is an epimorphism, there is a Nisnevich coveringš

Vi Ñ U , where Vi are irreducible smooth k-schemes such that every composite Vi ÑU Ñ SpX q lifts to a morphism Vi Ñ Sing˚X .

X

��Sing˚X

��š

Vi

;;

55

// U //

99

πs0pSing˚X q

Since the sheaf at simplicial level 0 of Sing˚X is X0, and since any map from a spaceof simplicial dimension 0 to another space is determined by a map at the 0th simpliciallevel, this map factors through the monomorphism X Ñ Sing˚X . Thus, it suffices toprove that for any irreducible, smooth k-scheme V and a morphism φ : V Ñ X , thecomposition V Ñ X Ñ πs0pSing˚X q is trivial.

It suffices to prove that for any irreducible smooth k-scheme V and a dense openset W ãÑ V , the sheaf V {W is A1-chain connected (in other words, Sing˚pV {W q issimplicially connected). For, since

limÝÑ

W ãÑV nonempty open

πs0pSing˚X pW qq

is trivial, there exists a dense open subset W ãÑ V such that the composite W Ñ

V Ñ X is simplicially homotopic to the trivial morphism. We may then replace X by asimplicially fibrant replacement of it and use the right lifting property of the projectionmap X Ñ ˚ with respect to simplicially trivial cofibrations to see that φ : V Ñ X issimplicially homotopic to a morphism φ1 : V Ñ X whose restriction to W is trivial.Applying Lemma 4.3.1 now finishes the proof.

Corollary 4.3.3. Let X be an R-trivial smooth proper scheme over a field k of charac-teristic 0. Then SpXq (and hence, πA1

0 pXq) is trivial.


Proof. Since X is smooth proper, we can invoke Theorem [1, Theorem 2.4.3] and Lemma4.1.9 to conclude that X satisfies the condition p1q of Proposition 4.3.2. Therefore, itfollows that SpXq (and hence, πA1

0 pXq) is trivial.

Thus, in particular, smooth proper rational varieties over a field of characteristic 0are A1-connected.

Real algebraic spheres

Let Σn denote the usual real sphere, namely the affine variety

Σn :“ SpecRrX1, . . . , Xn`1s{xX21 ` . . .`X

2n`1 ´ 1y.

It is quite easy to see that there is no non-constant morphism A1R Ñ Σn. Thus, the

following result is somewhat counterintuitive:

Theorem 4.3.4. For every n ą 1, we have S2pΣnq “ πA1

0 pΣnq “ ˚.

Remark 4.3.5. Clearly, for n “ 1, the statement cannot hold. Indeed, the varietyΣ1 ˆSpecpRq SpecpCq over SpecpCq is isomorphic to Gm,C and is thus A1-rigid. Hence, Σ1

is also A1-rigid and consequently, πA1

0 pΣ1q “ Σ1.

Proof of Theorem 4.3.4. In view of Proposition 4.3.2, it suffices to prove that

S2pΣnqpF q “ ˚,

for every finitely generated field extension F of R and every n ě 2. This would imme-diately imply that πA1

0 pΣnqpF q “ ˚, for every finitely generated field extension F of R,and consequently, that πA1

0 pΣnq “ ˚. We first reduce the theorem to the case n “ 2.For any real hyperplane H (i.e. a hyperplane given by a linear polynomial with real

coefficients) in AnR passing through the origin, the intersection of H with Σn is called a

great circle of Σn. Using the action of SOpn,Rq on Σn, it is easy to see that any greatcircle of Σn is isomorphic to Σn´1. It follows from this that the theorem will be proved ifwe just prove it for the case n “ 2. Indeed, suppose we already know that any two realpoints of Σ2 are A1-connected. For any real point P on Σ3, choose a great circle passingthrough P and the point p1, 0, 0, 0q. This great circle is isomorphic to Σ2 and thus Pis A1-connected to p1, 0, 0, 0q. Thus, every real point of Σ3 is connected to p1, 0, 0, 0q,which proves the result for n “ 3. Proceeding inductively in this manner, we can deducethat πA1

0 pΣnqpRq “ t˚u.We now consider the case n “ 2. Let us write Q :“ Σ2 “ SpecRrX1, X2, X3s{xX

21 `

X22 `X

23 ´ 1y. Let Q denote the closure of Q in P3

R. Let F be a finitely generated fieldextension of R. Observe that the points of QpF qzQpF q are in one to one correspondencewith nontrivial solutions of the quadratic form X2

1 ` X22 ` X2

3 “ 0 over the field F .Suppose that such a solution exists. Then it is easy to see that the projective quadric

4.3. A1-connectedness of R-trivial smooth proper varieties 41

QF is isomorphic to P1F ˆ P1

F and that QF is isomorphic to P1F ˆ P1

F z 4P1F

, which

is covered by affine spaces and hence, is A1-chain connected (see [1, Definition 2.2.10,Lemma 2.2.11]). Therefore, it follows that SpQqpF q “ ˚.

On the other hand, suppose that QpF q “ QpF q. Since any two F -rational pointsp, q of XF lie on a great circle, it follows that there exists a rational map h : P1

F 99K QF

defined at 0 and 1 such that hp0q “ p and hp1q “ q. The rational map h can be uniquelyextended to a morphism h : P1

F Ñ QF . Since QpF q “ QpF q, the morphism h maps allthe F -rational points of P1

F into QF . Thus, we see that the rational map h is defined atall the F -rational points of P1

F .The rest of the proof is along the same lines of the proof of Theorem 4.2.7. Define

V :“ h´1pQq X A1

F which is a Zariski open subscheme of A1F . Let A1

F zV “ tp1, . . . , pnu.Let E be a quadratic field extension of F such that QE is isotropic. Then QE »

P1E ˆ P1

E z 4P1E

and is A1-chain connected. Each pi has precisely two preimages under

the natural map A1E Ñ A1

F ; denote them by p1i and p2i . Define W :“ A1Eztp

21, . . . , p

2nu

and let hW : W Ñ Q denote a constant map sending W to a closed point of Q.Define pV : V Ñ A1

k to be the inclusion and pW : W Ñ A1F to be the composition

W ãÑ A1E Ñ A1

F . Since p´1W pA1

F zV q “ tp11, . . . , p

1nu, it is easy to see that tpV , pW u is an

elementary Nisnevich cover of A1F . In order to apply Lemma 3.1.1, we need to show that

the morphisms hV ˝prV and hW ˝prW from X :“ V Â1FW to Q are A1-chain homotopic.

Since QE is A1-chain connected, we have an A1-chain homotopy of XE connecting themaps phV ˝ prV qE and phW ˝ prW qE from XE to QE. Now we observe that X is anopen subscheme of A1

E. Thus, the projection XE “ X ˆSpecpF q SpecpEq has a sections : X Ñ XE. The morphisms h1V “ s ˝ phV ˝ prV qE and h1W “ s ˝ phW ˝ prW qE from Wto XC lift of hV ˝prV and hW ˝prW respectively with respect to the projection QE Ñ Q.

XE

��

// QE

��X //

s

CC

Q

Since QE » P1E ˆ P1

E z 4P1E

, there exists an A1-chain homotopy connecting h1V to h1W .

By composing this with the projection QE Ñ QF , we obtain an A1-chain homotopyconnecting the maps hV ˝ prV and hW ˝ prW . Thus, by Lemma 3.1.1, p and q map tothe same point in S2pQqpF q “ πA1

0 pQqpF q.Thus, we have shown that over every finitely generated field extension F of R, we

have S2pQqpF q “ πA1

0 pQqpF q “ ˚. By [34, Lemma 6.1.3] and Proposition 4.3.2, it followsthat

S2pQq “ πA1

0 pQq “ ˚.

Chapter 5

A1-connected components of smoothproper surfaces

In this chapter, we use the method suggested by results in Chapter 2 to investigate theA1-connected components of surfaces over an algebraically closed field. In Section 5.1,we prove Morel’s and Asok-Morel’s conjectures for proper non-uniruled surfaces over analgebraically closed field. In Section 5.2, we show that Asok-Morel’s conjecture fails tohold in general for ruled surfaces.

5.1 The Morel and Asok-Morel conjectures for non-

uniruled surfaces

In this section, we give a proof of Morel’s and Asok-Morel’s conjectures (see Conjectures2.1.5 and 2.1.13) for a non-uniruled surface over a field k. For convenience, we willassume throughout the chapter that k is algebraically closed. With some more work, itseems that it will be possible to generalize the results of this section to arbitrary fields.

Definition 5.1.1. A variety X of dimension n over a field k is said to be ruled (re-spectively, uniruled) if there exists a scheme Y of dimension n´ 1 over k and a rationalmap

φ : Y ˆ P1 99K X

which is birational (respectively, dominant). If X is proper over k, then we may assumethat φ is a morphism by shrinking Y , if necessary.

Remark 5.1.2. Ruled varieties are clearly uniruled. If k is an algebraically closed fieldof characteristic zero and X is a uniruled variety over k with dimX ď 2, then X isruled.

In the case of non-uniruled surfaces, our strategy to prove Conjectures 2.1.5 and2.1.13 is given by Lemma 2.3.4. The key observation in this case is that the problemcan be reduced to 1-dimensional schemes. Thus, we first observe the following:

44 Chapter 5. A1-connected components of smooth proper surfaces

Lemma 5.1.3. Let X be a (possibly singular) reduced, separated 1-dimensional properscheme over k. Then SpXq “ S2pXq.

Proof. We will show that SpXqpUq “ S2pXqpUq, for any smooth Henselian local schemeU over k.

If X is smooth, then it is a disjoint union of irreducible smooth curves tCiuiPI whereI is a finite set. We need to verify that SpCiqpUq “ S2pCiqpUq for every i P I. If Ci isnot rational, then its genus is ą 0 and hence, it is A1-rigid (see Definition 1.2.9), by theHurwitz formula. Thus, SpCiq “ Ci. So, in this case, SpCiqpUq “ S2pCiqpUq as desired.If Ci is rational, it is isomorphic to a smooth plane conic. Any morphism f : U Ñ Cifactors as

UΓfÑ pCiqU :“ Ci ˆSpecpkq U Ñ Ci

where Γf is the graph of f . If the morphism pCiqU Ñ U has no section, then clearly

SpCiqpUq “ S2pCiqpUq “ H

which verifies our claim in this case. On the other hand, if there exists a section forthe morphism pCiqU Ñ U , then pCiqU is a P1-bundle over U . As U is a Henselian localscheme, it is a trivial P1-bundle. Thus, in this case

SpCiqpUq “ S2pCiqpUq “ ˚

as desired. Thus we have proved the lemma in case X is smooth.

Suppose now that X is not smooth. Let U be a smooth irreducible scheme over kand let H be an A1-homotopy or an A1-ghost homotopy of U in X. With the notationof Definition 2.3.7, if the image of hH dense in X, then hH must factor through thenormalization of X. On the other hand if hH is not dominant, then its image is a singlepoint of X. Since k is algebraically closed, it is easy to see that even in this case, hHfactors through the normalization rX of X. Thus, we can reduce to the case of smoothproper curves and the result follows.

Theorem 5.1.4. Let X be a proper, non-uniruled surface over k. Then we have

SpXq “ S2pXq.

Proof. We will make use of the following observation. Since X is not uniruled, we know(see [27], Chapter IV, 1.3) that for any variety Z over k, if we have a rational mapP1 ˆ Z 99K X, then either

(1) this rational map P1 ˆ Z 99K X is not dominant, or

(2) for every z P Z, the induced map P1z 99K X is constant.

5.1. The Morel and Asok-Morel conjectures for non-uniruled surfaces 45

Let U be a smooth, Henselian, local scheme over k. We will show that SpXqpUq ÑSpXqpU ˆ A1q is a bijection. If dimpUq “ 0, this has already been proved in Theorem2.4.7. Thus we now assume that dimpUq ě 1.

We will use the notation of Definition 2.3.2 in the following arguments. So, let

H :“´

V Ñ A1U ,W Ñ V Â1

UV, h, hW

¯

be an A1-ghost homotopy of U in X connecting morphisms t1, t2 : U Ñ X. As discussedbefore H determines an A1-homotopy h : A1

U Ñ SpXq. We will show that either h liftsto an A1-homotopy of U in X or that the given A1-ghost homotopy factors through a1-dimensional closed subscheme of X.

We write V “š

iPI Vi for some indexing set I where each Vi is irreducible. We willdenote the morphism h|Vi by hi. Also, we write

W “ž

i,jPI

ž

lPKij

W ijl

for indexing sets Kij depending on pairs i, j P I, so that each W ijl is irreducible and

´

š

lPKijW ijl

¯

Ñ Vi Â1UVj is a Nisnevich cover.

Now for any index j P I, the scheme Vi Â1UVj is non-empty (since A1

U is irreducible

and Vi, Vj are etale over A1U). Suppose that for some i, j P I and l P Kij, the morphisms

hi|W ijl

and hj|W ijl

are not identical. Then they are A1-chain homotopic via non-constant

A1-homotopies. Then by (1) above, these A1-homotopies must factor through some 1-dimensional closed subvariety C Ă X. In particular, the morphisms hi|W ij

land hj|W ij

l

also factor through C ãÑ X. Thus as we assumed that Vi and Vj are irreducible, bothhi and hj also factor through C ãÑ X.

On the other hand, if for every l, the morphisms hi|W ijl

and hj|W ijl

are identical,

then since Vj is irreducible, again by (1) above, we see that hj also factors through the1-dimensional subvariety C obtained earlier.

Thus we see that if for some i, j P I and some l P Kij, we have hi|W ijl‰ hj|W ij

l, then

the entire A1-ghost homotopy factors through some 1-dimensional closed subscheme Dof X. Thus, in this case, the result follows because of Lemma 5.1.3. Indeed, then we seethat the t1, t2 : U Ñ X factor through D and, being connected by an A1-ghost homotopywithin D, are connected by an A1-chain homotopy in D.

On the other hand, suppose that for every i, j and every l P Kij, we have hi|W ijl“

hj|W ijl

. Then this implies that hi|ViÂ1UVj “ hj|ViÂ1

UVj . But this implies that the hi give

a morphism A1U Ñ X which lifts h : A1

U Ñ SpXq which again proves y1 “ y2.

This immediately implies the following:

Corollary 5.1.5. The Morel and Asok-Morel conjectures (Conjectures 2.1.5 and 2.1.13)hold for a proper non-uniruled surface over an algebraically closed field.


5.2 The case of ruled surfaces

In this section, we exhibit examples of ruled surfaces over an algebraically closed fieldof characteristic 0 for which the Asok-Morel conjectures (Conjectures 2.1.13 and [1,Conjecture 6.2.7]) fail to hold.

A classical result [20, Chapter V, Remark 5.8.4] (and references cited there) saysthat every ruled surface has for its minimal models P2, P1 ˆ P1 or a P1-bundle over C,where C is a curve of genus ą 0 (recall that by a P1-bundle over C we mean a smooth,proper morphism f : X Ñ C such that for every c P C, the fibre f´1pcq is a rationalcurve) 1. Conjectures 2.1.5 and 2.1.13 hold in the case of rational surfaces (see Corollary4.3.3). Thus, it only remains to analyze the case when the minimal model is P1 ˆ C,where C is a curve of genus ą 0. By Hurwitz formula, such curves are A1-rigid, in thesense of Definition 1.2.9. This along with the following simple observations, leads to theproof that the sheaf of A1-connected components of a P1-bundle over a smooth curve Cof genus ą 0 is just the sheaf represented by C (see also [32, Remark 1.13]).

Definition 5.2.1. Let U be a scheme. Let φ : F Ñ G be a morphism of sheaves of setson Sm{k.

(1) We say that an A1-homotopy h : U ˆ A1 Ñ F respects fibres of f if there existsa morphism γ : U Ñ G such that φ ˝ h “ γ ˝ pr1 where pr1 : U ˆ A1 Ñ U is theprojection on the first factor. We say that h lies over γ.

(2) Let H :“ pV , tW ijui,j, thiui, thijkui,j,kq be an A1-ghost homotopy of U in F . We

say that H respects fibres of f if there exists a morphism γ : U Ñ G such thatφ ˝ hH “ γ ˝ fH. We say that H lies over γ.

Remark 5.2.2. An A1-ghost homotopy of U in F gives rise to an A1-homotopy of Uin SpFq. However, to say that an A1-ghost homotopy respects fibres of F Ñ G is notexactly the same as saying that the corresponding A1-homotopy in SpFq respects thefibres of SpFq Ñ G. For instance, if H is an A1-ghost homotopy of U in F such that thecorresponding A1-homotopy of U in SpFq respects the fibres of the morphism SpFq Ñ G,it need not be true that φ˝hH “ γ ˝fH on every copy of W ij

k Â1 that occurs in SppHq.

Lemma 5.2.3. Let φ : F Ñ G be a morphism of sheaves of sets on Sm{k. Assume that Gis A1-invariant. Let U be any scheme. Then any A1-homotopy (resp. n-ghost homotopy)of U in F respects fibres of φ. Thus there exists a morphism γ : U Ñ G such that thegiven A1-homotopy (resp. n-ghost homotopy) of U factors through F ˆG,γ U Ñ F .

Proof. Let h : U ˆ A1 Ñ F be an A1-homotopy of U in F . Since G is A1-invariant,the A1-homotopy f ˝ h is constant, i.e. there exists a morphism γ : U Ñ G such thatφ ˝ h “ γ ˝ pr1. This proves the lemma for A1-homotopies.

1Compare [20, Chapter V, Section 2].

5.2. The case of ruled surfaces 47

Now, suppose we have an A1-ghost homotopy

H :“ pV , tW ijui,j, thiui, th

ijkui,j,kq

of U in F which represents an A1-homotopy h of U in SpFq. Since we have provedthe result for A1-homotopies, the fact that G is A1-invariant implies that there exists aγ P GpUq giving a commutative diagram

U ˆ A1 pr1 //

h��

U

γ

��SpFq // G.

Combining this with the commutative diagram

Vi //

hi��

U ˆ A1

h��

F // SpFq

we obtain a proof of the equality φ ˝ hH “ γ ˝ fH when the morphisms are restricted toVi Ă SppHq for i P I.

Let i, j P I and k P Kij. Since G is A1-invariant, for every A1-homotopy hijkl :W ijk ˆ A1 Ñ F occurring in hijk, there exists a morphism γl : W ij

k Ñ G such that thesquare

W ijk ˆ A1 //

��

W ijk

��F // G

is commutative. We need to show that γl is the same as the morphism

W ijk Ñ U ˆ A1 pr1

Ñ UγÑ G.

For t “ 0, 1, we have φ ˝ hijkl ˝ σt “ γl ˝ pr1 ˝ σt “ γl. However, by the definition of an

A1-chain homotopy, we have hijkl ˝ σ0 “ hijkl´1 ˝ σ1 and hijkl ˝ σ1 “ hijkl`1 ˝ σ0. Thus we seethat the γl “ γ1 for every l. However, γ1 is equal to the composition of morphism

W ijk

σ0Ñ W ij

k ˆ A1 hijklÑ F Ñ G

which, in turn, is equal to the composition

W ijk Ñ Vi Â1

UVj

pr1Ñ Vi

hiÑ F φ

Ñ G.


Since the morphism φ ˝ hi is the same as the composition

Vi Ñ U ˆ A1 pr1Ñ U

γÑ G,

we see that γ1 is equal to the composition

W ijk Ñ Vi Â1

UVj

pr1Ñ Vi Ñ U ˆ A1 pr1

Ñ UγÑ G

as desired. Proceeding inductively, we can prove the result for n-ghost homotopies.

Proposition 5.2.4. Let E Ñ C be a P1-bundle over a smooth curve C over a field k ofgenus ą 0. Then SpEq “ πA1

0 pEq “ C.

Proof. We fix a smooth Henselian local ring pR,mq, U “ SpecpRq. We fix a morphismγ : U Ñ C and consider the pullback E with respect to γ which we denote by Eγ. Sincea P1-bundle is etale locally trivial, we see that Eγ » P1

U . Since C is A1-rigid, by Lemma5.2.3, it follows that SpEqpUq “ CpUq, and hence, it follows that

SpEq “ πA1

0 pEq “ C.

We now state the main theorem of this section. Throughout the remainder of thissection, we will assume that k is an algebraically closed field of characteristic 0.

Theorem 5.2.5. Let E be a P1-bundle over a smooth projective curve C of genus ą 0.Let X be the blowup of E at a single closed point. Then SpXq ‰ S2pXq and S2pXq “S3pXq.

The strategy to prove Theorem 5.2.5, suggested by Lemma 5.2.3, is as follows: we fixa smooth Henselian local scheme U over a k and fix a map γ : U Ñ C. We will considerthe pullback Xγ Ñ U of the morphism X Ñ C and analyze the n-ghost homotopies of Uinside Xγ. We will determine explicit criteria for A1-homotopies and n-ghost homotopiesto exist. We begin by setting up some notation.

Notation 5.2.6. The following notation will stay in effect for the remainder of thissection:

(1) Let R denote the henselization of a ring of the form OV,v where V is a smoothscheme over k and v is a point of V . Let m be the maximal ideal of R and letK “ R{m. We will denote SpecpRq by U and the closed point of U by u. Forr P R, Uprq will denote the closed subscheme of r corresponding to the ideal xry.

(2) We will denote by X0, X1 the homogeneous coordinates on P1R. Thus, we have

P1U “ ProjpRrX0, X1sq.


(3) For any r0 P mzt0u, we define

Ipr0q :“ ideal sheaf on P1U corresponding to the

homogeneous ideal xr0, X0y.

Let T pr0q denote the closed subscheme of P1U corresponding to this ideal sheaf. Let

θr0 : Xpr0q Ñ P1U denote the blowup of P1

U at T pr0q.

We wish to examine the sections of morphisms Xpr0q Ñ U . The scheme P1U is

covered by affine patches isomorphic to A1U given by the conditions X0 ‰ 0 and X1 ‰ 0.

We will denote these by SpecpRrX0{X1sq and SpecpRrX1{X0sq respectively. Using thefact that U is a local scheme it is easy to see that any section of P1

U factors throughSpecpRrX0{X1sq or SpecpRrX1{X0sq. The sections of SpecpRrX0{X1sq Ñ U are givenby R-algebra homomorphisms RrX0{X1s Ñ R. Such a homomorphism is determined bya choice for the image of X0{X1.

Notation 5.2.7. We denote by αr : U Ñ SpecpRrX0{X1sq Ñ P1U the section of P1

U Ñ Udetermined by X0{X1 ÞÑ r. Similarly, we denote by βr : U Ñ SpecpRrX1{X0sq Ñ P1

U

the section of P1U Ñ U determined by X1{X0 ÞÑ r. Thus we see that if r is such that

r, 1{r R m, then αr “ β1{r.

A section of Xpr0q Ñ U can be composed with Xpr0q Ñ P1U to give a section of

P1U Ñ U . Of course, not every section of P1

U Ñ U can be lifted to give a section ofXpr0q Ñ U . However, if a lift exists, it must be unique since the image of any section ofP1U Ñ U maps the generic point of U outside the scheme T pr0q (since we assume r0 ‰ 0).

Notation 5.2.8. Let r0, P mzt0u. For r P R, if a lift of αr : U Ñ P1U to Xpr0q exists, it

will be denoted by αr0r . Similarly, we define βr0r to be the lift of βr to Xpr0q, if it exists.

A section α : U Ñ P1U of P1

U Ñ U , lifts to Xpr0q if and only if the ideal sheafα˚pIpr0qq is principal. Since U is local, this ideal sheaf corresponds to an ideal of R,which we continue to denote by α˚pIpr0qq by abuse of notation. We shall next determinethis ideal. Unless α is of the form αr where r P m, we have α˚pIpr0qq “ OU . Whenα “ αr for r P m, the ideal sheaf α˚pIpr0qq corresponds to the ideal xr0, ry in R. We nownote an elementary observation about principal ideals in local rings, which allows us todetermine the generator of the ideal α˚pIpr0qq of R.

Lemma 5.2.9. Let A be a local domain and let x1, x2 P A. Then the ideal xx1, x2y isprincipal if and only if x1|x2 or x2|x1.

Proof. Clearly, if x1|x2 or x2|x1, the given ideal is principal. Conversely, supposexx1, x2y “ xzy. Then z “ x1γ1 ` x2γ2 for γ1, γ2 P R. Also, for i “ 1, 2, xi “ δizfor δi P R. Thus γ1δ1 ` γ2δ2 “ 1. Thus one of the elements γ1, γ2 must be a unit. Thisproves the lemma.


Lemma 5.2.10. Let r0 P mzt0u. Let α : U Ñ P1U be a section of P1

U Ñ U which admitsa lift to Xpr0q. Then one of the following must hold:

(1) α˚pIpr0qq “ x1y. In this case, α is of the form βr where r P R.

(2) α˚pIpr0qq “ xr0y. In this case, α is of the form αr where r0|r.

(3) α˚pIpr0qq “ xry for some r P R such that r|r0 but r0 ffl r. In this case, α is of theform αr1 where r1{r is a unit in R.

Proof. We have α˚pIpr0qq “ x1y if and only if α factors through the complement of thescheme T pr0q of P1

U . This is so if and only if α is of the form βr where r P R. Thisproves (1).

We now have to consider the case that α is of the form αr : U Ñ P1U for some r P m.

In this case, α˚pIpr0qq “ xr0, ry. Thus, by Lemma 5.2.9, this ideal is equal to xr0y if r0|ror is equal to xry if r|r0. This proves (2) and (3).

Notation 5.2.11. Let r0 P mzt0u. Then for r P R, Ar0r denotes the set of sectionsα : U Ñ P1

U such that the ideal sheaf α˚pIpr0qq corresponds to the ideal xry in R. ByLemma 5.2.10, Ar0r is non-empty if and only if r|r0.

Notation 5.2.12. If pA, nq is a local ring, pA will denote the completion of A with respect

to m. For any ideal I of A, pI will denote the completion of I with respect to m, whichwe view as an ideal of pA.

We briefly describe our strategy about computing A1-homotopy and n-ghost homo-topy classes of sections. We will mainly be interested in morphisms from the total spaceof some n-ghost homotopy of a smooth Henselian local scheme U into P1

U that lift toXpr0q. We want to use Lemma 5.2.10 to determine the local generator of pullback ofIpr0q along the n-ghost homotopy in question. We shall then “spread” this local gener-ator at a point over a closed subscheme of the closed fibre containing that point. Thisis done in Lemma 5.2.13 and Lemma 5.2.14. This allows us to precisely determine thegenerator of the pullback of Ipr0q along the ghost homotopy that connects two sectionsin Ar0r (see Lemma 5.2.16 below).

Lemma 5.2.13. Let r0 P mzt0u. Let f : V Ñ U be a smooth morphism. Let h : V Ñ P1U

be a morphism over U which lifts to Xpr0q. Let v be a point of V such that fpvq “ uand such that κpvq “ K. Then the ideal h˚pIpr0qqv is generated by an element of theform f˚prq for some r P R such that r|r0.

Proof. Since R is Henselian, there exists a section α : U Ñ V such that αpuq “ v.By Lemma 5.2.10, α˚ph˚pIpr0qqvq “ xry where r P R is such that r|r0. We will showthat h˚pIpr0qqv is generated by f˚prq. We know that the ideal h˚pIpr0qvq is principal,generated by some element ρ P OV,v. Thus we need to show that f˚prq is a unit multipleof ρ.


The sequence of homomorphisms Rf˚Ñ OV,v

α˚Ñ R gives rise to the sequence of homo-

morphisms on the completions pRpf˚Ñ pOV,v

pα˚Ñ pR and pα˚˝ pf˚ is the identity homomorphism

on pR. (Here pf and pα are the induced morphisms of schemes Specp pOV,vq Ñ Specp pRq and

Specp pRq Ñ Specp pOV,vq, respectively.) We denote by φR : Specp pRq Ñ SpecpRq and

φv : Specp pOV,vq Ñ SpecpOV,vq the morphisms induced by the canonical homomorphismsof local rings into their completions.

Suppose n is the Krull dimension of R and m is the dimension of the fibres of f .Then there exists a Zariski local neighbourhood W of v in V such that f |W factors asW Ñ An

U Ñ U where W Ñ AnU is an etale morphism taking v to the origin in An

K Ă AnU .

Thus there exists a commutative diagram as follows where the horizontal morphisms areisomorphisms:

pR //

xf˚��

Krrsss� _

��pOV,v // Krrs, tss.

Here s “ ps1, . . . , snq and t “ pt1, . . . , tmq are tuples of variables.

The ideal {h˚pIpr0qqv “ φ˚vph˚pIpr0qvqq in pOV,v is principal. Suppose that it is gener-

ated by some element ρ P pOV,v. Then ρ| pfpr0q in pOV,v and hence by the above commuta-

tive square, we see that ρ is a unit multiple of some element in the image of pf˚. Thus,we may assume that ρ “ pf˚pr1q, where r1 P pR.

The morphism α ˝ φR : Specp pRq Ñ V , is equal to φv ˝ pα. Thus we have

pα ˝ φRq˚ph˚pIpr0qqvq “ α˚pφ˚vph

˚pIpr0qqvqq

“ pα˚px pf˚pr1qyq

“ xr1y.

However, we also have pα˝φRq˚ph˚pIpr0qqvq “ φ˚Rpxryq, showing that r1 is a unit multiple

of φ˚Rprq in pR. Thus, the ideals φvph˚pIpr0qvqq “ x

xf˚pr1qy and φvpxf˚prqyq “ xxf˚pφ˚Rprqqy

are equal. Since φv is a faithfully flat morphism, this shows that h˚pIpr0qvq “ xf˚prqy

in R.

Lemma 5.2.14. Let r0 P mzt0u. Let f : V Ñ U be a smooth morphism. Let h : V Ñ P1U

be a morphism which lifts to Xpr0q. Let z be a point of U and let v be a point off´1pzq “ V Û,z Specpκpzqq such that the ideal h˚pIpr0qqv in OV,v is generated by anelement of the form f˚prq where r P R. Let Z be the irreducible component of f´1pzqcontaining v. Then there exists an open subscheme W0 of V such that Z Ă W0 and theideal sheaf h˚pIpr0qq|W0 is generated by f˚prq.

Proof. In the local ring OV,v, we have the equality of ideals h˚pIpr0qq “ xf˚prqy which

must hold in a neighbourhood of v. Thus, there exists an open subscheme W of V


with v P W such that the ideals h˚pIpr0qqpW q and xf˚prqy of OV pW q are equal. Letv1 P ZzW0. We wish to show that the ideal h˚pIpr0qqv1 in OV,v1 is also generated byf˚prq.

Let z be of codimension m in U . There exists a (non-unique) K-algebra isomorphism

κpzqrrtss Ñ xRz, where t “ pt1, . . . , tmq is an m-tuple of variables. We fix such anisomorphism and using the ring homomorphism f˚ : κpzq Ñ κpv1q, we define the ring

R1 :“ κpv1q bκpzqxRz – κpvqrrtss.

Let ψU denote the morphism SpecpR1q Ñ SpecpRq. Let rV “ V Û SpecpR1q. Let ψVand rf denote the projections rV Ñ V and rV Ñ SpecpR1q respectively.

We have a commutative square

Specpκpv1qqv1 //

��

V

��SpecpR1q // SpecpRq

which gives a morphism rv1 : Specpκpv1qq Ñ rV . By the argument in the proof of Lemma5.2.13, we see that the ideal ph˝ψV q

˚pIpr0qqrv1 in OrV ,rv1

is generated by an element of the

form rf˚pρq, for some ρ P R1. Thus, there exists an open subscheme ĂW of rV containing

rv1 such that ph ˝ ψV q˚pIpr0qqpĂW q is generated by rf˚pρq.

Since κpv1q is a finitely generated field extension of κpzq, the morphism ψV is open.

Hence, the image of ĂW is an open subscheme containing v1. consequently, there existsa point rv2 in ĂW such that the point v2 :“ ψV prv2q is in W0 and we have

ph ˝ ψV q˚pIpr0qqrv2 “ ψ˚V ph

˚pIpr0qqv2q “ ψ˚V pxf

˚prqyq “ x rf˚pψ˚Uprqqy.

But since rv2 P ĂW , we also have

ph ˝ ψV q˚pIpr0qqrv2 “ x

rf˚pρqy.

Thus, ρ is a unit multiple of ψ˚Uprq, and the ideal ψ˚V ph˚pIpr0qqv1q in O

rV ,rv1is equal to

ψ˚V pxf˚prqyq. Since ψ˚V : OV,v1 Ñ OrV ,rv1

is faithfully flat, this implies that that the idealh˚pIpr0qqv1 in OV,v1 is equal to xf˚prqy.

Lemma 5.2.15. Let r0 P mzt0u. For i “ 1, 2, let fi : Vi Ñ U be smooth morphismsand let hi : Vi Ñ U be morphisms over U which lift to Xpr0q. Let f : V1 Ñ V2 andg : V2 Ñ V1 be morphisms over U such that g is a a closed embedding, h2 “ h1 ˝ gand f ˝ g “ IdV2. Let vi be a point of Vi for i “ 1, 2 such that gpv1q “ v2. Supposeh˚2pIpr0qqv2 “ xf

˚2 prqy for some r P R. Then h˚1pIpr0qqv1 “ xf

˚1 prqy.


Proof. This proof is along the same lines as the proof of Lemma 5.2.14. Let z “ f1pv1q “

f2pv2q. Let Rz be the localization of R at the prime ideal corresponding to the point

z. We fix a K-algebra isomorphism xRz Ñ κpzqrrtss where t “ pt1, . . . , tmq is an m-tuple of variables. Using the ring homomorphism f˚ : κpzq Ñ κpv1q, we define R1 “

κpv1q bκpzqxRz. Let ψU be the morphism SpecpR1q Ñ SpecpRq. For i “ 1, 2, we define

rVi “ ViÛ SpecpR1q and denote by rfi and ψVi the projections rVi Ñ SpecpR1q and rVi Ñ Virespectively. Also, the morphisms rf and rg are the pullbacks of f and g. We also havepoints rvi on rVi for i “ 1, 2 such that ψViprviq “ vi and rgprv2q “ rv1.

Then by the argument in the proof of Lemma 5.2.14, there exists an element ρ P R1

such that the ideal ph1 ˝ ψV1q˚pIpr0qqrv1 in O

rV ,rv1is generated by rf˚1 pρq. Thus, we have

ph2 ˝ ψV2q˚pIpr0qqrv2 “ ph1 ˝ g ˝ ψV2q

˚pIpr0qqrv2

“ ph1 ˝ ψV1 ˝ rgq˚pIpr0qqrv2

“ rg˚px rf˚1 pρqyq

“ x rf˚2 pρqy.

However, by assumption, ph2 ˝ ψV2q˚pIpr0qqrv2 “ ψ˚V2pxf

˚2 prqyq “

rf˚2 pψ˚Uprqq. This proves

that ρ is a unit multiple of ψ˚Uprq in R1. As in the proof of Lemma 5.2.14, this showsthat h˚1pIpr0qqv1 “ xf

˚1 prqy.

Lemma 5.2.16. Let r0 P mzt0u and let n ě 0 be an integer. Let α and α1 be sectionsof P1

U Ñ U which are connected by an n-ghost homotopy. Then there exists r P R suchthat r|r0 and α, α1 P Ar0r . Also, in this case h˚HpIpr0qq is generated by f˚Hprq.

Proof. This will be proved by induction on n. We begin with the case n “ 1. Thus,suppose α and α1 are connected by a single A1-homotopy h : A1

U Ñ U . Let r be suchthat α P Ar0r , i.e. α˚pIpr0qq “ xry. Then by Lemma 5.2.15, we see that h˚pIpr0qq|σ0puq isgenerated by pr˚2 prq where pr2 is the projection A1

U “ A1 ˆ U Ñ U . By Lemma 5.2.14,there exists an open subscheme W0 Ă A1

U containing the closed fibre A1k Ă A1

U suchthat h˚pIpr0qq|W0 is generated by pr˚2 prq. In particular, h˚pIpr0qqσ1puq is generated bypr˚2 prq. Thus, pα1q˚pIpr0qq “ σ˚1 phpIpr0qqq is generated by σ˚1 ppr

˚2 prqq “ r and hence, α1

is contained in Ar0r . This completes the proof in the case n “ 0.Now suppose that the result has been proved for m-ghost homotopies where m ď n.

Suppose α and α1 are connected by an n-ghost homotopy



q.

Let r, r1 P R be such that α˚pIpr0qq “ xry and pα1q˚pIpr0qq “ xr1y. Then, by lemma5.2.15, we see that h˚HpIpr0qqĂσ0puq “ xfHprqy and h˚HpIpr0qqĂσ1puq “ xfHpr

1qy. Let Z0 andZ1 be the irreducible components of the closed fibre of f´1

H puq of V containing rσ0puq andrσ0puq respectively. By Lemma 5.2.14, there exist open subschemes Wi Ą Zi of V fori “ 0, 1 such that h˚HpIpr0qq|W0 is generated by f˚Hprq and h˚HpIpr0qq|W0 is generated by


f˚Hpr1q. The morphisms Zi Ñ A1

k are etale and thus there exists a point z P A1k Ă A1

U

lying in the image of both Zi Ñ A1k for i “ 0, 1. Thus, there exist points zi P Zi which

map to z under the morphism V Ñ A1U . Thus, we obtain the point pz1, z2q P V Â1

UV and

there exists a point z3 P W which maps to pz1, z2q under the morphism W Ñ V Â1UV .

Let g1, g2 denote the two compositions

W Ñ V Â1UV

priÑ V

where pri : V Â1UV Ñ V is the projection on the i-th factor for i “ 1, 2. Thus

g˚1 ph˚HpIpr0qqz3 is generated by g˚1 pf

˚Hprqq and g˚2 ph

˚HpIpr0qqz3 is generated by g˚2 pf

˚Hpr

1qq.Since the morphisms hH˝g1 and hH˝g2 are pn´1q-ghost chain homotopic, the inductionhypothesis implies that the ideals xg˚1f

˚Hprqy and xg˚2f

˚Hpr

1qy ofOW,z3 are equal. It is easilyseen that the morphisms fH ˝ g1 and fH ˝ g2 from W Ñ U are identical; denote thismorphism by ϕ. Since ϕ is a smooth morphism, the induced map ϕ7 : OU Ñ ϕ˚OW isfaithfully flat and thus the equality xg˚1f

˚Hprqy “ xg

˚2f˚Hpr

1qy implies that xry “ xr1y in Ras desired.

Now we need to prove that the ideal sheaf h˚HpIpr0qq is generated by f˚Hprq. The abovearguments show that there exists an open subscheme V0 Ă V containing the closed fibref´1H puq such that h˚HpIpr0qq|V0 is generated by f˚Hpr0q. Of course, it is possible that V zV0

is non-empty. Suppose v P V zV0 and let fHpvq “ z P U . We define Uz “ SpecpOhU,zq.Applying the above arguments for Uz instead of U , we see that there exists an elementrr P OhU,z such that hHpIpr0qqv1 is generated by f˚Hprrq for every v1 in f´1

H pzq (in particular

for v1 “ v). But since f´1H pzq X V0 is non-empty, we see that xrry “ rOhU,z. Thus we see

that h˚HpIpr0qq|V is generated by f˚Hprq. Now, by Lemma 5.2.15, it easily follows thath˚HpIpr0qq|SppHiq is also generated by f˚Hprq for every pn´ 1q-ghost homotopy appearingin HW . This completes the proof.

Remark 5.2.17. We note a simple consequence of Lemma 5.2.16 for later use. Let Hbe an n-ghost homotopy of sections of P1

U which lifts to Xpr0q for r0 P mzt0u. Then ifHp0q “ α : U Ñ P1

U is such that the image of α is disjoint from the closed subschemeT pr0q (cut out by the ideal xr0, X0y), then H factors through the complement of T pr0q.

Explicit description of A1-homotopies and ghost homotopies

When we take n “ 0, Lemma 5.2.16 gives us a necessary condition for two elements αr1and αr2 to be A1-homotopic. However, we will see now that it is not strong enough tocharacterize the A1-homotopy classes of the sections of Xpr0q Ñ U . The following tworesults give a complete description of A1-homotopy classes of the sections of Xpr0q Ñ U .

Lemma 5.2.18. Let r0 P mzt0u. Any two elements of Ar0r are A1-homotopic via homo-topies that lift to Xpr0q if r “ 1 or r0.

Proof. First consider the case r “ 1. Suppose α, α1 are in Ar01 . Then it is easy to seethat α “ βr1 and α1 “ βr2 for some r1, r2 P R. Then these two elements are homotopic


in P1U via the homotopy given by the R-algebra homomorphism

RrX1{X0s Ñ RrT s, X1{X0 ÞÑ r1p1´ T q ` r2T.

The resulting morphism A1U Ñ P1

U factors through the open immersion P1UzT pr0q ãÑ P1

U .Thus, it lifts to Xpr0q.

If α, α1 are in Ar0r0 , they are of the form αr1 and αr2 where r0|r1 and r0|r2. Thenconsider the homotopy h : A1

U Ñ P1U given by the R-algebra homomorphism

RrX0{X1s Ñ RrT s, X0{X1 ÞÑ r1p1´ T q ` r2T.

Then h˚pIpr0qq is the sheaf corresponding to the ideal xr0, r1p1´T q`r2T y “ xr0y, whichis principal. Thus, h lifts to Xpr0q as claimed.

Proposition 5.2.19. Let r0 P mzt0u. Suppose r1, r2 P R are non-units such that ri|r0

but r0 ffl ri for i “ 1, 2. Let r1 “ r0{r1. Then αr0ri for i “ 1, 2 are A1-chain homotopic inXpr0q if and only if the following two conditions hold:

(1) r1{r2 is a unit.

(2) r1{r2 ´ 1 is in radpxr1yq ` radpxr1yq Ă radpxr1, r2yq.

Proof. First we prove the necessity of the conditions. The fact that r1{r2 is a unit isalready known to us by Lemma 5.2.16. An A1-homotopy h : A1

U Ñ P1U is given by the

choice of an invertible sheaf on A1U along with the choice of two generating sections.

Since U is essentially smooth, on both U and A1U , the Picard group is isomorphic to

the group of Weil divisors, that is, PicpUq – ClpUq and PicpA1Uq – ClpA1

Uq. Theprojection A1

U Ñ U induces an isomorphism ClpUq – ClpA1Uq. Since U is local, this

implies that any invertible sheaf on A1U is isomorphic to OA1

U. Thus we see that the

homotopy h can be given by two generating global sections of OA1U

, that is, two poly-nomials ppT q, qpT q P RrT s such that xppT q, qpT qy is the unit ideal. If Dpqq denotes theopen subscheme of A1

U where q is a unit, then h maps Dpqq into the open subschemeA1U “ SpecpRrX0{X1sq Ă P 1

U and the morphism h|Dpqq is given by the R-algebra homo-morphism X0{X1 ÞÑ ppT q{qpT q. Notice that pp0q{qp0q “ r1 and pp1q{qp1q “ r2. By thefirst equality, we see that r1|pp0q and thus, in particular, pp0q is a non-unit. However,xpp0q, qp0qy is the unit ideal in R (since xp, qy is the unit ideal in RrT s). Thus qp0q is aunit in R. Thus qpT q is a primitive polynomial, that is, its content, denoted by contpqq,is a unit.

Now suppose that the ideal sheaf h˚pIpr0qq is locally principal. Since RrT, q´1s isalso a UFD, this means that the ideal h˚pIpr0qqpDpqqq is principal. By the formuladefining h, this ideal is equal to xr0, p{qy. We know that this ideal is actually equal tor1RrT, q

´1s. Thus, we have

p P r1RrT, q´1s XRrT s “ Yně0pr1 : qnq.


We claim that pr1 : qnq “ xr1y for every n ě 0. Indeed, suppose f P pr1 : qnq. If n “ 0,there is nothing to prove. So, we assume n ą 0. There exists an element gpT q P RrT sand an integer n such that r1g “ fqn in RrT s. But then, qn|r1g in RrT s. As q is aprimitive polynomial, qn|pr1g{contpr1gqq “ g{contpgq. In particular, qn|g and thus r1|fin RrT s, as claimed. This shows that r1|p in RrT s and we write p “ r1p

1 where p1 P RrT s.The ideal xr1p

1, qy is a unit ideal. Thus, modulo r1, we see that the polynomial qpT qmust be a unit. Thus if qi denotes the coefficient of T i in qpT q, then we see that q0 is aunit (as we have already seen before) and qi is nilpotent modulo r1 for i ě 1.

In RrT, q´1s, we have the equality of ideals xr0, r1p1{qy “ xr1y, which implies that

xr1, p1{qy “ x1y. Thus there exists an integer n ě 0 such that qn lies in the ideal xr1, p1yin RrT s. We know that xp, qy “ 1. Thus, xpn, qny “ 1 which implies xpp1qn, qny “ 1.Thus, xr1, p1y, which contains xpp1qn, qny is equal to the unit ideal. Thus if p1i denotes thecoefficient of T i in p1pT q, we see that p10 is a unit and p1i is nilpotent modulo r1.

Using pp1q{qp1q “ r2, we see that r2 “ r1pu1 ` t1q{pu2 ` t2q where ui are units, t1is nilpotent modulo r1 and t2 is nilpotent modulo r1. Using pp0q{qp0q “ r1, we see thatu1{u2 “ 1. Thus we may actually assume that u1 “ u2 “ 1. Let t3 “ ´t2{p1` t2q. Thenwe see that t3 is nilpotent modulo r1 and r2 “ r1p1` t1qp1` t3q. Thus

r2{r1 ´ 1 “ t1 ` t3 ` t1t3

and it is obvious that this element lies in radpxr1yq ` radpxr1yq as claimed. This provesthe necessity of the conditions.

Now we prove the sufficiency. Let s1 and s1 be the square-free parts of r1 andr1 :“ r0{r1. Then radpxr1yq “ xs1y and radpxr1yq “ xs1y. Thus r2{r1´ 1 “ s1δ` s

1δ1. Wedefine r3 “ r1p1` s1δ1q.

Consider the rational function f1pT q “ r3{p1 ` s1δ1T q. The ideal xr3, 1 ` s1δ1T y isthe unit ideal since s1δ1 is nilpotent modulo r1 and thus nilpotent modulo r3. Thusf1 defines a homotopy h1 : A1

U Ñ P1U by X0{X1 ÞÑ r3{p1 ` s1δ1T q. Since f1p0q “ r3

and f1p1q “ r1, this homotopy connects αr3 to αr1 . Now, let p be a point of A1U which

projects to z P U under the projection A1U Ñ U . We see that

h˚pIpr0qqp “ xr0,r3

1` s1δ1Ty

is principal if 1 ` s1δ1T is a unit at p. If 1 ` s1δ1T is not a unit, then s1, δ1 and T areall units at p. Thus r1 is also a unit at p. Since r3 is a unit multiple of r1, we see thatr3 is a unit at p. Thus r3{p1 ` s1δ1T q “ 8 modulo z. Thus hppq “ pz, p1 : 0qq which isnot in the closed subscheme defined by Ipr0q. Thus h˚pIpr0qqp is the unit ideal in thiscase. Thus h˚pIpr0qq is a locally principal ideal which allows us to conclude that h liftsto Xpr0q.

Consider f2pT q “ r3 ` r1s1δ1T “ r3p1 ` s1δ1T q{p1 ` s1δ1q. This defines a homotopy

h2 : A1U Ñ P1

U byX0{X1 ÞÑ r3p1` s

1δ1T q{p1` s1δ1q.


This homotopy lifts to Xpr0q as the ideal xr0, r3p1` s1δ1T q{p1` s1δ1qy is principal (since

s1δ1 is nilpotent modulo r0{r3 “ r1p1`s1δ1q). We compute that f2p0q “ r3 and f2p1q “ r2

and thus h2 connects αr3 to αr2 .

Notation 5.2.20. We will use the symbol Zp´q to denote the closed subschemes definedby ideals of rings, ideal sheaves or homogeneous ideals of graded rings.

We are now set to obtain a necessary condition for two sections to be n-ghost homo-topic. We will then show that this is sufficient for the existence of 1-ghost homotopiesin this case.

Proposition 5.2.21. Let r0 P mzt0u. Let r1 P R be such that r1|r0 but r0 ffl r1. Letr1 “ r0{r1. Let r2 be a unit multiple of r1.

(1) If there exists an n-ghost homotopy H, for an integer n ą 0, connecting αr1 to αr2which lifts to Xpr0q, then r2{r1 ´ 1 P radpxr1, r

1yq.

(2) If r2{r1 ´ 1 P radpxr1, r1yq, there exists a 1-ghost homotopy H connecting αr1 and

αr2 which lifts to Xpr0q.

Proof. Let H denote an n-ghost homotopy connecting αr1 to αr2 which lifts to Xpr0q.By Lemma 5.2.16, the ideal sheaf h˚HpIpr0qq is generated by f˚Hpr1q. However, any pointv of SppHq, the ideal h˚HpIpr0qqv is equal to xf˚Hpr0q, h

˚HpX0{X1qy. Thus, we see that

the ideal xf˚Hpr1q, h˚HpX0{X1qy of OSppHq,v is principal. Thus the ideal sheaf h˚HpIpr1qq

is locally principal. Let Xpr0, r1q denote the scheme obtained by blowing up P1U at

the closed subscheme ZpIpr0qIpr1qq. The scheme Xpr0, r1q can also be constructed byfirst blowing up the closed subscheme ZpIpr1qq to construct Xpr1q and then blowing upthe total transform of the ideal sheaf Ipr0q on Xpr1q. We will now compute this totaltransform.

We view Xpr1q as a closed subscheme of P1 ˆ P1U “ P1 ˆ P1 ˆ U where we use the

homogeneous coordinates Y0, Y1 for the first copy of P1 and X0, X1 for the second copy.Then, Xpr1q is given by the equation r1Y0X1 “ Y1X0. It suffices to compute this in theopen patch X1 ‰ 0 (since both Ipr1q and Ipr0q have support within this patch). Wecompute this total transform on the open patch Y1 ‰ 0 as follows:

xr0, X0{X1y “ xr0, X0{X1y x1, Y0{Y1y

“ xr0{r1, Y0{Y1y xr1, r1pY0{Y1qy

“ xr1, Y0{Y1y xr1, X0{X1y.

On the patch Y0 ‰ 0, we have r1 “ pX0{X1qpY1{Y0q. Thus as r1|r0 we have the equalityof ideals

xr0, X0{X1y “ xX0{X1y

“ xr1, X0{X1y.


Note that xr1, X0{X1y is nothing but the ideal corresponding to the exceptional divisorof the blowup morphism Xpr1q Ñ P1

U . Therefore, we see that Xpr0, r1q is isomorphic tothe scheme obtained by blowing up Xpr1q at the closed subscheme Zpxr1, Y0yq.

We observe that the projection map θ1 : Xpr0, r1q Ñ ProjpRrY0, Y1sq is just theblowup of ProjpRrY0, Y1sq at the closed subschemes Zpxr1, Y1yq and Zpxr1, Y0yq. Let H1be the n-ghost homotopy θ1 ˝ H. Then if αr0,r1r1

denotes the lift of αr1 to Xpr0, r1q,we see that θ1 ˝ αr0,r1r1

is the morphism corresponding to the R-algebra homomorphismRrY0{Y1s Ñ R, Y0{Y1 ÞÑ 1. Thus, by the observation in Remark 5.2.17, we see that H1avoids the closed subschemes Zpxr1, Y1yq and Zpxr1, Y0yq. Thus, the restriction of then-ghost homotopy H1 to the subscheme Upr1, r

1q of U cut out by the ideal xr1, r1y factors

through

P1Upr1,r1q

zptp0 : 1q, p1 : 0qu ˆ Upr1, r1qq – Gm ˆ Upr1, r

1q.

As Gm is A1-rigid, the induced homotopy of the underlying reduced subscheme of Uis constant. In particular, since θ1 ˝ αr0,r1r2

: U Ñ ProjpRrY0, Y1sq corresponds to thehomomorphism Y0{Y1 ÞÑ r2{r1, we see that r2{r1 “ r1{r1 “ 1 modulo any prime of Rcontaining the ideal xr1, r

1y. Thus r2{r1´1 lies in the ideal radpxr1, r1yq. This completes

the proof of (1).

Now we prove (2) - we assume δ “ r2{r1 ´ 1 P radpxr1, r1yq and construct a 1-ghost

homotopy

H1 :“ pV Ñ A1U ,W Ñ V Â1

UV, h, rσ0, rσ1,HW

q

of U in ProjpRrY0, Y1sq connecting θ1 ˝ αr0,r1r1to θ1 ˝ αr0,r1r2

which avoids the closed sub-schemes Zpxr1, Y1yq and Zpxr1, Y0yq.

We first construct a Zariski open cover V of A1U “ SpecpRrSsq. Define

V1 :“ A1UzZpx1` δSyq;

V2 :“ A1UzUpr1q

;

V3 :“ A1UzUpr1q.

This is a Zariski open cover of A1U . Indeed, to show this, it is enough to see that if

p is a prime ideal of RrSs containing p1 ` δSq, then δ is a unit modulo p. Thus asδ P radpr1, r

1q, we see that p cannot be a prime containing xr, r1y. But then p must failto contain either r or r1. Thus the point of A1

U corresponding to p must lie in V2 or V3.We define V “ V1 > V2 > V3. We will now define morphisms hi : Vi Ñ ProjpRrY0, Y1sq fori “ 1, 2, 3 and obtain h : V Ñ ProjpRrY0, Y1sq by defining hVi “ hi.

The morphism h1 : V1 Ñ ProjpRrY0, Y1sq is given by Y0{Y1 ÞÑ p1 ` δSq. Notice thatthe assignment Y0{Y1 ÞÑ 1`δS actually defines a morphism A1

U Ñ ProjpRrY0, Y1sq whichfactors through the complement of Zpxr1, Y1yq. However, it fails to avoid Zpxr1, Y0yq.Indeed the preimage of this subscheme is the closed subscheme of Zpxr1, 1 ` δSyq ofA1U . Hence, we have cut out the scheme Zpx1 ` δSyq to ensure that the morphism

h1 : V1 Ñ ProjpRrY0, Y1sq gives rise to a morphism V1 Ñ Xpr0, r1q.


The morphism h2 : V2 Ñ ProjpRrY0, Y1sq is given by composing the projection V2 Ñ

UzUpr1q with the morphism UzUpr1q Ñ P1U given by Y0{Y1 ÞÑ 1, i.e. it is the “constant

section at 1”. In other words, it factors through Y1 ‰ 0 and is given by Y0{Y1 ÞÑ 1 PRrS, r´1

1 srT s. Similarly, the morphism h3 : V3 Ñ P1U also factors through Y1 ‰ 0 and is

given by Y0{Y1 ÞÑ 1 P RrS, pr1q´1srT s.The morphisms σ0, σ1 : U Ñ A1

U “ SpecpRrSsq factor through the open immersionV1 ãÑ A1

U . We choose the induced morphisms U Ñ V1 ãÑ V as the morphisms rσi.We choose W to be equal to V Â1

UV “

š

pi,jqpVijq where Vij “ ViXVj. The 0-ghost

chain homotopy HW will be a single A1-homotopy which we will define separately oneach piece Vij. For i “ j, we simply define it to be the constant homotopy on ViXVi “ Vi.The remaining are given by explicit formulas below.

We define a homotopy between h1|V1XV2 and h2|V1XV2 . This morphism will be designedto factor through Y0 ‰ 0 and thus will avoid Zpxr1, Y0yq. On the other hand, by thedefinition of V2 it is clear that it will also avoid Zpxr1, Y1yq Ă Zpxr1yq Ă ProjpRrY0, Y1s.We define this morphism by Y1{Y0 ÞÑ p1` δSq´1p1´ T q ` T P RrS, r1

´1p1` δSq´1srT s.The restriction of HW to V12 is taken to be this homotopy. The restriction of HW to V21

is be the inverse of this homotopy.We define a homotopy between h1|V1XV3 and h3|V1XV3 . This morphism will be designed

to factor through Y1 ‰ 0 and thus will avoid Zpxr1, Y1yq. On the other hand, by thedefinition of V3, it is clear that it will also avoid Zpxr1, Y0y Ă Zpxr1yq Ă ProjpRrY0, Y1s.We define this morphism by Y0{Y1 ÞÑ p1 ` δSqp1 ´ T q ` T P RrS, pr1q´1p1 ` δSq´1srT s.The restriction of HW to V12 is taken to be this homotopy. The restriction of HW to V21

is the inverse of this homotopy.The restriction of HW to V23 is just taken to be the constant homotopy between

h2|V2XV3 and h3|V2XV3 . The restriction of HW to V32 is the inverse of this homotopy.Thus we have successfully constructed an 1-ghost homotopy in ProjpRrY0, Y1sq which

lifts to Xpr0, r1q. It is easy to see that it connects θ1 ˝ αr0,r1r1and θ1 ˝ αr0,r1r2

. The lift ofthis 1-ghost homotopy to Xpr0, r1q will connect αr0,r1r1

and αr0,r1r2. Thus, on composing

with the blowup morphism Xpr0, r1q Ñ Xpr0q, we get the desired homotopy in Xpr0q.This completes the proof of (2).

Thus we now have a complete description of n-ghost homotopy classes of the sectionsof Xpr0q and we are now set to finish the proof of Theorem 5.2.5.

Proof of Theorem 5.2.5. Fix a smooth Henselian local ring pR,mq, U “ SpecpRq. Wefix a morphism γ : U Ñ C and consider pullbacks of X and E with respect to γ whichwe denote by Xγ and Eγ. Thus Xγ is obtained by blowing up a closed subscheme ofEγ. But since a P1-bundle is etale locally trivial, we see that Eγ – P1

U . Thus we mayassume that E Ñ C is the trivial P1-bundle P1

C .Thus, let us assume that E “ P1

C “ P1k ˆ C and X has been obtained from P1

C byblowing up the point p :“ pc0, p0 : 1qq where c0 is a closed point in C. Suppose the imageof γ : U Ñ C does not contain c0. Then Xγ is isomorphic to P1

U . On the other hand, if γ


maps the whole of U into the point c0, then Xγ “ U ˆ T where T consists of two copiesof P1

k intersecting transversally in a single point. In both these cases, any two sectionsof Xγ Ñ U are A1-chain homotopic and thus the required result follows immediately.Thus we may now focus on the case in which γ maps the closed point of U to c0 butdoes not map the whole of U into c0.

Let ω be a uniformizing parameter in the ring OC,c0 . Then Xγ is obtained from P1U “

ProjpRrX0, X1sq by blowing up the closed subscheme corresponding to the homogeneousideal xγ˚pωq, X0y. Lemmas 5.2.16, 5.2.18 and Proposition 5.2.19 give a classification ofthe A1-homotopy classes of sections of Xγ Ñ U . The classification of 1-ghost homotopyclasses as well as the verification that these agree with the n-ghost homotopy classes forany n ě 1 follows from Proposition 5.2.21. This shows that S2pXqpUq “ S3pXqpUq,for every smooth Henselian local ring U over k, as desired. It also follows that SpXq ‰S2pXq, as is shown by the following concrete example.

Let R “ krx, yshxx,yy. Let r0 “ xpy2 ` xq, r1 “ x. Thus r1 “ y2 ` x. We have

radpr1q ` radpr1q “ xx, y2 ` xy “ xx, y2y while radpr1, r1q “ xx, yy. We take r2 “

xp1 ` yq “ x ` xy. Thus r2{r1 ´ 1 “ y R radpr1q ` radpr1q. By Proposition 5.2.19,αr0r1 and αr0r2 are not A1-chain homotopic. But by Proposition 5.2.21, they are n-ghosthomotopic, for any n ą 1. Therefore, SpXqpUq ‰ S2pXqpUq, for U “ SpecpRq. Thiscompletes the proof of the theorem.

Remark 5.2.22. Note that the characterization of A1-homotopies and n-ghost homo-topies given above shows that SpXqpUq » SnpXqpUq, where U is the Spec of a Henseliandiscrete valuation ring, for all n. Therefore, we have SpXqpUq » πA1

0 pXqpUq for one-dimensional smooth Henselian local schemes.

Remark 5.2.23. It is interesting to examine the question of whether, for any smoothprojective variety X, the sequence of sheaves pSnpXqqně1 stabilizes at some finite valueof n. If it does, one may ask whether it stabilizes to πA1

0 pXq, since that is what ispredicted by Morel’s conjecture (see Corollary 2.2.5).

Appendix A

Model categories

In this appendix, we gather definitions and some basic facts about model categoriesthat are relevant in the context of this thesis. Model categories were first introduced byQuillen [38] in order to axiomatize homotopy theory. A basic reference for the resultsstated here (which contains detailed proofs) is the book by Hirschhorn [21].

Definition A.1. Let C be a category. Let f : AÑ B and g : C Ñ D be two morphismsin C. We say that f is a retract of g if there is a commutative diagram as follows:

A //

1A

##

f��

C //

g��

A

f��

B //

1B

;;D // B

Definition A.2. A category C is said to be complete if it is closed under small limits.It is said to be cocomplete if it is closed under small colimits.

Definition A.3. A model category is a category C with three distinguished classes ofmorphisms, called the weak equivalences, the cofibrations and the fibrations. Fibrationsthat are weak equivalences are called trivial fibrations and cofibrations that are weakequivalences are called trivial cofibrations. This data is required to satisfy the followingaxioms:

(M1) C is complete and cocomplete.

(M2) Let f, g P MorpCq be such that g ˝ f is defined. If two out of f , g and g ˝ f areweak equivalences, then so is the third.

(M3) Let f, g PMorpCq be such that f is a retract of g. If g is a weak equivalence, or acofibration, or a fibration, then so is f .

62 Appendix A. Model categories

(M4) Given a commutative diagram in C (shown by solid arrows)

A //

i��

X

p��

B //

>>

Y

the dotted arrow that makes the diagram commutative exists if either

(1) i is a cofibration and p is a trivial fibration.

(2) i is a trivial cofibration and p is a fibration.

In such a case, we say that i has the left lifting property with respect to p and thatp has the right lifting property with respect to i.

(M5) Every f PMorpCq has two functorial factorizations:

(1) f “ q ˝ i, where i is a cofibration and q is a trivial fibration.

(2) f “ p ˝ j, where j is a trivial cofibration and p is a fibration.

Definition A.4. Let C be a model category. Since C is complete and cocomplete, ithas an initial object H and a final object ˚. An object X of C is said to be fibrant ifthe canonical map X Ñ ˚ is a fibration. X is said to be cofibrant if the canonical mapHÑ X is a cofibration.

Definition A.5. Let C be a model category and let X be an object of C. A fibrantreplacement of X is a pair p pX, jq, where pX is a fibrant object and j : X Ñ pX is a weak

equivalence. A cofibrant replacement of X is a pair p rX, iq, where rX is a cofibrant object

and j : rX Ñ X is a weak equivalence. Functorial fibrant and cofibrant replacementsalways exist (see [21, Chapter 8] for details).

Definition A.6. A model category C is said to be left proper if weak equivalences inC are preserved by pushouts along cofibrations. It is said to be right proper if weakequivalences in C are preserved by pullbacks along fibrations. It is said to be proper ifit is both left and right proper.

Remark A.7. The axioms satisfied by a model category imply that any two of the threeclasses of maps weak equivalences, cofibrations and fibrations determine the third. Also,model categories satisfy the following duality principle: given any statement followingfrom these axioms that holds for all model categories, we can obtain a dual statement byreversing all arrows, interchanging fibrations and cofibrations, and interchanging limitsand colimits. See [21, Chapter 7] for details.

Examples A.8. The categories of simplicial sets, topological spaces, chain complexesof modules over rings, simplicial sheaves on a site are all examples of model categories.See [21], [22], [38] for details and more examples.

63

Localization of model categories

We begin by introducing the notion of the homotopy category of a model category. Wethen introduce the concept of Quillen functors between two model categories. Theseare the functors between model categories for which it is possible to define the inducedfunctors on the associated homotopy categories, as we shall see later.

Definition A.9. Let C be a model category and let W be a class of morphisms in C. Alocalization of C with respect to W is a category LW pCq with a functor γ : C Ñ LW pCqsuch that

(1) if w P W , then γpwq is an isomorphism, and

(2) if D is a category and ϕ : C Ñ D is a functor such that ϕpwq is an isomorphismfor every w P W , then there exists a unique functor δ : LW pCq Ñ D such thatδ ˝ γ “ ϕ.

If a localization of C with respect to W exists, then it is unique. If C is a smallcategory, then the localization of C with respect to any class of morphisms exists.

Definition A.10. Let C be a model category. Then the homotopy category associatedwith C is the localization of C with respect to the class of weak equivalences. It is usuallydenoted by γ : C Ñ HopCq.

The existence of the homotopy category associated to a model category is proved in[21, Theorem 8.3.5], for example.

Definition A.11. Let C, D be model categories. Let F : C Ñ D and G : D Ñ C be apair of adjoint functors. We say that F is a left Quillen functor, that G is a right Quillenfunctor and that pF,Gq is a Quillen pair if

• the left adjoint F preserves both cofibrations and trivial cofibrations, and

• the right adjoint G preserves both fibrations and trivial fibrations.

It follows from definitions that a left Quillen functor preserves weak equivalences be-tween cofibrant objects and a right Quillen functor preserves weak equivalences betweenfibrant objects.

Definition A.12. Let F : C Õ D : G be a Quillen pair. The total left derived functorof F , LF : HopCq Ñ HopDq, is defined by LF pXq “ F p rXq, where rX is a functorialcofibrant replacement of X. The total right derived functor of G, RG : HopDq Ñ HopCq,is defined by RGpXq “ F p pXq, where pX is a functorial fibrant replacement of X.

64 Appendix A. Model categories

For a proof of that total derived functors are well-defined, see [21, Chapter 8].We now define the notion of left Bousfield localization. This is generally defined in the

context of a simplicial model category or an even more general setup (see [21, Chapter4]). For the sake of brevity, we omit the definition of a simplicial model category andrefer the reader to [21, Chapter 9]. In these categories, given a pair of objects X andY , one can associate a mapping simplicial set MappX, Y q. Bousfield localization can beperformed on these categories after putting some additional conditions (see [21, Theorem4.1.1]) (such as, for example, the model category is question is left proper and cellularor combinatorial).

Definition A.13. Let C denote a simplicial model category and let S be a class ofmorphisms on C. An object X of C is said to be S-local if

• X is fibrant, and

• for every element f : AÑ B of S, the induced map MappB,Xq Ñ MappA,Xq isa weak equivalence of simplicial sets.

A morphism g : AÑ B in C is said to be an S-local equivalence if for any S-local objectX, the induced map MappB,Xq ÑMappA,Xq is a weak equivalence.

The left Bousfield localization of C with respect to S is a model category structureLSC on the underlying category of the original model category C such that

• the class of weak equivalences of LSC is the same as the class of S-local equivalencesof C;

• the class of cofibrations of LSC is the same as the class of cofibrations of C; and

• the class of fibrations of LSC is the class of morphisms having the right liftingproperty with respect to the morphisms that are both cofibrations and S-localequivalences.

If the left Bousfield localization exists, then it has the following properties (see [21,Chapter 3] for proofs and more details):

(1) If X and Y are S-local objects of C, then f : X Ñ Y is a fibration in LSC if andonly if it is a fibration in C.

(2) If C is left proper, then an object is S-local if and only if it is fibrant in LSC.

(3) Let X be a cofibrant object of C and let Y be an S-local object. Then two mapsin HomCpX, Y q are homotopic in C if and only if they are homotopic in LSC.

Appendix B

Nisnevich topology

Fix a a noetherian base scheme S of finite dimension. We will denote by Sch{S thecategory of (separated, finite type) schemes over S and by Sm{S its subcategory ofsmooth schemes over S.

Definition B.1. Let X P Sch{S. A family of morphisms tpi : Ui Ñ XuiPI is said to bea Nisnevich covering if the following hold:

• tpi : Ui Ñ XuiPI is an etale cover of X.

• For any point x P X, there exists i P I and u P Ui such that pipuq “ x and themap induced by pi on the residue fields κpxq Ñ κpuq is an isomorphism.

The notion of Nisnevich coverings satisfies the axioms for a Grothendieck pretopol-ogy ([19]) and the corresponding topology is called Nisnevich topology. We denote thecorresponding site by pSm{SqNis. Nisnevich topology was first introduced in [37]. TheNisnevich topology is strictly stronger than the Zariski topology and strictly weaker thanthe etale topology.

Example B.2. Let k be a field with char k ‰ 2 and let a P kˆ. Let U0 :“ A1kztau ãÑ A1

k

and U1 :“ A1kzt0u Ñ A1

k be the map x ÞÑ x2. Then tU0, U1u is always an etale coveringof A1. However, it is a Nisnevich covering of A1

k if and only if a P pkˆq2.

Let X be a scheme over S. Then the presheaf on Sm{S represented by X is alwaysa sheaf in Nisnevich topology (see [19, VII.2] for a proof). Hence, the canonical functor

Sm{S Ñ ShvpSm{SqNis

is a fully faithful embedding. We often identify the category Sm{S with its image viathis functor.

Definition B.3. A local ring pA,mq is said to be Henselian local if any finite A-algebraB is a product of local rings.

66 Appendix B. Nisnevich topology

For various interesting characterizations of Henselian local rings, see [31, I.4.2]. LetpA,mq be a Henselian local ring. If tUi Ñ SpecpAqui is a Nisnevich covering of SpecpAq,then there exists i such that Ui Ñ SpecpAq is finite etale, and consequently, Ui ÑSpecpAq splits. Therefore, every Nisnevich covering of SpecpAq has the trivial coveringas a refinement. Thus, Henselian local schemes determine points for the Nisnevichtopology.

Lemma B.4. If tpi : Ui Ñ XuiPI is a Nisnevich covering, then there exists a nonemptyopen subscheme V ãÑ X and i P I such that pi|V has a section.

Proof. Let η be a generic point of X. By definition, there exists i P I and a generic pointu P Ui such that pipuq “ x and κpxq Ñ κpuq. Hence, Ui

piÝÑ X is a birational morphism

and the lemma follows.

Definition B.5. For any scheme X, an elementary Nisnevich cover of X consists oftwo morphisms p1 : U1 Ñ X and p2 : U2 Ñ X such that:

(1) p1 is an open immersion.

(2) p2 is an etale morphism and its restriction to p´12 pX z p1pU1qq is an isomorphism

onto X z p1pU1q.

The resulting cartesian diagram

U1 ˆX U2//

��

U2

p2��

U1p1 // X

is called an elementary distinguished square.

The following (see [35, §3, Lemma 1.6]) is a formal consequence of definitions.

Lemma B.6. Every elementary distinguished square is a cocartesian square in thecategory of Nisnevich sheaves of sets ShvpSm{SqNis. In particular, for every elementaryNisnevich cover tp1 : U1 Ñ X, p2 : U2 Ñ Xu of X,

U2{U1 ˆX U2 Ñ X{U1

is an isomorphism of Nisnevich sheaves.

We end this section by giving a proof of a criterion for presheaves on Sm{S tobe Nisnevich sheaves (taken from [35, §3, Proposition 1.4] or [30, Chapter 12]), whichsubstantially simplifies matters.

67

Proposition B.7. A presheaf of sets F on Sm{S is a sheaf in the Nisnevich topologyif and only if for every object X of Sm{S and for every elementary Nisnevich covertp1 : U1 Ñ X, p2 : U2 Ñ Xu of X, the square

FpXq //

��

FpU1q

��FpU2q // FpU1 ˆX U2q

is cartesian.

Proof. We first prove the only if part. Let F be a Nisnevich sheaf and let tp1 : U1 Ñ

X, p2 : U2 Ñ Xu be an elementary Nisnevich cover of a scheme X. Therefore, the square

FpXq //

��

FpU1

š

U2q

��FpU1

š

U2q // FpU1

š

U2q ˆX FpU1

š

U2q

is cartesian. We need to show that the diagram

FpXq //

��

FpU1q

��FpU2q // FpU1 ˆX U2q

is cartesian, that is, the map FpXq Ñ FpU1q ˆFpU1ˆXU2q FpU2q is an isomorphism. Itonly remains to prove the surjectivity, since the map is already injective, tU1, U2u beinga cover of X. We have

pU1

ž

U2q ˆX pU1

ž

U2q “ pU1 ˆX U1qž

pU1 ˆX U2qž

pU2 ˆX U1qž

pU2 ˆX U2q.

Write V :“ U1 ˆX U2 and observe that tU24ÝÑ U2 ˆX U2, V Û1 V Ñ U2 ˆX U2u is a

Nisnevich cover of U2 ˆX U2. Hence, we have an injection

FpU2 ˆ U2q Ñ FpV Û1 V q ˆ FpU2q.

Since F is a Nisnevich sheaf, we have an equalizer diagram

FpXq Ñ FpU1q ˆ FpU2q Ñ FpU1q ˆ FpV q ˆ FpU2 ˆX U2q.

Now, a diagram chase finishes the proof of the only if part.We now prove the if part. Let F be a presheaf that takes every elementary distin-

guished square to a cartesian square. Fix a Nisnevich covering tUi Ñ XuiPI . We needto prove that

FpXq Ñź

FpUiq Ñź

FpUi ˆX Ujq

68 Appendix B. Nisnevich topology

is an equalizer diagram. We will call an open subscheme V ãÑ X good for the coveringtUi Ñ XuiPI if

FpV q Ñź

FpUi ˆX V q Ñź

FpUi ˆX Uj ˆX V q

is an equalizer diagram. In order to complete the proof, we need to show that X is goodfor the covering tUi Ñ XuiPI .

By noetherian induction, we may assume that there is a largest good open subschemeV ãÑ X for the covering tUi Ñ XuiPI . Suppose, if possible, that V ‰ X. Set Z :“ XzV .By Lemma B.4, there is a nonempty open subscheme W ãÑ Z and i P I such thatUi|W Ñ W splits. Let X 1 :“ XzpZzW q. Then tV, U 1i :“ Ui|X 1u is an elementaryNisnevich cover of X 1. Pulling back along each U 1j :“ Uj|X 1 also gives an elementaryNisnevich covering of U 1j. By hypothesis, we have cartesian squares

FpX 1q //

��

FpU 1iq

��FpV q // FpU 1i ˆX 1 V q

FpU 1jq //

��

FpU 1i ˆX U 1jq

��FpU 1j|V q // FpU 1i ˆX U 1j|V q

and it follows that X 1 is also good for the covering tUi Ñ XuiPI , contradicting themaximality of V . Therefore, X has to be good for each of its Nisnevich coverings.Consequently, F is a Nisnevich sheaf and the proof is complete.

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