Seminar series on etale cohomology

P. Brito, J. TaylorTypeset by Aaron Chan (aaron.chan@abdn.ac.uk)

Last update: December 15, 2011

This is the notes taken for the series of seminars, given by members of the Institute of Mathematics atUniversity of Aberdeen at Fall 2011. Most of the seminars are given by P. Brito and J. Taylor. Thisnotes is intended to give a picture and introduce the tools around the etale cohomology and does notprovide the detailed mathematical rigour.

1 Motivations (from algebraic group)

Let X be a smooth projective variety over field K = Fp for p > 0We usually denote q = pa, and so Fq ⊂ Fp

When we say X is defined by Fq, this implies we have Frobenius map F : X → X induced by

φ : K → Kx 7→ xq

(and so Kφ = Fq)

Note that |XF | = |X(Fq)|

It turns out that Fm : X → X corresponds to X defined over Fqm ; and X =⋃m>0X

Fm.

Define the Zeta function

Z(X, t) := exp(∞∑n=1

|XFn | tn

n) ∈ Q [[t]]

Weil Conjecture(s):

(1) Z(X, t) is a rational function (i.e. Z(X, t) = P (t)/Q(t) for some polynomials P,Q)

(2) P (t) =b∏

j=1

(1− βjt)

Q(t) =a∏i=1

(1− αit)

where αi, βj are algebraic integers with modulus qi/2 and a+ b = 2 dimX

Weil’s idea:Use some “cohomology”, over field of characteristic 0.But this is not possible for Zariski topologySolution: Etale cohomology

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Application:

`-adic cohomology H ic(X,Q `) := H i

c(X,Z `)⊗Q ` `(6= p) > 0 primeNote Frobenius map F indeuces F ∗ : H i

c → H ic

Main Theorem:

|XF | =∑i

(−1)i Tr(F ∗, H ic(X,Q `))

|XFm | =a∑i=1

αmi −b∑

j=1

βmj

2 Sheaves and Categories

2.1 Limits and colimits

Let I be a category

Definition 2.1Let β : Iop → C be a functor (e.g. when C = k-vector space, this functor can be thought of as rightI-module where I is considered as a k-algebra)A cone C over β is denoted by the diagram C → β(i) which means:∀s : i→ j in I, the following diagram commutes

β(j)

φj !!

β(s) // β(i)

φiC

i.e. C is the collection of natural transformations from (i 7→ C)∀i to β.The limit, limβ is the terminal object in the category of cone. i.e. we have following commutativediagram:

β(j)

##φj

β(s) // β(i)

||φi

limβ

C

∃!

OO

The dual notion of cone is cocone C of α : I → C, represented by diagram α(i)→ C such that thereis commutative diagram as above.The dual notion of limit is colimit, which is the initial object in the category of cocone.

When C = Sets:

limβ =xi∈

∏i∈I

β(i)∣∣β(s)(xj) = xi ∀s : i→ j

(2.1)

colimα =∐i∈I

α(i)/ ∼ (2.2)

where ∼ is generated by α(i) 3 x ∼ y ∈ α(y) if α(s)(x) = y

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Example 2.2

(1) I = •, •, . . . (category with objects and no arrows)β : Iop → Sets α : I → Sets

limβ =∏i∈I

β(i) =: product

colimα =∐i∈I

α(i) =: coproduct

(2) I = i← j → k

lim(Af−→ C

g←− B) = (a, b)A×B|f(a) = g(b)=: pullback =: A×C B

colim(Af←− C g−→ B) = A tB/f ∼ g

=: pushout =: A tC B

(3) I = •⇒ •

lim( Af //g// B ) = (x, y) ∈ A×B | f(x) = y = g(x)

= equaliser “ = ” kernel

(in abelian categories, this is ker(f − g))

colim( Af //g// B ) = B/f ∼ g

= coequaliser “ = ” cokernel

We denote K → A⇒ B as the equaliser above, where K is the limit object;and A⇒ B → C as the coequaliser above, with C being the colimit object.

2.2 Grothendieck topology and sheaves

Definition 2.3A functor F : Cop → Sets is representable if F = HomC(−, X) for some X ∈ C; and X is called therepresentative of F

A presheaf is a functor F : Cop → Sets.The category of presheaves is denoted as PSh(C)

There is an obvious functorC → PSh(C)X 7→ HomC(−, X)

Yoneda LemmaNat(HomC(−, X), F ) ' F (X)

Consequently, the functor above is an embedding r : C → PSh(C)We usually denote the image of X ∈ C as rX or even just X itself if there is no ambiguity.

Definition 2.4A category which has all small limits (resp. colimits) is called complete (resp. cocomplete).

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When C is not cocomplete, we want to approximate C by a cocomplete category C such that

C //

∀γ

C

∃ colim. preserving∀D

It turns out that C = PSh(C))The slogan is PSh(C) is the “free cocompletion” of C

Note: The embedding r : C → PSh(C) does not necessarily commute with colimitsi.e. r(U

∐V ) 6= (rU)

∐(rV ) in general. In another words, by embedding C into PSh(C), we lost some

geometric information (See later).

Definition 2.5Let C be a category. A Grothendieck topology T on C is an assignment of a set of covering familiesCovT (X) = U = Ui → Xi∈I (where U are called cover) for each object X ∈ C satisfying:

(1) If Ui → Xi∈I is a cover and Vi,j → Uij∈J is a refinement ∀i ∈ Ithen Vi,j → Xi,j is a cover

(2) If f : Y → X and Ui → X is a coverthen Ui ×X Y exists and Ui ×X Y → Y is a cover of Y

(3) If f : Y → X is an isomorphismthen Y → X is a cover of X

Example 2.6Let T be a topological space, and C = open sets of TCovT (X) = usual open cover of X gives Grothendieck topology T

The following example demonstrate how colimits are lost through embedding to PSh(C)

Example 2.7C = Top, Ui → X be cover of X ∈ C∐

i,j

Ui ∩ Uj ⇒∐i

Ui → X (2.3)

is an coequaliser in C but NOT coequaliser in PSh(C)

In order to turn (2.3) into a coequaliser, we need the Grothendieck topology T on C:

(C 3) X 7→ CovT (X) =

covers of XU := Ui → Xi∈I

Given cover U = Ui → Xi∈I , can form the distinguished cocone∐

i,j

Ui ×X Uj ⇒∐i

Ui → X

which will be (see later) our coequaliser in the enlarged category

Proposition 2.8∃ category Sh(C) together with a functor C → Sh(C) which sends distinguished cocones into colimitsand universal with respect to this property. i.e. we have commutative diagram:

d.cocones

&&

C

%%

// Sh(C)

∃ !( up to isom.)

colimits D

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Definition 2.9A sheaf is a presheaf satisfying:∀ cover U := Ui → Xi∈I , we have an equaliser diagram:

F (X)→∏i

F (Ui)⇒∏i,j

F (Ui ×X Uj)

More explicitly,

(1) if s, t ∈ F (X) with the same image in∏i F (Ui), then s = t

(2) (Gluing property)∀ collection si where si ∈ F (Ui)∀i, such that si|Ui×XUj = sj |Ui×XUj

then they extend to s ∈ F (X)

Terminology

Let D be category. We say that Z ∈ D sees a cocone D(i) → X as a colimit, if HomD(X,Z)∼−→

limi HomD(D(i), Z)

Note:If every object Z sees a cocone D(i)→ X as a colimit, then X is the limit itself.

Claim: Sheaf = presheaf which sees distinguished cocones as colimitsNote: This is actually obvious from the definition, but we want to demonstrate more.ProofWe write Ui,j := Ui ×X Uj . The definition of sheaf gives:

F (X) ∼= lim

∏F (Ui)⇒∏i,j

F (Ui,j)

∼= lim

∏HomPSh(C)(rUi, F )⇒∏i,j

HomPSh(C)(rUi,j , F )

(by Yoneda)

By applying Yoneda on the left hand side:

HomPSh(C)(X,F ) ∼= lim

HomPSh(∐

rUi, F )⇒ HomPSh(∐i,j

rUi,j , F )

i.e. F sees distinguished cocones as colimits

∐Ui,j ⇒

∐i Ui → X. we now get:

HomPSh(X,F )∼−→ HomPSh(SU , F )

where SU := colim(∐i,j rUi,j ⇒

∐i rUi) ∈ PSh(C). Also note that

SU (V ) ∼= f : V → X|f factors through Ui → X some i

because this is∐i,j Hom(V,Ui,j)⇒

∐i Hom(V,Ui)

SU is called the sieve generated by U . It keeps all the essential information from topology.

Question: When can we say (Grothendieck) topologies T and T ′ have the same sheaves?

2.3 Refinements and Sheafification

Definition 2.10Let U := Ui → X and V := Vj → X

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we say V refines U if we have commutative diagram:

Vj∀f //

∃

X

Ui

??

We say T ′ refines T if every cover in T has refinement by a cover in T ′and say T ′ and T are equivalent if they refines each other.

Proposition 2.11Suppose T ′ refines T . If F is a sheaf for T ′, then F is a sheaf for T

(Recommended exercise: prove this)

As there is the obvious (forgetful) functor Sh(C) → PSh(C), we take the left adjoint of this and thisis in fact the sheafification functor. The consequence of the existence of this functor is that we cancompute colimits of sheaves in the category of presheaves and then sheafify.

2.4 Etale motivation

Let C = Top (or category of manifolds, etc.)

Take a “local homeomorphism” (this is like taking a covering map) Ep−→ X

E ×X E ⇒ E → X coequaliser in C r(E ×X E)⇒ rE → rX coequaliser in PSh(C) (using the fact that p is a local homeo.)Any sheaf F sees the last coequaliser as a colimit.

However, if C is the category of schemes, where the analogue of the “covering map” is the etale map

then the etale map Ep−→ X giving rE ×X E ⇒ rE → X is not a coequaliser in PSh(C)

In order to force this to be colimits, we need Grothendieck topology, where we include E p−→ X as acover.

The etale topology is the smallest topology which contains Zariski covers Ui → X and E p−→X|p etale map

2.5 Derived functor

Reader could refer to Fall-2011 graduate course notes on “Derived Equivalence of Blocks of FiniteGroups”. Commonly used source for this section is Gelfand,Manin’s Methods of Homological Algebra.

The reader needs to know the following keywords:

(1) Additive and abelian categoriese.g. PShAb(C) := Cop → Ab and ShAb(C)

(2) Complex

(3) Category of complexes over C, denoted C(C)

(4) Homotopy of complex, and chain homotopies, quasi-isomorphism (quism)

(5) Homotopy category K(C), constructed by taking homotopy classes of maps in C(C)

(6) Derived functors:

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Let F : C → A be additive functor between abelian categories. e.g. The global section functor:

ΓX : PShAb(C) → Ab

F 7→ F(X)

Sometimes we will write Γ(−, X) for the global section functor. We often ask following questions forthese kind of functors:

(1) Is the functor exact?

(2) If not, can you approximate it by exact functor?

(3) F induces functor on homotopy categories, K(F ) : K(C)→ K(A)How can you approximate K(F ) by a functor which sends quasi-isomorphism (quism) to quasi-isomorphism?

The answer is to use derived functors and consider the corresponding derived categories D(C), D(A)

K(C)K(F ) //

K(A)

D(C) RF // D(A)

RF above is the (right) derived functor. This is somehow the best (initial) approximation to K(F ).The formula we often use is:

RF (X) = F (I)

where I is injective resolution of X (assuming it always exists for all X ∈ K(C))i.e., ∀X ∃X → I which is a quism and I ∈ I where I is a subcategory of C s.t. F |I exact.

The reason for such formula is as follows:As F |I exact⇒ K(F )|K(I) sends quism to quismAs ∀X ∈ C, ∃I ∈ I with X → I quism⇒ D(C) ∼= K(I) (this is non-trivial)Hence we have following diagram:

X_

K(C)

F // K(A)

I D(C) ∼= K(I)

F // D(A)

with the F on the second row sending quism to quism. As well as giving the above formula. Wedenote RiF (X) := H i(F (X))

The sheaf cohomology is the (right) derived functor of ΓX(= Γ(−, X)) above.

3 Varieties and Schemes

Fix a field K = K. An = Kn affine n-space.Projective space Pn is defined to be the sapce An+1 \ 0 s.t. x ∼ y iff ∃λ ∈ K× s.t. xi = λyi ∀i (i.e.x = λy)All rings are commutative with 1.

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3.1 Affine and projective algebraic sets

Theorem 3.1 (Hilber’s Basis Theorem)If R is a Noetherian ring, then R[X] is Noetherian.(Note: R is Noetherian iff every ideal of R is finitely generated.)

Definition 3.2Let S ⊆ K[X1, . . . , Xn] be a set of polynomials, we define

V (S) := x ∈ An|f(x) = 0 ∀f ∈ S

to be the affine algebraic set asociated to S. Conversely, for any subset V ⊆ An, we define

I(V ) := f ∈ K[X1, . . . , Xn]|f(x) = 0∀x ∈ V

to be the vanishing ideal associated to V . We define

K[V ] := K[X1, . . . , Xn]/I(V )

to be the affine coordinate ring of V .

Example

(1) V (1) = ∅V (0) = An

(2) x = (x1, . . . , xn) ∈ An then V ((X1 − x1, . . . , Xn − xn)) = x

(3) Twisted cubic. Take f = X2 −X21 , g = X3 −X3

1

⇒ f, g ∈ [X1, X2, X3]⇒ V (f, g)(a, a2, a3) ∈ A3|a ∈ A1

RemarkIf S ⊆ K[X1, . . . , Xn] and J = 〈S〉, then V (S) = V (J)

Hilbert Basis= V (f1, . . . , fr).

Assume R is a (positively) graded ring R =⊕

d≥0Rd of abelian groups such that RdRe ⊆ Rd+e ∀d, e ≥0. Then an element ofRd is a homogeneous element of degree d. We say an ideal JCR is a homogeneousif J =

⊕d≥0(J ∩Rd)

Check that the product, sum, intersection and radical of homogeneous ideals is again homogeneous.

Let R := K[X0, . . . , Xn] then R is graded where Rd is all linear combinations of monomials inX0, . . . Xn with total weight d.

Definition 3.3S ⊆ K[X0, . . . , Xn]H then

V (S) = p ∈ Pn|f(p) = 0 ∀f ∈ S

to be the projective algebraic set associated to S. Conversely, V ⊆ Pn, define

I(V ) = 〈f ∈ K[X0, . . . , Xn]H |f(x) = 0 ∀x ∈ V 〉

to be the homogeneous vanishing ideal associated to V .Then K[V ] := K[X0, . . . , Xn]/I(V ) is homogeneous coordinate ring

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Remark: “f(p) = 0 ∀p ∈ Pn” is only well-defined if f is homogeneous.f ∈ K[X0, . . . , Xn]H

f(p) =

0 if f(p) = 0

1 if f(p) 6= 0

Proposition 3.4Let S, TSi ⊆ R (or RH), i ∈ I a (possibly infinite) indexing set. Then

(1) V (ST ) = V (S) ∪ V (T )

(2) V (⋃i∈I Si) =

⋂i∈I V (Si)

Corollary 3.5The affine/projective algebraic sets in An (or Pn resp.) form the closed sets of a topology called theZariski topology

Remark:

(1) S1 ⊆ S2 ⊆ R or RH , then V (S2) ⊆ V (S1)

(2) S ⊆ R or RH then S ⊆ I(V (S))

(3) V1 ⊆ V2 ⊆ An or Pn, then I(V2) ⊆ I(V1)

(4) V ⊆ An or Pn, then V ⊆ V (I(V ))

So I and V are order reversing operators between subsets of An (Pn resp.) and ideals (homog. idealsresp.) of R (RH resp.).

Lemma 3.6If V ⊆ An or Pn, then V (I(V )) = V the topological closure of V in the Zariski topology.

So this says I takes injective affine algebraic sets to ideals of R.What is I(V (S)) for S ⊆ R? Or when is I injective?

Theorem 3.7 (Hilbert’s Nullstellensatz)If S ⊆ K[X1, . . . , Xn] then I(V (S)) = rad(S) :=

√S := f ∈ R|fm ∈ S for some m ≥ 1

This says I and V are inverse bijection between

affine algebraic sets of An ↔ radical ideals of K[X1, . . . , Xn]

(Radical ideals means ideal S such that S = rad(S))

In the projective case, we have the following. Let R+ =⊕

d>0Rd, we have

projective algebraic sets of Pn ↔ homogeneous radical ideals of K[X0, X1, . . . , Xn] except R+

Reason: If S ⊆ RH , then V (S) = ∅⇒ S = R or R+ ⇒ I(∅) = R

We say a topological space is Noetherian if it satisfies DCC on closed sets.

Proposition 3.8V a non-empty algebraic set

(1) V is Noetherian topological space: ∃ unique decomposition V = V1∪· · ·∪Vr where Vi are maximal(wrt includsion) closed irreducible subsets of V .

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(2) V is irreducible ⇔ I(V ) is prime.

Example

(1) a ∈ An. Then I(a) = f |f(a) = 0If fg ∈ I(a) then f(a)g(a) = 0, so f(a) or g(a) = 0, so f ∈ I(a) or g ∈ I(a)

(2) p(X1, X2) = X1X2

V (〈p〉) = (a1, a2) ∈ A2|p(a1, a2) = 0= V (X1) ∪ V (X2)

I(V (p)) = 〈X1X2〉

Let V be an algebraic set and f ∈ R or RH a polynomial, then the principal open set determined by fis

Vf = V \ V (f) = x ∈ V |f(x) 6= 0

THe principal open sets form a basis for the topology. If U ⊆ V is open, then the closed set V \U closed,so there is an algebraic set W s.t. V \U = W ∩V . By Hilbert’s basis theorem W ∩V = V (f1, . . . , fr),so U =

⋃ri=1 Vfi .

ExampleLet = C and let f ∈ K[X], then V (f) is a finite set consisting of the roots of f . Then everyclosed set is finite because they are a finite union of closed sets V (f). Conversely, the open sets arelarge (they are dense). This is main problem when we use Zariski topology. Etale topology (or evenGrothendieck topology) is the way to (try to) resolve this issue (by including more open sets than theZariski topology).

3.2 Morphisms

Definition 3.9If V ⊆ An and W ⊆ Am affine algebraic sets, then a map φ : V →W is a morphism if ∃f1, . . . , fm ∈K[X1, . . . , Xn] s.t. φ(x) = (f1(x), . . . , fm(x)) ∀x ∈ V . We say φ is an isomorphism if φ is bijectiveand φ−1 is a morphism.

Remark: V (S) ⊆W , then φ−1(V (S)) = V (ψ φ|ψ ∈ S)So φ is continuous w.r.t. the Zariski topology

φ : V →W is a morphism. Take φ∗ : K[W ]→ K[V ] s.t. φ∗(f) = f φ

Exampleφ : V → A1, then ∃f ∈ K[X1, . . . , Xn] s.t. φ(x) = f(x) ∀x ∈ Vbut if g ∈ K[X1, . . . , Xn] s.t. φ(x) = g(x) ∀x ∈ V , then (f − g)(x) = 0 ∀x ∈ V , so f − g ∈ I(V )In particular, f = g. Given f + I(V ) ∈ K[V ], we get a well-defined morphism φ(x) = f(x) ∀x ∈ V

Proposition 3.10Let φ : V →W be a morphism then φ 7→ φ∗ gives a bijection

ψ : V →W morphisms ↔ -algebra hom.ψ∗ : K[W ]→ K[V ]

Furthermore, φ is isom ⇔ φ∗ is isom.

Remark: V → K[V ] can be extended to a contravariant functor between

irred. aff. alg. sets + morphisms ↔ f.g. int. dom. over K & K-alg. hom.

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which is an equivalence of categories.

Example

(1) Let = Fp for p > 0 a prime. Then take φ : A1 → A1 to be φ(x) = xp. Then φ∗ : K[X]→ K[X]is the map X 7→ Xp.So φ∗ is the embedding K[Xp] → K[X]

(2) = C . Then φ : A1 → A2. φ(x) = (x2, x3). Then image is V (p) where p = 〈X31 − X2

1 〉.φ∗ : K[X2, X3] → K[X]

Question: Do we have correspondence ? ↔ commutative rings over KAnswer: ? = Affine schemes

Affine variety is a pair (V,K[V ]) where V is an irreducible affine algebraic set and K[V ] is its affinecoordinate ring.

Note: we always assume all rings are commutative with 1 and K algebraically closed.

irred. aff. alg. sets + morphisms

oo // f.g. int. dom. over K & K-alg. hom.

aff. alg. sets + morphisms

oo // f.g. nilpotent free rings over K

? oo // commutative rings with 1

3.3 Spectrum of Rings

Let R be a ring then we define the spectrum of R to be

SpecR = pCR|p is a proper prime idea of R

We say a prime ideal p is a point of the spectrum which we denote [p] ∈ SpecR

If p ∈ SpecR then R/p is an integral domain so we can consider the quotient field of R/p which wedenote κ(p). If f ∈ R then we can consider the image of f under the sequence of maps

R→ R/p→ κ(p)

and we define this to be f(p) or f evaluated at p. If S ⊆ R, we define

V (S) = [p] ∈ SpecR|f(p) = 0 ∈ κ(p) ∀f ∈ S= [p] ∈ SpecR|S ⊆ p

These V (S) forms closed set of a topology on SpecR, which we call Zariski topology (c.f. previoussection)

ExampleR = K[V ] of some irreducible affine algebraic set V . A consequence of the Nullstellensats is thatS ⊆ V is a singelton (point) ⇐⇒ I(S) maximalIf p is maximal in K[V ] then V (p) = [p], and K[V ]/p ∼= K, and under this association f(q) (forsome prime ideal q) is really what we expect.

ExampleR = K[X]. Every prime ideal in R is of the form 〈fa〉 where fa = X − a for a ∈ K = K, or is the zero

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ideal. So V (〈fa〉) = [〈fa〉], which corresponds to a point a ∈ K which is a point in the affine 1-spacein the sense we used previously. However, the zero ideal [0] is not closed because 0 is contained inevery ideal. It is also not open because its complement from A1 is infinite. We call [0] ∈ SpecR ageneric point of the spectrum. (This is a subset whose topological closure is the whole space). Wedenote A1 to mean SpecK[X].

Letf ∈ R, then we define the principal open set of SpecR determined by f as

Xf = SpecR \ V (f)= all proper prime ideal of R not containing f

Recall Rf is the ring localised at f . If T = fn|n ≥ 0 then

Rf = a/t|a ∈ R, t ∈

where (a1/t1) ∼ (a2/t2) ⇐⇒ ∃s ∈ T s.t. s(a1t2 − a2t1) = 0

Assume [p] ∈ Xf then these correspond one-to-one to the prime ideals of Rf by sending p(CR) 7→pRf (CRf ). So we can associate Xf to SpecRf .

3.4 Affine schemes and schemes

Definition 3.11An affine scheme X is a pair (|X|,OX) where |X| is SpecR of some ring and OX is a sheaf of ringson |X| which we will define in the following.

Recall to define a sheaf (OX) we must associate to every open set U ⊆ X a ring OX(U) and for everypair U ⊆ V ⊆ X a restruction map resV,U : OX(V )→ OX(U) which satisfies

(1) resU,U = idU

(2) resW,V resV,U = resW,U for all U ⊆ V ⊆W ⊆ X

(3) sheaf axiom: U ⊆ X and an open cover U = ∪a∈IUa then for all fa ∈ OX(Ua) s.t. fa and fbagrees on restriction to Ua ∩ Ub, then ∃!f ∈ OX(U), s.t. f |Ua = fa ∀a ∈ I

If Xf is a principal open set of X = SpecR, then we want to set OX(Xf ) = Rf .Let’s assume Xg is a principal open set of X s.t. Xg ⊆ Xf , then g is divisible by f . We take therestriction map to be the natural map resXf ,Xg : OX(Xf ) = Rf → (Rf )g = Rg = OX(Xg)

Proposition 3.12Using the description above, OX extends uniquely to a sheaf of rings on |X| called the structure sheafof X (or sheaf of regular functions).

Definition 3.13A scheme X is a pair (|X|,OX) where |X| is a topological space and OX is a sheaf of rings on |X|which is locally affine. By this, we mean |X| is covered by open sets Ui ⊆ |X| s.t. ∃ ring Ri andhomeomorphisms Ui ∼= |SpecRi| such that OX |Ui

∼= OSpecRi

If X is a scheme and U ⊆ X then a regular function on U is a section of OX over U , i.e. an elementof OX(U)

A global regular function is a global section of OX

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3.5 Subschemes

If X is a scheme and U ⊆ X an open set, then we define (|U |,OX |U ) to be the open subschemedetermined by U . If Xf is a principal open set of X contained U , then (|Xf |,OXf

) = (|Xf |,OX |Xf) =

SpecRf . As U =⋂a∈I Ua for some finite indexing set I. Then use the fact that these principal open

sets contained in U cover U , we get (|U |,OX |U ) as a scheme.

Assume near X = SpecR and I CR is an ideal, then we can canonically identify V (I) of SpecR with|SpecR/I|. A closed subscheme is defined to be SpecR/I for some ideal I C R. Let Y = SpecR/I,then we define the ideal sheaf JY/X of Y in X to be the sheaf of ideals of OX gives an distinguishedopen sets by J (Xf ) = IRf .

We have an inclusion j : |Y | → |X| then the pushforward of OY is the sheaf j∗OY on |X| given byj∗OY (U) = OY (j−1(U)) for any open set U ⊆ |X|We can identify OY with j∗OY , which can be identified with the quotient sheaf OX/J (this may onlybe a presheaf, we need to take the its sheafification if necessary)

Remark: Not all sheaves of ideals in OX come from ideals of R. If J comes from an ideal in R, wecall it a quasi-coherent sheaf of ideals. If X is a scheme, we say J ⊆ OX is a quasi-coherent sheaf ofideals if for every affine open subset U ⊆ X then restriction J |U is quasi-coherent on U .

Definition 3.14A closed subscheme Y of X is a closed subspace |Y | ⊆ |X| together with a sheaf of rings OY that isthe quotient sheaf of OX by a quasi-coherent sheaf of ideals.such that the intersection of J with anyaffine open set U ⊆ X is the closed subscheme associated to J (U).

ExampleK[X], ideal 〈X〉V (〈X〉) = [〈X〉] = V (〈Xi〉)So the closed set is not uniquely determined by one ideal, and the structure sheaf of K[X]/〈Xi〉 isdifferent for different i. Therefore, a closed subscheme does not uniquely determine the ideal, but itdoes uniquely determine the ideal sheaf.

Definition 3.15If X is a scheme we can specifty the structure sheaf by its so called stalk. Let x ∈ |X| then the stalkof OX at x is

OX,x = colimU3xOX(U) (3.1)

=⊗U3xOX(U)/I as Z -module (3.2)

where I is an ideal generated by the following relations: Assume U → V inclusion of open sets andlet r ∈ OX(U) and s ∈ OX(V ) be elements, then we identify r with s if resV,U (s) = r

ExampleX = SpecR and Xg → Xf principal open sets, so f divides g and

Rf → (Rf )g (3.3)

a 7→ a/1 (3.4)

ExampleX = SpecR. The principal open sets form a basis for the topology on |X|. Let [p] ∈ SpecR. If U ⊆ |X|containing p then U = ∪Xf s.t. f /∈ p. Considering the stalk

OX,[p] = colimf /∈pOX(Xf ) (3.5)

= colimf /∈pRf = Rp (3.6)

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which has elements of form r/f i for some f /∈ p. In fact, Rp is a local ring, i.e. has unique maximalideal. The unique maximal ideal is the image of p under the embedding R → Rp.

3.6 Morphisms

If F and G are sheaves on a space |X|, then a morphism of sheaves φ : F → G is uniquely determinedby its induced map on stalks (by sheaf axiom).

In classical algebraic geometry, if V,W are affine algebraic sets over algebraically closed field K, thena morphism φ : V →W induces a morphism φ∗ : K[W ]→ K[V ] of K-algebras.

IfX and Y are schemes, there is no embedding of the structure sheaf into some bigger sheaf of functions.In particular, a continuous map ψ : |X| → |Y | does not induce a map of sheaves OY → ψ∗OX . Soto define morphism, we need to define a pair (ψ,ψ#) : X → Y , with ψ : |X| → |Y | continuous andψ# : OY → ψ∗OX with some appropriate conditions.

What we would like is to say that the value of a section f ∈ OY (U) at a point q ∈ U agrees with thepullback ψ#(f) = f ψ at a point p ∈ OX(ψ−1U)

Definition 3.16A morphism between schemes X and Y is a pair (ψ,ψ#), s.t. ψ : |X| → |Y | continuous and ψ# is

a map of sheaves ψ# : OY → ψ∗OX s.t. for any point p ∈ X and any nbhd U of q = ψ(p) in Y , asection f ∈ OY (U) vanishes at q if and only if ψ#(f) of ψ∗OX(U) vanishes at p.

Assume X = SpecR and Y = SpecS, then ψ# : OY → ψ∗OX induces a map

OY,q = colimU3qOY (U)→ colimU3qOX(ψ−1U)→ colimV 3pOX(V ) = OX,p

If p = [p] ∈ SpecR and q = ψ([p]) = [q] ∈ SpecS, then this says we have induced map

OY,q = Sq → Rp = OX,p

The condition of a morphism says f ∈ OY (U)vanishes at q ⇔ ψ#f vanishes at p. Hence this justsays f ∈ q ⇔ ψ#f ∈ p which says Sq → Rp sends the unique maximal ideal to maximal ideal. (Thisis called a local homomorphism)

Theorem 3.17For any scheme X and R a ring , the morphisms (ψ,ψ#) : X → SpecR are in one-to-one correspon-dence with ring homomorphism R→ OX(X) = ψ∗(OX)(SpecR) given by (ψ,ψ#)→ ψ#

Corollary 3.18The category of affine schemes with morphisms is equivalent to the (opposite) category of commutativerings

Remark: There is a unique ring homomorphism Z → R. By Theorem, there is a unique morphismof schemes X → SpecZ for all sheaves X. Therefore SpecZ is a terminal object in the category ofschemes.

Definition 3.19Assume X,Y, Z are sets with maps ψ : X → Z and φ : Y → Z, then the fibre product or pullbackX ×Z Y is defined to be

X ×Z Y = (x, y) ∈ X × Y |ψ(x) = φ(y)

It is universal with respect to the existence of properties X ×Z Y → X and X ×Z Y → Y , i.e. the

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diagram commutes:X ×Z Y //

X

Y // Z

Assume X = SpecA, Y = SpecB and Z = SpecR, then the tensor product A⊗R B is universal w.r.t.the existence of A → A⊗R B and B → A⊗R B making the diagram commutes:

A⊗R B Aoo

B

OO

Roo

OO

From these diagrams we can see that the fibred product of affine schemes should be X×Z = Spec(A⊗RB). Also note that the pullback defined above is the pullback in category of schemes as introduced inExample 2.2, and A⊗R B is the pushout in the category of commutative rings.

3.7 Types of morphisms

There are two main problems with morphisms of schemes:

(1) ψ : X → Y then the image of a closed set need not be closed or open.

(2) If φ : X → Y is another morphism of schemes, then the set of points where ψ and φ agree is notnecessarily closed

Let ψ : X → Y be a morphism, we define the diagonal morphism to be the unique morphism

∆ : X → X ×Y X

whose composition with the canonical projection is the identity.

Definition 3.20We say ψ is separated if ∆(|X|) is closed in |X×Y X|. We say X is separated if the unique morphismψ : X → SpecZ is separated.

Note: A topological space is Hausdorff if and only if the image of the space under the diagonal mapis closed in the product space. So separatedness is the analogue of Hausdorff (and in the language ofcategories, the two definitions, using diagonal morphism, are formally the same).

In the case of affine schemes, X = SpecR, then it is always separated. Because ∆(|X|) is closedcorresponding to Spec(R⊗Z R/I), where I = ker(multiplication) = 〈a⊗ 1− 1⊗ a|a ∈ R〉.

Let ψ : A → B ring homomorphism, this induces ψ : Y = SpecB → X = SpecA (by equivalence ofcategories). Take distinguished open set Yg in SpecB and Xf in SpecA such that ψ(Yg) ⊆ Xf

then this induces OY (Yg) = Bg → OX(Xf ) = Af . By the equivalence of categories, we can just aswell start from a morphism Y → X.

For general schemes X,Y , we want a morphism f : X → Y such that ∀x ∈ X, ∃ affine nbhd U 3 x ∈ Xand affine V 3 f(x) ∈ Y such that f |U is a morphism of affine schemes.

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4 Etale morphisms

Theorem 4.1 (Inverse function theorem)M,N be smooth manifolds of the same dimension, and f : M → N morphism. Then dxf : TxM →Tf(x)N is isomorphism ⇔ f local isomorphism at x ∈ X, i.e. there exists nbhd U 3 x such that f |Uis a diffeomorphism onto its image.

So differential df encodes local behaviour, which is less apparent in the construction of morphisms ofschemes.

Definition 4.2An etale morphism f : Y → X is a flat and unramified morphism of schemes.

Roughly, a flat morphism f is a morphism which has its fibre f−1(x) vary continuously w.r.t. x, i.e.we want dim f−1(x) = dimY − dimxX. An unramified morphism is something more complicated.But all these will be explained in the following.

Definition 4.3Let X be a scheme and x ∈ |X|, recall the residue field κ(x) of OX,x is OX,x/mx where mx is the(unique) maximal ideal in OX,x.The inclusion map x → |X| and the map OX,x → κ(x) define a morphism of schemes Specκ(x)→ X.The fibre of a morphism f : Y → X is the pullback Y ×X Specκ(x), which we denote f−1(x).

The notation can be justified by showing a fibre is homeomorphic to the preimage of x of the underlyingtopological space. With this construction, the fibre f−1(x) is a scheme over Specκ(x) (via the mapf−1(x)→ Specκ(x)).

Definition 4.4Let φ : A→ B be ring homomorphism, then we say B is of finite type over A if B ∼= A[x1, . . . , xn]/I(i.e. B is a finitely generated A-algebra)For schemes, φ : Y → X is of finite type if ∃ affine covers Ui = SpecBi of Y and affine coversVj = SpecAj of X such that ∀i,∃j with φ(Ui) ⊆ Vj, and Bj is a Ai-algebra of finite type.

Recall that a morphism of ring φ : A→ B is flat if the functor M ⊗A − : A-Mod→ B-Mod is exact.

Definition 4.5A morphism f : Y → X is flat if for all affine open sets U ⊆ Y and V ⊆ X with f(U) ⊆ V , we haveOX(V )→ OY (U) is flat.

Remark: this happens if and only if the local homomorphism OX,f(y) → OY,y are flat for all y ∈ Y

4.1 Relation of finiteness to dimension

(Krull) Dimension of a ring R, dimR, can be defined as supn|p0 ( · · · ( pn.Hence we can define dimOX,xLet f : Y → X be a flat morphism of schemes, x ∈ X, consider the scheme f−1(x) over Specκ(x), andwe have

dimOf−1(x),|κ(x)| = dimOY,y − dimOX,xwhere y ∈;−1 (x), note that |κ(x)| is a one-point-space, which is identified with y.

Let F be a finitely generated filed extension of field K

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Definition 4.6We say that xi ∈ K is transcendentally independent if xi do not satisfy any polynomial relationswith coefficients in K

A transcendental basis is a collection xi of transcendentally independent elements such that F isalgebraic over K(xi); i.e. every x ∈ F is a solution of a polynomial with coefficients in K(xi)

The transcendental dimension trdimK F := cardinality of the transcendental basis.

Theorem 4.7 (Krull)Let A be integral domain which is finitely generated K-algebra, then dimA = trdimK A

Definition 4.8Let A be a ring, rad(A) = nilpotent elements of A. We say A is reduced if rad(A) = 0. DenoteArad as A/rad(A)

Note that the canonical homomorphism A→ Arad induces Spec(Arad)∼−→ SpecA

Definition 4.9A scheme X is reduced if for all open U ⊆ X, OX(U) is reducedIf X is reduced irreducible scheme X, then it is called integral

ExampleX = SpecA, then X is integral ⇔ A is integral domain (Exercise)

Definition 4.10X scheme, radX : U 7→ rad(OX(U)) defines a sheaf of ideals.Define Xred with structure sheaf OX/radX .

It can be shown that Xred → X is a closed immersion (equivalently, Xred is isomorphic to a closedsubscheme of X)

Definition 4.11Let ψ : A → B be homomorphism of local rings. Then ψ is unramified if B/ψ(mA)B is a finite,separable field extension of A/mA. We say B is unramified over A.For f : Y → X morphism of schemes of finite type, we say that it is unramified if OX,f(y) is unramifiedover OY,y

The idea to keep in mind is that, in a “nice” situation, unramified morphism implies the preimageof any point is a finite set of the same size. In summary, an etale morphism is a morphism which iswell-behaved with respect to tensor products, and preimage of each point is relatively nice and easyto work with.

4.2 Tangent Spaces

Assume V ⊆ An is an affine algebraic set. Assume f ∈ I(V )CK[X1, . . . , Xn], then we associate to fthe linear polynomial given by

dp(f) =

n∑i=1

∂f

∂Xi(p)Xi ∈ K[X1, . . . , Xn]

Definition 4.12We define the tangent space of V at p to be

Tp(V ) = v ∈ An|dp(f)(v) = 0 ∀f ∈ I(V )

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Examplef(X,Y ) = X2 + Y 2 − 1 and V = V (〈f〉), I(V ) = 〈f〉. Let p = (0, 1), then

dp(f) =∂f

∂X1(0, 1)X1 +

∂f

∂X2(0, 1)X2 = 2X2

Tp(V ) = (v1, 0) ∈ A2 (dimTp(V ) ≥ dimV ∀p ∈ V )

If A is an R-algebra and M is an A-module then A-derivation is a map D : A→M s.t.

• D(a+ a′) = D(a) +D(a′)

• D(aa′) = aD(a′) +D(a)a′

• D(r · 1) = 0 ∀r ∈ R

The space of such map is denoted DerR(A,M).

Lemma 4.13We have a K-linear isomorphism

Φ : Tp(V ) → DerK(K[V ],Kp)

v 7→ Dv

where DV (f) = dp(f)(v) ∀f ∈ K[V ]So here Kp (a copy of the field K) is a module for K[V ] where the action is given by f · x = f(p)x.

N.B. This isomorphism is also true for singular points.

4.3 The differential of a morphism

If φ : V →W is a morphism of algebraic sets, then for p ∈ V, q = φ(p) ∈W , we denote the differentialof φ at p to be

dp(φ) : DerK(K[V ],Kp) → DerK(K[W ],Kq)

D 7→ D φ∗

on the level of Tp(V ) and Tq(W ), we have the following. Assume φ = (f1, . . . , fm), then the mapdp(φ) : Tp(V )→ Tq(W ) is given by multiplication of the Jacobian

Jp(φ) =

(∂fi∂Xj

(p)

)1≤i≤m;1≤j≤n

Proposition 4.14Assume V,W are irreducible and dp(φ) is surjective where p and q = φ(p) are non-singular, then φ is

dominant. i.e. φ(V ) = W

4.4 A “local” version of tangent space

Assume V ⊆ An is irreducible and p ∈ V then we form the local ring

K[V ]p = f/g|f, g ∈ K[V ], g(p) 6= 0

Let εp : K[V ]p → K be the evaluation homomorphism, i.e.εp(f) = f(p), then mp = ker(εp)

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Let κ(p) = K[V ]p/mp then mp/m2p is a κ(p)-vector space. Note κ(p) = K[V ]p/mp then the action of

κ(p) on mp/m2p is given by

(f + mp)(g + m2p) = fg + m2

p

Proposition 4.15There is a well-defined K-linear isomorphism

Homκ(p)(mp/m2p, κ(p)) → DerK(K[V ],Kp) (4.1)

µ 7→ Dµ (4.2)

such that Dµ(a) = µ(a+ m2p) for all a ∈ K[V ] with a(p) = 0.

Corollary 4.16There is a well-defined K-linear isomorphism Tp(V )→ Homκ(p)(mp/m

2p, κ(p))

4.5 Tangent spaces for schemes

Assume A is an R-algebra then the module of relative differentials of A over R is an A-module ΩA/R

together with an R-derivation D : A→ ΩA/R which satisfies the following universal property:For any A-module M and any R-derviation D′ : A→M , ∃! A-module homomorphism f : ΩA/R →Ms.t. D′ = f D.

This does exist. Define the free module generated by all symbols D(a) and quotient by the appropriaterelations for a derivation.

ExampleIf A = K[X1, . . . , Xn] and R = K, then ΩA/R is just the free module generated by dX1, . . . , dXn.

Assume I CA, let A = A/I, then we have an exact sequence

I/I2 → ΩA/R ⊗A A→ ΩA/R → 0 (4.3)

where the first map is induced by f 7→ D(f)⊗ 1

ExampleAgain A = K[X1, . . . , Xn], then (4.3) shows ΩA/R is the quotient of ΩA/R ⊗A A =

⊕ni=1A · dXi by

the submodule generate by df =∑n

i=1∂f∂Xi

dXi for all f ∈ I (Note I is a point).

Corollary 4.17If A is local with residue field k = A/m contained in A then we have an isomorphism

m/m2 → ΩA/R ⊗A R

induced by (4.3).

Definition 4.18Let x ∈ X be a point of a scheme of finite type over K, write mx for the maximal ideal of the stalkOX,x. Then we define the tangent space at x to be

TX,x = HomK(mx/m2x,K)

4.6 Alternative definition of etale morphisms

Previously, etale morphism is motivated by the attempting to acquire an analogue of Inverse FunctionTheorem on schemes. Now we also want to have an analogue of Implicit Function Theorem for schemes.

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Recall Implicit Function Theorem says that:

Let x ∈ R n+k, f1, . . . , fk continuous s.t. det1≤i,j≤k

(∂fi∂Xj

(x))6= 0, then the restriction

Locus of f1 = · · · = fk → R n (4.4)

(x1, . . . , xn+k) 7→ (xk+1, . . . , xk+n) (4.5)

is locally an isomorphism at x

Example(2X = Y ) ∼= A1

φ : V (X2 − Y )→ A1 via (x, y) 7→ yLet p = (1, 1). For all but a finite number of values a contains both (

√a, a) and (−

√a, a).

Definition 4.19Let S := R[X1, . . . , Xn]/〈f1, . . . , fn〉 and X := SpecS. The morphism

φ : X → SpecR (4.6)

is etale at a point x ∈ X if the

det

(∂fi∂Xj

(x)

)6= 0 (4.7)

(Equivalently the Jacobian is a unit in R) So (4.7) means exactly that ΩS/R = 0. Indeed, ΩS/R =

module gen by dXi quotient out b relations∑n

i=1∂fi∂Xj

dXj = 0

ΩS/R is zero when det( ∂fi∂Xj) is a unit.

Nevertheless, in general situation, we should use the original definition of etale morphism, i.e. aflat and unramified morphism. For convenience, we recall the definition of unramified morphism:a morphism φ : Y → X such that the induced map on stalks φy : OX,f(y) → OY,y with maximalideals mX,f(y) and mY,y respectively satisfy (1) φy(mX,f(y))OY,y = mY,y and (2) OY,y/mY,y is finiteseparable field extension of OX,f(y)/mX,f(y). The following propositions says the two definitions ofetale morphism is the same:

Proposition 4.20φy is unramified at all y ∈ Y ⇔ ΩX/Y = 0

Proposition 4.21A morphism φ : Y → X is flat and unramified if and only if it locally is of the form of in Definition4.19. Namely, for all y ∈ Y , then ∃V = SpecS 3 y and U = SpecR 3 f(y), such that φ|V is of thatform, and S ∼= R[X1, . . . , Xn]/(f1, . . . , fn) with det(∂fi/∂Xj) is a unit in R.

This justifies that (4.6) is etale in the original definition.

Flatness of (4.6) comes from R → S = R[X1, . . . , Xn]/(f1, . . . , fn) (i.e. that S is a finitely generatedR-algebra). If R is a field, then S is an R-vector space, which is free and hence implies flatness.

Unramified comes from the Proposition 4.20: It suffices to see ΩS/R = 0. ΩS/R is a S-module with

generators dX1, . . . , dXn with relations∑ ∂fi

∂XjdXj = 0, this is equivalent to saying det( ∂fi∂Xj

) is a unit.

4.7 Examples

More general examples:(1) Open immersions are etale (since they induce an isomorphism on stalks)

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(2) Closed immersions are usually NOT etale . The problem that usually comes up is flatness (seelater).

Note that etale morphism has finite fibre, as unramified morphism have finite fibres

Unramified but not flat(1) Assume K a field. Inclusion of a point in A1.

Geometric map: Spec(K[X]/(X))→ Spec(K[X])Ring hom: K[X]→ K[X]/(X) via f 7→ f(0)

Claim: K[X]/(X)⊗K[X] − is not exactProof :The functor is exact ⇔ K[X]/(X) as a K[X]-module is not flat⇔ K[X]/(X) is torsion. (Note this is true for K PID)But K ∼= K[X]/(X) is a torsion K[X]-module as K[X]/(X) = f ∈ K|f · r = 0 for some non-zeror ∈ K[X].

Alternatively, we can use the short exact sequence 0 → K[X]·X−→ K[X] → K → 0, then tensor with

K[X]/(X) we get 0→ K[X]/(X)0−→ K[X]/(X)→ K → 0 which is not exact.

(2) A1 → y2 = x2 + x3 ⊂ A2.

The later space has a singularity at 0 (The fibre at 0 contains 2 points and everywhere else is 1).Therefore this is not flat as flatness preserve the size of fibre everywhere.

Algebraically:Geometric map: Spec(K[t])→ Spec(K[X,Y ]/(Y 2 −X2(X + 1)))Ring hom: S = K[X,Y ]/(Y 2 −X2(X + 1))→ B = K[t] via X 7→ t2 − 1 and Y 7→ t3 − t.

Claim: K[t] is not a flat S-module.Proof :We use the fact that an A-module M is flat ⇔ ∀ f.g. ideal I of A, I ⊗M →M is injective.

Take I = 〈X,Y 〉, then

I ⊗K[t] → K[t]

x⊗ tn 7→ (t2 − 1)tn

y ⊗ tn 7→ (t3 − t)tn

This is not injective as the non-zero element x⊗ t− y ⊗ 1 7→ (t2 − 1)t− (t3 − t) = 0.So now we have the map A1 → SpecS is not flat (at 0).

However, this map is unramified as we will show that ΩB/S = 0. Note that ΩB/S = db|b ∈ B etc.⇒ we need 0 = dx = d(t2 − 1) = 2tdt and 0 = dy = d(t3 − t) = (3t2 − 1)dtBut the gcd of 2t and 3t2 − 1 is 1, so dt = 0, hence ΩB/S = 0.

To conclude, A1 → SpecS is etale everywhere but 0.

A ramified example(1) Suppose char K 6= n, A1 → A1 via X 7→ Xn. This is ramified at 0 as dXn

dX |0 = nXn−1|0 = 0.

IntuitionEtale maps X → SpecK are simply

∐i SpecKi → SpecK where Ki are finite separable extension of

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KIf X is affine, i.e. we have SpecA→ SpecK (induced by/to K → A).

5 Etale site and cohomology

5.1 Sites

Definition 5.1A site is a category C with Grothendieck topology T on C. Recall T is an assignment of Ob(C)→ Sets,assigning object X to a covering families of X.

Zariski site is C = Op(|X|) category of open sets of X where X is a scheme. Coverings in the Zariskisite are given by fi : Ui → Ui∈I such that

⋃i fi(Ui) = U .

Etale site for X a scheme is given by the category C, where objects are etale morphisms U → X (i.e.given by a pair (U, f) s.t. f : U → X is etale ), the morphisms of C are given by the commutativediagram U //

V

~~X

where all the arrows in this diagram are etale .

The Grothendieck topology of C are defined by coverings etale fi : Ui → Ui∈I and⋃i fi(|Ui|) = |U |.

If Ui Zariski, then Ui → U are open immersion, which are etale .∴ any Zariski covering is an etale covering.

Given a Grothendieck topology, we can define sheaf F : Cop → D (where D can be category of abeliangroups or category of rings, or etc.) by F (U)

∼−→ equaliser(∏i F (Ui) ⇒

∏i,j F (Ui ×U Uj)) for all

U = Ui → U covering of C.

The following proposition gives a very easy criterion to check whether a sheaf is defined on the etalesite of X.

Proposition 5.2F is an etale sheaf (i.e. a sheaf on the etale site of X) if and only if:

(1) it is a sheaf for Zariski coverings (sheaf of Zariski site of X)

(2) it is a sheaf for coverings etale V → U (where V and U affine)

In particular, F is an etale sheaf, then it is a Zariski sheaf.

5.2 Etale cohomology

Let S be a scheme and Set be its etale site, where U = Uiφi−→ Ui∈I denote cover where U =⋃

i∈I φi(Ui). The functorΓ(S,−) : Sh(Set) → Ab

F 7→ Γ(S,F)is left exact, and its right derived functor

gives the etale cohomology (group) H∗et(S;F). Explicitly, form an injective resolution

0→ F → I0 → I1 → · · ·

then H∗et(S;F) = H∗(Γ(S), I∗).

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Example 5.3The following are examples of sheaves we should always keep in mind

(1) Z /nZ as the constant sheaf with value in Z /nZ

(2) Ga be sheaf U 7→ Γ(U,OU )

(3) Gm be sheaf U 7→ Γ(U,O×U )

Property 1: H0et(S;F) = Γ(S,F)

Property 2: If I is injective, then H∗et(S; I) = 0 for all ∗ > 0

In what follows, we will look at(1) etale cohomology of a sheaf on a point H∗et(Spec(K);F) for K field(2) The Cech cohomology H∗(S;F), which is more “geometeric” and “computable”(3) Show H∗(S;F) = H∗et(S;F) for ∗ = 0, 1.(4) Mayer-Vietoris sequence for H∗et

(5) Compute H∗(S;Gm)

5.3 Etale cohomology of a point

Let K be a field, so we can regard Spec(K) as a point.Define Spec(K)et be category with objects being

∐i∈I Spec(Ki) where Ki/K are finite separable

extension and the indexing set I is finite. Spec(K)et is a site with covers being surjective families.Take Ksep be separable closure of K, and G := Gal(Ksep/K) (strictly speaking, this should be theK-invariant automorphism group of Ksep)If we take intermediate field Ksep/L/K s.t. L is (finite) Galois over K, we get homomorphismG→ Gal(L/K).We equip G with the profinite topology (the smallest topology making all of these homomorphismscontinuous)

Galois theory gives a bijection:intermediate field L/K

Ksep/L/K finite separable

↔ H ⊆ G open

L 7→ StabG(L)

(Ksep)H ←[ H

and equivalence of sites:

Spec(K)et → G-Sets

X 7→ Spec(Ksep)→ X which fixes SpecK∐i∈I

Spec((Ksep)Hi) ←[∐i∈I

G/Hi

where G-Sets is a category of sets acted on by G for which G acts continuously and objects arefinite union of orbits and morphisms are G-equivariant maps. This further gives equivalence of sheafcategories:

Sh(Spec(K)et) ↔ continuous G-modules ↔ Sh(G-Sets)F 7→ MF := colimLF(Spec(L))

M 7→ (A 7→ MapG(A,M))

where the colimit is taken over intermediate fields L which are finite Galois over K.

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Theorem 5.4H∗et(Spec(K),F) ∼= H∗(G;MF ) where the later (group) cohomology is computed by resolving the G-module MF by continuous G-modules. (It is called the continuous group cohomology of profinitegroups)Spec(K) is a “geometric point” if K is separably closed. In this case Ksep = K, then G = 1, thenH∗et(Spec(K),F) = 0 ∀∗ > 0.

5.4 Cech Cohomology

We now work over general site C. Let F ∈ Sh(C), U ∈ C, H∗(U ;F) = H∗(Γ(U, I∗)), with F → I0 →I1 → · · · is an injective resolution.

Definition 5.5Let U = Ui → Ui∈I be a cover of U , and let F be a presheaf. The Cech complex C∗(U ;F) is∏

i∈IF(Ui)

d0−→∏

i0,i1∈IF(Ui0 ×U Ui1)

d1−→ · · ·

where dl =∑l

k=1(−1)kdlk, where

dlk :∏F(Ui0 ×U · · · ×U Uil)→

∏F(Ui0 ×U · · · ×U Uil+1

)

is induced by the morphism

Ui0 ×U · · · ×U Uil+1→ Ui0 ×U · · · × Uik × · · · × Uil+1

Definition 5.6The Cech cohomology of U with coefficients in F is H∗(U ;F) = H∗(C∗(U ;F))

5.5 Changing the cover

Recall that, we say V = Vj → Uj∈U refines U if for j ∈ J , we can find

Vjφj //

Uτj

~~U

τj ∈ I. The morphismsVj0 ×U · · · ×U Vjl → Uτj0 ×U · · · ×U Uτjl

induces C∗(U ;F)→ C∗(V;F) which induces H∗(U ;F)→ H∗(V;F), which is independent of choices.

Definition 5.7The Cech cohomology of U with coefficient in F is H∗(U ;F) = colimUH

∗(U ;F)

5.6 The properties

Lemma 5.8If F is a sheaf then H0(U ;F) = Γ(U,F) = F(U).More generally, if F is a presheaf, H0(U ;F) = aF(U) where aF is the sheafification of F .

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Proof

H0(U ;F) = ker(∏F(Ui0)

d0=d00−d01−−−−−−→∏F(Ui0 ×U Ui1)) = equaliser(

∏F(Ui0)⇒

∏F(Ui0 ×U Ui1))

If F is a sheaf, then this equaliser is precisely F(U). Generally, aF(U) is the colimit of all theseequalisers.

Lemma 5.9If I is injective then H∗(U ; I) = 0 for ∗ > 0.

ProofIn fact H∗(U;I) = 0 for all ∗ > 0 whenever U is a cover and I is an injective presheaf.

Define a chain complex of presheaves

ZU∗ := (Z∐Ui0← Z∐

Ui0×UUi1

← · · · )

where Z V is presheaf defined by Z V (W ) := ZHom(W,V )⇒ C∗(U ; I) = Hom(ZU∗, I)As I injective, Hom(−, I) exact. So it suffices to show that ZU∗ is exact in positive degrees.As we are working with presheaves, it suffices to prove ZU∗(V ) is exact in positive degrees.

ZU∗(V ) = Z [∐

Hom(V,Ui0)]← Z [∐

Hom(V,Ui0 ×U Ui1)]← · · ·

= Z [∐

Hom(V,Ui0)]← Z [∐

Hom(V,Ui0)×Hom(V,U) Hom(V,Ui1)]← · · ·

=⊕

φ∈Hom(V,U)

(Z [∐

Homφ(V,Ui0)]← Z [∐

Homφ(V,Ui0)×Homφ(V,Ui1)]← · · ·)

where Homφ(V,Uj) means space of homomorphism for which φ factors through Uj .The complex then becomes a sum of complexes

Z [S]← Z [S × S]← Z [S × S × S]← · · ·

which are always exact in positive degrees.

Theorem 5.10There is a spectral sequence Hs(U ;Ht

et(F)) ⇒ Hs+tet (U ;F). In fact, this is true if we replace etale

cohomology H∗et with cohomology on any site H∗.

Remark: For F sheaf, Hq(F) is the presheaf V 7→ Hq(V ;F)

Corollary 5.11H∗(U ;F) = H∗et(U ;F) for ∗ = 0, 1 (Again, we can replace Het by H)

5.7 Introduction to spectral sequences

Definition 5.12A (first quadrant, cohomological) spectral sequence, denote (Er, dr) consists of

• Abelian groups Es,tr for s, t ≥ 0, r ≥ 1For a fixed r, the collection Es,tr is called an Er-page

• Differentials ds,tr : Es,tr → Es+r,t−r+1r , which satisfy ds+r,t−r+1

r ds,tr = 0

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We can the form cohomology Hs,t(Er) = ker ds,tr

Im ds−r,t+r−1r

These satisfy Hs,t(Er) = Es,tr+1

Each page is the cohomology of the previous one.

Definition 5.13Observe that if r > max(t+ 1, s), then ds,tr = 0 and ds−r,t+r−1

r = 0. So Hs,t(Er) = Es,tr .In other words, for s, t fixed, Es,tr = Es,tr+1 = Es,tr+2 = · · · . The common value is called Es,t∞ . The

collection of Es,t∞ is called the E∞-page

Definition 5.14Let F ∗ be a graded abelian group. We say that (Er, dr) converges to F ∗, and write Es,t1 ⇒ F s+t, if

there is a filtration F ∗ = FF ∗0 ≥ F ∗1 ≥ F ∗2 ≥ · · · with F pq = 0 for q > p, such that Es,t∞ = F s+ts /F s+ts+1.

Theorem 5.15Let E

π−→ B be a fibration, fibre F , and π,B = 0. There is a spectral sequence H∗(B;H∗F )⇒ H∗(E)

5.8 Double complexes

A double complex is an array of abelian groups Cs,ts,t∈N0 and maps dh : Cs,∗ → Cs+1,∗ (thehorizontal differential) and dv : C∗,t → C∗,t+1 (the vertical differential) such that d2

h = 0 = d2v and

dvdh = dhdv.

Definition 5.16H∗,•h is the bigraded group obtained by taking vertical cohomology, then horizontal cohomology, of C∗,•

Tot(C)∗ =⊕

s+t=∗Cs,t equipped with differential dh + (−1)tdv is the total complex of C

Theorem 5.17There is a spectral sequence H∗,•h (C)⇒ H∗(Tot(C))

Remark We can flip Cs,t (diagonally) to obtain another double complex Ct,s, which we get us anotherspectral sequence converging to the same thing. Equivalently, the second spectral sequence comes fromtaking H∗,•v of C∗,•. This will help us to obtain our Theorem 5.10.

Proof of Theorem 5.10Let F → I∗ be an injective resolution and consider the double complex C∗(U , I∗).

(1) What is Hs,th (C(U ; I))?

Vertical part:...

...

∏i0I1(Ui0)

OO

∏i0,i1

I1(Ui0 ×U Ui1)

OO

· · ·

∏i0I0(Ui0)

OO

∏i0,i1

I0(Ui0 ×U Ui1)

OO

· · ·

26

Vertical cohomology

......

∏i0H1(Ui0 ,F)

OO

∏i0,i1

H1(Ui0 ×U Ui1 ;F)

OO

· · ·

∏i0H0(Ui0 ,F)

OO

∏i0,i1

H0(Ui0 ×U Ui1 ;F)

OO

· · ·

which is:...

...

C0(U , H1(F))

OO

C1(U , H1(F))

OO

· · ·

C0(U , H0(F))

OO

C1(U , H1(F))

OO

· · ·Take horizontal cohomology:

......

H0(U , H1(F))

OO

H1(U , H1(F))

OO

· · ·

H0(U , H0(F))

OO

H1(U , H1(F))

OO

· · ·

which is H∗(U , H•(F))

(2) What is H∗(Tot(C(U ; I)))?To answer this, we map flip the double complex diagonally. Taking vertical cohomology:

......

H1(U , I0)

OO

H1(U , I1)

OO

· · ·

H0(U , I0)

OO

H0(U , I1)

OO

· · ·

By Lemma 5.8, the bottom row is I0(U), I1(U), I2(U), . . .; and second row is H1(U ; I0), H1(U , I1), . . ..By Lemma 5.9, every rows above the bottom one are zero. Hence when we take the horizontalcohomology we get H0(U ;F), H1(U ,F), . . .. These are E2-term of a spectral sequence converging toH∗(Tot(C)). But E2 = E∞ for positioning reason (all differential are zero, each diagonal has only onenon-zero term). i.e. E∗,0∞ = H∗(Tot(C)) since E∗,t∞ = 0 for t > 0⇒ H∗(Tot(C)) = H∗(U ;F)

(3) Conclusion:The spectral sequence of the original double complex now has the form

H∗(U ;H∗(F))⇒ H∗(U ;F)

which is our theorem.

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Remark: Taking colimits, we can replace U by U . (Exercise/Think carefully)

Proof of Corollary 5.11Use H∗(U ;H∗(F))⇒ H∗(U ;F)First, H0(U ;Ht(F)) = aHt(F)(U) = 0 for t > 0. This is because the sheaf aHt(F) is zero, the reasonfor this is that

aHt(F) = aHt(iI∗) = Ht(aiI∗) = Ht(I∗) = 0

where i is the inclusion functor sending sheaves to presheaves.

Also H0(U ;H0(F)) = H0(F)(U) = H0(U ;F).Now H0(U ;H∗(F)) = 0 for ∗ > 0and H∗(U ;H0(F)) = H∗(U ;F)So we get E∞ has

......

... · · ·

0 ? ? · · ·

H0(U ;F) H1(U ;F) ? · · ·

So H0(U ;F) = H0(U ;F) and H1(U ;F) = H1(U ;F).

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Index

Cech cohomolgy, 24etale cohomology, 22etale morphism, 16

algebraic setaffine, 8projective, 8

cocomplete, 3cocone, 2

distinguished, 4coequaliser, 3colimit, 2complete, 3cone, 2converge, 26coordinate ring

affine, 8homogeneous, 8

coproduct, 3cover, 4covering families, 4

defined by Fq, 1derivation, 18differential, 18dominant, 18double complex, 26

E∞-page, 26equaliser, 3Er-page, 25etale topology, 6

fibre, 16fibre product, 14finite type, 16Frobenius map, 1

Ga, 23generic point, 12global section ΓX , 7gluing property, 5Gm, 23graded ring, 8Grothendieck topology, 4

equivalent, 6

homogeneouscoordinate ring, 8element, 8ideal, 8vanishing ideal, 8

ideal sheaf, 13quasi-coherent, 13

integral scheme, 17

limit, 2local homomorphism, 14locally affine, 12

module of relative differentials ΩA/R, 19morphism

diagonal, 15fibre of, 16finite type, 16flat, 16of affine algebraic sets, 10of schemes, 14separated, 15unramified, 16, 17

mrophismetale , 16

Noetherian topological space, 9

presheaf, 3princiapl open set, 10principal open set, 12product, 3profinite topology, 23pullback, 14pullback A×C B, 3pushout A tC B, 3

quasi-coherent, 13

reduced ring, 17reduced scheme, 17refine, 6regular function, 12

global, 12representable functor, 3representative, 3rX, 3

schemeaffine, 12integral, 17reduced, 17separated, 15tangent space of, 19

sees as colimit, 5sheaf, 5

etale , 22structure, 12

29

sheaf cohomology, 7sheafification, 6sieve, 5site, 22

Etale , 22Zariski, 22

spectral sequence, 25converge, 26

spectrum, 11point of, 11

stalk, 13structure sheaf, 12subscheme

closed, 13open, 13

tangent space, 17, 19topology

profinite, 23Zariski, 9, 11

total complex, 26transcendental basis, 17transcendental dimension, 17transcendentally independent, 17twisted cubic, 8

vanishing ideal, 8homogeneous, 8

varietyaffine, 11

Weil conjectures, 1

Yoneda Lemma, 3

Zeta function, 1

30