Bloch–Ogus properties for topological cycle theory

Ann. Scient. Éc. Norm. Sup.,4e série, t. 33, 2000, p. 57 à 79.

BLOCH–OGUS PROPERTIESFOR TOPOLOGICAL CYCLE THEORY

BY ERIC M. FRIEDLANDER 1

ABSTRACT. – We reformulate and extend “morphic cohomology” developed by the author andH.B. Lawson so that, together with “Lawson homology”, it satisfies the axioms of S. Bloch and A. Ogusfor a “Poincaré duality theory with supports” on complex quasi-projective varieties. 2000 Éditionsscientifiques et médicales Elsevier SAS

RÉSUMÉ. – Nous reformulons et étendons la cohomologie morphique développée par l’auteur etH.B. Lawson de telle sorte qu’elle satisfasse, avec l’homologie de Lawson, les axiomes de S. Bloch etA. Ogus pour une théorie de la dualité de Poincaré à supports sur les variétés complexes quasi-projectives. 2000 Éditions scientifiques et médicales Elsevier SAS

In this paper, we re-formulate “morphic cohomology” as introduced by the author andH.B. Lawson [8] in such a way that it and “Lawson homology” satisfy the list of basic propertiescodified by S. Bloch and A. Ogus [3]. This reformulation enables us to clarify and unify ourprevious definitions and provides thistopological cycle theorywith foundational propertieswhich have proved useful for other cohomology theories. One formal consequence of theseBloch–Ogus properties is the existence of a local-to-global spectral sequence which should provevaluable for computations (a simple example of which is given in Corollary 7.3).

The basic result of this paper is thattopological cycle cohomology theory(which agreeswith morphic cohomology for smooth varieties) in conjunction withtopological cycle homologytheory(which is shown to always agree with Lawson homology) do indeed satisfy the Bloch–Ogus properties for a “Poincaré duality theory with supports” on complex quasi-projectivevarieties. As we demonstrate in the final section of this paper, our techniques also suffice toprove that topological cycle theory satisfies the stronger axioms of H. Gillet [15] (other thanhomotopy invariance of cohomology on singular varieties).

We view the challenge of verification of the Bloch–Ogus properties as worthy for severalreasons. First, the properties require certain definitions and constructions whose developmentadd substance to morphic cohomology/Lawson homology. For example, considerable effortis required to extend earlier definitions to a cohomology theory defined and contravariantlyfunctorial on all quasi-projective varieties. As another example, our cap product pairing (whosecontinuity is established in [14]) leads to a natural extension of earlier duality theorems of theauthor and Lawson [9,6,11].

Second, the properties constrain the formulation of our theory, thereby giving us a good basisfor choosing the definitions we propose. Third, the fact that these properties can be verified tells

1 Partially supported by the N.S.F. and the N.S.A.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE. – 0012-9593/00/01/ 2000 Éditions scientifiques etmédicales Elsevier SAS. All rights reserved

58 E.M. FRIEDLANDER

us that this theory, although originating as it does in differential geometry and topology, behavesvery much as other theories familiar to algebraic geometers and thus might be more readilyapplicable to geometric problems. Finally, the constructions we present are closely related tothose involved in formulating a suitable motivic cohomology theory as in [13] (for example,we use V. Voevodsky’s qfh-topology to extend definitions from normal varieties to all varieties),so that our topological point of view may serve as an accessible entry into that more algebraictheory.

Over the past decade, Lawson homology and morphic cohomology have been reformulatedseveral times, with each reformulation providing either a simplification of definitions and/or anextension of the class of varieties for which the theory is applicable (cf. [5,6,8,9,12,16,17]). Asnow presented our topological cycle theory is an appropriate extension to all quasi-projectivevarieties of Lawson homology/morphic cohomology on smooth varieties.

Throughout this paper, “variety” will be taken to mean a locally closed subset (in the Zariskitopology) of some complex projective space (i.e., a reduced but not-necessarily irreduciblescheme of finite type over the complex fieldC which admits a Zariski locally closed embeddingin a complex projective space).

I am grateful to Robert Lateveer for asking whether the Bloch–Ogus properties hold fortopological cycle theory. I also thank my colleagues Andrei Suslin, Vladimir Voevodsky, andMark Walker for valuable insights into this formalism. Finally, I thank the referee who suggestedconsideration of Gillet’s axioms as well as those of Bloch and Ogus.

1. Spaces of continuous algebraic maps

In this section, we investigate the topological mapping spaceMor(X ,Cr(Y )) of continuousalgebraic mapsfrom a varietyX to the abelian monoidCr(Y ) of effective r-cycles on aprojective varietyY . We begin by considering cycles onX × Y equi-dimensional of relativedimensionr over a normal varietyX as considered in [6]. We recall that such “effective cocycles”on a normal varietyX can be reinterpreted as morphisms fromX to Cr(Y ). With the aid of theqfh topology, we are led to consider continuous algebraic maps from a general varietyX to aprojective varietyW . This leads to a suitable topological monoidMor(X ,Cr(Y )) for generalX .

If Y is a projective variety provided with a given embeddingY ⊂ PN into some projectivespace, then the Chow varietyCr,d(Y ) is a projective variety (i.e., a Zariski closed subset of someprojective space) whose points naturally correspond to effectiver-cycles onY of degreed. We letCr(Y ) denote the abelian monoid

∐d>0Cr,d(Y ). We provideCr(Y ) with the analytic topology;

so defined, the topological abelian monoidCr(Y ) is independent of the projective embeddingof Y . (Indeed, the algebraic structure ofCr(Y ) does not depend upon the projective embeddingof Y as shown by D. Barlet [2].) IfV ⊂ Y is a Zariski open subset of a projective varietyYwith complementY∞ = Y − V , then we defineCr(V ) to be the quotient topological monoidCr(Y )/Cr(Y∞); so definedCr(V ) is independent of the projective closureV ⊂ Y (cf. [7,17]).

We begin with what we consider a good definition of the topological abelian monoid of equi-dimensional cycles over a normal varietyX .

DEFINITION ([6, 1.6]). –Let X be a normal quasi-projective variety of pure dimensionmandY a projective variety. The topological abelian monoid of effective cocycles onX of relativedimensionr in Y is the following quotient monoid

Cr(Y )(X) =: Er(Y )(X)/Cr+m(X∞ × Y ).

4e SÉRIE– TOME 33 – 2000 –N◦ 1

BLOCH–OGUS PROPERTIES FOR TOPOLOGICAL CYCLE THEORY 59

Here,X ⊂ X is a projective closure,X∞ = X − X , and Er(Y )(X) ⊂ Cr+m(X × Y ) is theconstructible submonoid of effectiver +m-cycles onX × Y whose restrictions toX × Y areequi-dimensional overX .

To extend this to more generalX , it will be useful to recall the following reinterpretation ofCr(Y )(X).

PROPOSITION 1.1 ([9, C.3]). –LetX be a normal, quasi-projective variety andY projective.The topological abelian monoidCr(Y )(X) can be identified with the abelian monoid ofmorphisms fromX to Cr(Y ) provided with the following topology: a sequence{ fn :X→Cr(Y )}converges to some morphismf :X → Cr(Y ) provided that it converges with respect to thecompact open topology inHomcont(X ,Cr(Y )) (whereX ,Cr(Y ) are provided with the analytictopology) and provided that for some projective closureX ⊂X the closures of the graphs offnin X × Y have bounded degree.

We denote this topological abelian monoid of morphisms fromX toCr(Y ) byMor(X ,Cr(Y )).As shown in [6, 3.3], the mapMor(X ,Cr(Y ))→Mor(X ′,Cr(Y )) induced by a morphismf :X ′→X is continuous (with respect to the topology described above).

We next recall V. Voevodsky’s qfh-topology, a Grothendieck topology on the category ofvarieties. Recall that a continuous mapf :V → Y of topological spaces is said to be a universaltopological epimorphism if for allY ′ → Y the pull-backf ′ :V ×Y Y ′ → Y ′ satisfies theconditions thatf ′ is surjective and thatU ⊂ Y ′ is open if and only iff ′−1(U ) ⊂ V ×Y Y ′ isopen.

DEFINITION ([18]). – A qfh-covering of a varietyX is a finite family of quasi-finite morphisms{pi :Xi→X} such that

∐pi :∐Xi→X is a universal topological epimorphism in the Zariski

topology.A presheafF on the category of varieties is said to be a qfh-sheaf if it satisfies the sheaf axiom

for all qfh-coverings{Xi→X}:

F (X) = equalizer

{∏F (Xi)⇒

∏F (Xi ×X Xj)

}.

PROPOSITION 1.2. –The contravariant functorMor(−,Y ) sending a normal varietyX tothe set of morphisms to a fixed projective varietyY satisfies the qfh-sheaf condition for all qfh-coverings{Xi→X} with the property thatX and eachXi is normal.

Moreover, for any such qfh-covering, the inclusionMor(X ,Y ) ⊂∏Mor(Xi,Y ) identifies

Mor(X ,Y ) as a subspace of∏Mor(Xi,Y ).

Proof. –By [18, 10.3], it suffices to assume {Xi→X} is either a Zariski open covering or acovering of the formX ′→X whereX ′ is the normalization ofX in some finite Galois extensionof the field of fractionsk(X) of X . (Following [18], we call this second type of covering apseudo-Galois covering.) The (set theoretic) sheaf condition forMor(−,Y ) is evident for aZariski open covering.

If p :X ′→X is a pseudo-Galois covering with Galois groupG, then

equalizer{Mor(X ′,Y )⇒Mor(X ′×X X ′)

}=Mor(X ′,Y )G.

To check the sheaf condition for such a pseudo covering, consider a morphismf ′ :X ′→ Y whichisG-equivariant. Such a morphism determines a rational mapf :X→ Y sincek(X) = k(X ′)G.The graphΓf of this rational map (i.e., the closure inX × Y of the morphism defined on someZariski open subset ofX) has the property thatp−1(Γf ) = Γf ′ and is thus quasi-finite as well as

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60 E.M. FRIEDLANDER

birational overX . SinceY is projective,Γf →X is also finite and thus Zariski’s Main Theoremimplies thatΓf →X is an isomorphism. This implies thatf is a morphism as required to verifythe sheaf condition.

To check that the topology onMor(X ,Y ) is the subspace topology inherited from∏Mor(Xi,Y ), we must verify that a sequence {fn :X → Y } converges tof :X → Y if each

of the induced sequences {pi ◦ fn :Xi→ Y } converges topi ◦ f . Once again, it suffices to checkthis when {fn} is a Zariski covering and whenX ′→X is a pseudo-Galois covering with finiteGalois groupG. In the case of a Zariski covering, this is clear in view of the fact that a Zariskiopen covering is also a covering by open subsets in the analytic topology (so convergence withrespect to the compact-open topology of {fn} to f :X → Y is assured) and the finiteness ofthe covering {Xi ⊂X} (which assures boundedness of degree). In the case of a pseudo-GaloiscoveringX ′→X , we observe thatX is a quotient space ofX ′ in the analytic topology so thatconvergence of {fn} in the compact-open topology is clear. Moreover, as argued in [7, 1.6], theboundedness of the degrees of the closures of the graphs of {p ◦ fn} is equivalent to that of{ fn}. 2

The following extension technique is a technique of A. Suslin and V. Voevodsky.

PROPOSITION 1.3. –Let F be a contravariant functor defined on normal varieties whichsatisfies the qfh-sheaf axiom when restricted to qfh-coverings of normal varieties by normalvarieties. ThenF admits a unique extension as a qfh-sheaf on all varieties.

Moreover, ifF takes values in topological spaces and satisfies the condition that the inclusionF (X) ⊂

∏F (Xi) identifiesF (X) as a subspace of

∏F (Xi) for all qfh-coverings of normal

varieties by normal varieties, thenF becomes a qfh-sheaf defined on all varieties with values intopological spaces.

Proof. –The uniqueness of such an extension is clear, since the normalizationX˜→X of avarietyX is a qfh-covering. To construct the extension, we defineF (X) for a general varietyXwith normalizationX˜→X as the equalizer

F (X) =: equalizer{F (X˜)⇒ F (X˜̃ )

},

whereX˜̃ →X˜×XX˜ is the normalization ofX˜×XX˜. One readily checks that this satisfiesthe sheaf condition for any qfh-covering of any variety using the observation that an equalizer ofa diagram of equalizers is the equalizer of the associated large diagram.

To provide a topology onF (X) for X arbitrary, we give it the subspace topology with respectto the embeddingF (X)⊂ F (X˜), whereX˜→X is the normalization ofX . With this topology,if { Xi→X} is any qfh-covering, thenF (X) so defined has the subspace topology of

∏F (Xi)

in view of the following observation: ifT ′′ ⊂ T ′ andT ′ ⊂ T are subspaces, thenT ′′ ⊂ T is alsoa subspace (i.e., has the subspace topology).2

Recall that acontinuous algebraic mapf :X→ Y with Y projective is a Zariski closed subsetΓf ⊂X × Y with the property thatΓf is the graph of a set-theoretic map fromX to Y [5]. Sodefined, a continuous algebraic map fromX to Y is equivalent to a morphism fromXwn to Y ,whereXwn→X is the weak normalization ofX . (The weak normalizationpw :Xwn→X isa map which factors the normalizationp :X˜→X , which is a homeomorphism on underlyinganalytic topological spaces, and which satisfies the condition that the ring of analytic functionsof Xwn consists of those continuous functions onX whose composition withp are analytic onX˜.)

The following proposition gives some insight into why continuous algebraic maps occurred inthe formulation of morphic cohomology theory (cf. [8]).

4e SÉRIE– TOME 33 – 2000 –N◦ 1


PROPOSITION 1.4. –Let Y be a projective variety and letMor(−,Y ) denote the qfh-sheafon varieties with values in topological spaces whose value on a normal varietyX is the space ofmorphisms fromX to Y as topologized in Proposition1.1. Then for anyX , the underlying setofMor(X ,Y ) is the set of continuous algebraic maps fromX to Y .

Proof. –It suffices to show for any varietyX with normalizationX˜ that the set of continuousalgebraic maps fromX → Y is naturally identified with the equalizer ofMor(X˜ ,Y ) ⇒Mor(X˜̃ ,Y ), whereX˜̃ →X˜×X X˜ is the normalization ofX˜×X X˜. Since a morphismXwn→ Y induces a morphismX˜ → Y , there is a natural inclusion of this set of continuousalgebraic maps into the equalizer. On the other hand, given a morphismf˜ :X˜→ Y which liesin the equalizer, we readily verify that its graphΓf˜ ⊂X˜ × Y projects to a closed subvarietyof X × Y which maps birationally and bijectively toX and therefore represents a continuousalgebraic map. 2

DEFINITION. – The topological abelian monoid of effective relativer-cycles on a varietyXwith values in a projective varietyY is defined to be

Mor(X ,Cr(Y )

)=:∐d

Mor(X ,Cr,d(Y )

).

Eachf :X → Cr,d(Y ) has an associated graphΓf ⊂ X × Y which is equi-dimensional ofrelative dimensionr overX . Sendingf to its graphΓf is clearly 1–1. On the other hand, ifXwn

is not normal, then not every effective algebraic cycle onX × Y equi-dimensional of relativedimensionr overX arises as the graph of such a continuous algebraic map. Thus, our qfh-sheafification of relative cycles does not give the group of equidimensional cycles overX on anarbitrary varietyX .

2. Topological cycle cohomology and homology

In the preceding section, we constructed the qfh-sheafMor(−,Cr(Y )) which takes valuesin topological abelian monoids. The purpose of this section is to convertMor(X ,Cr(Y )) to abivariant functor which will lead us to definitions of bivariant topological cycle cohomology,topological cycle cohomology (with supports), and topological cycle homology. As the readerwill quickly see, this section consists primarily of formal definitions. Our somewhat sophisticatedformulation of this theory is justified by the properties we prove in the next two sections.

Notation. – Let Zar denote the big Zariski site consisting of quasi-projective complexalgebraic varieties (whose coverings are Zariski open coverings); for a given varietyX , letZarX denote the small Zariski site consisting of Zariski open subsets ofX . We denote byP(respectively,PX ) the category of chain complexes bounded below of abelian presheaves onZar (respectively,ZarX ); we denote byS (respectively,SX ) the category of chain complexesbounded below of abelian sheaves onZar (respectively,ZarX ).

If A is a topological abelian monoid, we denote byA˜ the normalized chain complexassociated to the group completion of the simplicial abelian monoidSing.A obtained by level-wise group completing the abelian monoidsSingkA. Here,Sing.T denotes the singular complexof a topological spaceT , a simplicial set which inherits the structure of a simplicial abelianmonoid provided thatT itself has the structure of a topological abelian monoid.

If C∗ is a chain complex andn an integer, we denote byC∗[n] the chain complex “shifted tothe left byn”; in other words, (C∗[n])i = Ci−n for eachi ∈ Z. In other words, ifC∗ is viewedas a (co-)chain complex with differential of degree+1 and denotedC∗ (with Ci ≡ C−i), then(C∗[n])j =Cn+j .


62 E.M. FRIEDLANDER

Example. – For any simplicial varietyU , let ZU . denote the presheaf defined to sendV tothe normalized chain complex of the simplicial abelian group whose value onn ∈ ∆ is the freeabelian group on the set Hom(V ,Un).

We now introduce the presheaves of chain complexes bounded below which we shall use todefine topological cycle cohomology and homology.

DEFINITION. – For a given non-negative integers, we define the complex of abelianpresheaves on Zar

M(−,s) =: U 7→ cone{Mor

(U ,C0(Ps−1)

)̃→Mor

(U ,C0(Ps)

)̃ }[−2s] ∈ P .

Consider a varietyX , closed subvarietyY ⊂X , and a non-negative integers. LetM(−,s)|Xdenote the restriction ofM(−,s) to ZarX . We define the following complexes of abelianpresheaves on ZarX (i.e., objects ofPX ):

MY (−,s) =:U 7→ cone{M(U ,s)|X→M(U ∩X − Y ,s)|X

}[−1],

L(−,r) =:U 7→ cone{Cr(U∞)̃ →Cr(X )̃

}[2r],

whereX ⊂ X is a projective closure andU∞ ⊂ X is the Zariski closed complement ofU ⊂X ⊂X .

Remark2.1. – The presheafL(−,r) ∈ PX applied to anyU ∈ ZarX is a chain complex oftorsion free abelian groups. This follows from the observation thatA˜ is a chain complex oftorsion free abelian groups whenever the topological abelian monoidA is torsion free.

These presheaves are presheaves of morphic cohomology and Lawson homology groups aswe observe in the following proposition.

PROPOSITION 2.2. –For a normal varietyX and anyn ∈ Z,

H−n(M(X ,s)

)= H−n

(M(X ,s)|X

)= LsHn(X),

whereLsHn(X) denotes the morphic cohomology ofX as introduced in[8] and formulatedin [6].

For any varietyX ,

Hm(L(X ,r)

)= LrHm(X),

whereLrHm(X) is the Lawson homology ofX as in[7].

Proof. –The formulation of morphic cohomology in [6, 2.4],

LsH2s−j(X ,s)≡ πj((C0(Ps)(X)/C0(Ps−1)(X)

)+),

is in terms of the homotopy groups of the “naïve topological group completion” of the topologicalquotient monoidC0(Ps)(X)/C0(Ps−1)(X). On the other hand, ifA is a topological abelianmonoid, thenSing.A→ (Sing.A)+ is a homotopy theoretic group completion [12, AppQ] and

πi((Sing.A)+

)= Hi(A˜), i> 0.

Proposition 1.2 provides a natural isomorphism of the topological abelian monoidsC0(Ps)(X)andMor(X ,C0(Ps)). Consequently, [6, 1.9, 2.2] establishes that the original formulation ofmorphic cohomology agrees with H−n(M(X ,s)).

4e SÉRIE– TOME 33 – 2000 –N◦ 1


Similarly, the formulation of Lawson homology in [7, 1.5],

LrH2r+i(X) = πi(Zr(X )/Zr(X −X)

),

is in terms of the homotopy groups of the quotient topological groupZr(X )/Zr(X−X), whereX ⊂X is a projective closure andZr(X ) is the naïve topological group completion of the ChowmonoidCr(X ). As argued above, the fact thatCr(X )→ Zr(X ) is a homotopy theoretic groupcompletion and thatZr(X −X)→ Zr(X )→ Zr(X) yields a distinguished triangle of chaincomplexes (cf. [7]) implies that

πi(Zr(X)

)= Hi+2r

(L(X ,r)

). 2

Our definition of cohomology and homology will be in terms of maps in the derived categoryDX associated toSX introduced in the following definition.

DEFINITION. – Let HP denote the homotopy category ofP obtained by identifying mapsof chain complexes related by a chain homotopy. ThenHP has the natural the structure ofa triangulated category whose distinguished triangles are triples inP , P → Q→ R→ P [1],which are isomorphic to a triple arising from a short exact sequence of chain complexes0→ P →Q→R→ 0. We similarly define the homotopy categoriesHPX ,HS,HSX .

LetD denote the localization ofP with respect to the thick subcategory of thoseP ∈ P withthe property that every stalk ofP is acyclic. So defined,D is isomorphic to the derived categoryof S. We similarly defineDX , a localization of bothPX andSX .

We say thatf :P →Q in eitherP ,PX,S, or SX is aquasi-isomorphismif each fibre of boththe kernel and cokernel off is acyclic.

DEFINITION. – We define the bivariant topological cycle cohomology ofX with values inYand weightr to be

Hj(X ;Y ,r) =: HomDX(ZX ,M(−;Y ,r)[j]

)whereM(−;Y ,r)(U ) =Mor(U ,Cr(Y ))̃ [2r].

We define the topological cycle cohomology ofX with supports inY ⊂X closed to be

HjY (X ,s)≡HomDX(ZX ,MY (−,s)[j]

).

In particular,

Hj(X ,s)≡HomDX(ZX ,M(−,s)[j]

).

We define the topological cycle homology ofX in degreei to be

Hi(X ,r)≡HomDX(ZX [i],L(−,r)

).

3. Topological cycle homology theory

We begin this section by verifying that topological cycle homology equals Lawson homology.This enables us to prove the Bloch–Ogus properties (including functoriality) of a “twistedhomology theory” for topological cycle homology theory. On the other hand, our definition interms of the derived category is so formulated to fit with topological cycle cohomology theory aswe shall see in later sections.


64 E.M. FRIEDLANDER

DEFINITION. –P ∈ PX is said to be pseudo-flasque provided that

P (U )→ P (U1)⊕ P (U2)→ P (U1∩U2)

is quasi-isomorphic to a distinguished triangle of chain complexes wheneverU1,U2 are Zariskiopens ofU ∈ ZarX with U = U1 ∪U2.

The following theorem is the derived category analogue of a theorem of Brown–Gerstenconcerning simplicial presheaves.

THEOREM 3.1. –Let HPX denote the homotopy category of the categoryPX of chaincomplexes(of presheaves of abelian groups on ZarX ). Then for anyP ∈ PX , anyi ∈ Z

Hi(P (X)

)= HomHPX

(ZX [i],P

).

If P ∈ PX is pseudo-flasque, then for anyi ∈ Z the natural map

HomHPX(ZX [i],P

)→HomDX

(ZX [i],P

)is an isomorphism.

Proof. –The equality Hi(P (X)) = HomHPX (ZX [i],P ) is essentially immediate.Let P → I in PX be a quasi-isomorphism withI injective (i.e.,Ii is an injective presheaf

on ZarX for eachi) and letT = cone{P → I}. Since an injective presheaf is flasque, theoctahedral axiom implies thatT is pseudo-flasque (assumingP is pseudo-flasque) as well asacyclic. Hence, the proof of [4, Theorem1′] implies thatT (U ) is acyclic for allU ∈ ZarX . Inparticular,P (X)→ I(X) is a quasi-isomorphism of chain complexes, so that

HomHPX(ZX [i],P

)= HomHPX

(ZX [i], I

)= HomDX

(ZX [i], I

)= HomDX

(ZX [i],P

).

2Theorem 3.1 and the localization theorem in Lawson homology easily imply the following

theorem.

THEOREM 3.2. –WheneverY ⊂ X is a closed subvariety, there is a “localization” distin-guished triangle inHPX

L(−∩ Y ,r)→L(−,r)→L(−∩ (X − Y ),r

)→L(−∩ Y ,r)[1](3.1)

for eachr > 0. Consequently, the presheafL(−,r) in HPX is pseudo-flasque, so that

Hi(X ,r) = Hi(L(X ,r)

)= LrHi(X),

whereLrHi(X) is the Lawson homology ofX as defined in earlier papers(e.g.,[7]).In particular,Hi(X ,r) = 0 for i < 0.

Proof. –For anyU ∈ ZarX , the localization theorem for Lawson homology [7, 1.6], [17]asserts that

L(U ∩ Y ,r)→L(U ,r)→L(U ∩ (X − Y ),r

)→L(U ∩ Y ,r)[1](3.2)

4e SÉRIE– TOME 33 – 2000 –N◦ 1


is a distinguished triangle of chain complexes. (We implicitly appeal to Proposition 2.2.) Thisimmediately implies that (3.1) is a distinguished triangle inPX .

To verify thatL(−,r) is pseudo-flasque, it suffices to show for anyU1⊂ U , U2⊂ U in ZarXwith U1 ∪U2 =U that

L(U ,r) L(U1,r)

L(U2,r) L(U1 ∩U2,r)

is homotopy Cartesian. This follows immediately from the localization sequence (3.2), sinceU −U2 = U1−U1− (U1∩U2). 2

As observed in [7], Lawson homology is contravariantly functorial for flat maps andcovariantly functorial for proper maps, thereby giving us the Bloch–Ogus properties (1.2) and(1.2.1) for topological cycle homology.

The following compatibility of this functorial behaviour is (a generalization of) the Bloch–Ogus property (1.2.2).

PROPOSITION 3.3. –Consider the following Cartesian square of varieties

X ′

p′

hX

p

Y ′g

Y

with flat horizontal maps of relative dimensione and proper vertical maps. Then for anyi ∈ Zandr > 0, the following square commutes

Hi(X ′,r+ e)

p′∗

Hi(X ,r)h∗

p∗

Hi(Y ′,r+ e) Hi(Y ,r)g∗

Proof. –By Theorem 3.2, it suffices to prove the commutativity of the following square

Cr+e(X′)/Cr+e(X

′ −X ′)

p′∗

Cr(X)/Cr(X −X)h∗

p∗

Cr+e(Y′)/Cr+e(Y

′ − Y ′) Cr(Y )/Cr(Y − Y )g∗

whereY ⊂ Y is a projective closure,X is the closure ofX in a projective closure ofX ×Y Y ,Y′

is the closure ofY ′ in a projective closure ofY ′ ×Y Y , andX′

is the closure ofX ′ in aprojective closure ofX ′ ×Y ′×YX Y

′ ×Y X . This follows directly from the definitions of flatpull-back and proper push-forward of cycles.2

Bloch–Ogus require of the localization sequence the following naturality, their property(1.2.4).


66 E.M. FRIEDLANDER

PROPOSITION 3.4. –Consider the following commutative square

Y ′

fY

X ′

f

Y X

whose horizontal maps are closed immersions, whose vertical maps are proper, and satisfyingthe condition thatfY (Y ′) = Y . Then the following is a commutative ladder of localization exactsequences

· · · Hi(Y ′,r)

fY ∗

Hi(X ′,r)

f∗

Hi(X ′− Y ′,r)

f∗◦g∗

· · ·

· · · Hi(Y ,r) Hi(X ,r) Hi(X − Y ,r) · · ·

whereg :X ′− f−1(Y )⊂X ′ − Y ′.

Proof. –Consider the following commutative diagram whose rows are distinguished trianglesof chain complexes

L(Y ′,r) L(X ′,r) L(X ′ − Y ′,r) L(Y ′,r)[1]

L(Y ,r) L(X ,r) L(X − Y ,r) L(X − Y ,r)[1].

It suffices to show that the compositionL(X ′,r)→ L(X ,r)→ L(X − Y ,r) factors throughL(X ′,r)→L(X ′ − Y ′,r). This in turn follows from the observation that

Cr(X′)→Cr(X )→Cr(X )/Cr

(X − (X − Y )

)factors throughCr(X

′)→ Cr(X

′)/Cr(X

′ − (X ′ − Y ′)), whereX ⊂X is a projective closureandX

′is the closure ofX ′ in a projective closure ofX ′×X X . 2

4. Topological cycle cohomology theory

We now proceed to consider topological cycle cohomology theory. The excision property(Theorem 4.3 below) justifies our formulation of this theory in terms of the derived categoryDX .

We begin with functoriality of topological cycle cohomology with supports.

PROPOSITION 4.1. –A Cartesian square

Y ′

fY

X ′

fX

Y X

4e SÉRIE– TOME 33 – 2000 –N◦ 1


of varieties whose horizontal arrows are closed immersions naturally induces a map of gradedgroups for allj ∈ Z, s> 0:

(fX ,fY )∗ :⊕j

HjY (X ,s)→⊕j

HjY ′ (X ′,s).

Frequently,fY will remain implicit,fX will be simply denotedf , and(fX ,fY )∗ byf∗.

Proof. –Sincef∗ :PX → PX′ is exact and preserves quasi-isomorphisms and sinceZX′ =

f∗ZX , we obtain a natural map

HomDX(ZX ,MY (−,s)[j]

)→HomDX′

(ZX′ ,f∗MY (−,s)[j]

).

The natural map

f∗M(−,C0(Ps)

)(U ′) ≡ colim

U ′⊂f−1(U)M(f−1(U ),C0(Ps)

)→M

(−,C0(Ps)

)(U ′)

induces a map inPX′ of presheaves of chain complexes onZarX′ , f∗MY (−,s)→MY ′ (−,s).Functoriality thus follows. 2

The next proposition is the assertion of the validity of the first two Bloch–Ogus properties forthe cohomology theory [3, 1.1.1, 1.1.2].

PROPOSITION 4.2. –LetZ → Y →X be closed immersions. Then for anys > 0, there is along exact sequence

· · ·HjZ (X ,s)→HjY (X ,s)→HjY−Z (X −Z,s)→Hj+1Z (X ,s) · · ·

Moreover, ifZ ′→ Y ′→X ′ is another sequence of closed immersions and if

Z ′

fZ

Y ′

fY

X ′

fX

Z Y X

(4.1)

is a commutative diagram with Cartesian squares, then(fX ,fZ)∗, (fX ,fY )∗, and(fX|,fY |)∗ fittogether to form a commutative ladder of long exact sequences, where

fX|,fY | : (X ′−Z ′,Y ′ −Z ′)→ (X −Z,Y −Z)

is the restriction offX ,fY .


68 E.M. FRIEDLANDER

Proof. –We consider the following commutative diagram inPX whose rows are distinguishedtriangles

M(−,s)

=

M(−∩ (X −Z),s) MZ(−,s)[−1]

M(−,s) M(−∩ (X − Y ),s) MY (−,s)[−1]

0 MX−Y (−∩ (X −Z),s)[−1] = MX−Y (−∩ (X −Z),s)[−1]

The octahedral axiom implies that the right vertical column is also a distinguished triangle,thereby implying the asserted long exact sequence. The naturality of this construction impliesthat the commutative diagram (4.1) determines a map of distinguished triangles and thus acommutative ladder of exact sequences.2

The following excision theorem is the Bloch–Ogus property (1.1.3). This is the property whichrequires our somewhat sophisticated definition of topological cycle cohomology in terms of mapsin the derived categoryDX of bounded complexes of presheaves.

THEOREM 4.3. –Let Y ⊂X be a closed subvariety andi :U ⊂X be a Zariski open subsetcontainingY . Then the natural map

HjZ (X ,s)→HjZ (U ,s)

is an isomorphism for allj ∈ Z,s> 0.

Proof. –Consider the following commutative square inPX :

M(−,s)

p

M(−∩U ,s)

pU

M(−∩ (X − Y ),s) M(−∩ (U − Y ),s).

If V ∈ ZarX is sufficiently fine thatV is either contained inU orX − Y , then the evaluation ofthis square onV leads either to a square whose horizontal maps are equalities of chain complexesor to a square whose vertical maps are equalities. This immediately implies that the map

MZ(−,s) = cone(p)[1]→ cone(pU )[1] =MZ(−,s)

is a quasi-isomorphism and hence the statement of the theorem.2

5. Cap product and duality

In this section, we relate our cohomology and homology theories via a cap product and verifythe final property of Bloch–Ogus: Poincaré duality for a smooth variety relating topologicalcycle cohomology with supports to topological cycle homology. Continuity of the cap productfor possibly singular varieties is somewhat subtle; the necessary arguments can be found in [14].On the other hand, the duality theorem (for smooth varieties) is a direct consequence of duality

4e SÉRIE– TOME 33 – 2000 –N◦ 1


proved in [6] following [9]. As a consequence, we show that topological cycle cohomology of asmooth variety equals morphic cohomology.

We begin by sketching the construction of cap product, based on techniques developped in[14].

PROPOSITION 5.1 (cf. [14, 2.6]). –There is a natural pairing of presheaves inPX

M(−,s)[2s]⊗L(−,r)→L(−×As,r),(5.1)

whereL(−×As,r) sendsU ∈ ZarX to

cone{Cr(U∞ ×Ps ∪X ×Ps−1

)̃→Cr

(X ×Ps

)̃ }[2r].

(Here,X ⊂X is a projective closure andU∞ =X −U .)

Proof. –ForU ∈ ZarX , we consider the pairing

Γ()| :Mor(U ,C0(Ps)

)× Cr(U )→C(U ×Ps)(5.2)

which sends the pair (f :U →C0(Ps), iW :W ⊂ U ) to the graphΓf |W ⊂ U × Ps of iW ◦ f forany irreducibler-cycleW on U . The continuity of this pairing is established (in much greatergenerality) in [14, 2.6].

The naturality of (5.2) with respect toU andPs−1 ⊂ Ps implies that this pairing induces apairing of the form (5.1). 2

Using the localization property of Lawson homology, we obtain the following cap productpairing. We implicitly use thehomotopy invarianceproperty of Lawson homology [7] whichasserts that flat pull-back

L(−,r− s)[2s]→L(−×As,r)(5.3)

is a quasi-isomorphism.

PROPOSITION 5.2. –If Y ⊂X is closed, then the pairing(5.1) induces a natural pairing

MY (−,s)⊗L(−,r)→L(−∩ Y ,r− s)

in DX which yields the following cap product pairing on cohomology/homology

∩ :HjY (X ,s)⊗Hn(X ,r)→Hn−j(Y ,r− s)

wheneverr > s> 0, n> j.Proof. –The naturality of (5.1) enables us to obtain the following commutative diagram inPX

whose columns are distinguished triangles:

MY (−,s)⊗L(−,r) L(−∩ Y ×As,r)[−2s]

M(−,s)⊗L(−,r) L(−×As,r)[−2s]

M(−∩ (X − Y ),s)⊗L(−,r) L(−∩ (X − Y )×As,r)[−2s]

(5.4)


70 E.M. FRIEDLANDER

The asserted pairing on cohomology/homology is obtained from the top row by applying thefollowing sublemma in conjunction with Remark 2.1.2

SUBLEMMA . – A pairingP ⊗Q→R of presheaves inPX induces a pairing

HomDX(ZX [i],P

)⊗HomDX

(ZX [j],Q

)→HomDX

(ZX [i+ j],R

),

provided thatQ(U ) is a complex of abelian groups without torsion for everyU ∈ ZarX .

Proof. –If P → P , Q → Q are quasi-isomorphisms inPX with P ,Q projective, thenP ⊗Q→ P ⊗Q is also a quasi-isomorphism since eachQ(U ) is torsion free. Thus, the assertedpairing is given by the natural pairing

HomHPX(ZX [i],P

)⊗HomHPX

(ZX [j],Q

)→HomHPX

(ZX [i+ j],P ⊗Q

).

2Observe that ifX is an irreducible variety of dimensiond, thenX itself viewed as an effective

d-cycle determines a point ofCd(X) and thus afundamental class

ηX ∈(L(X ,d)

)2d.(5.5)

(By an abuse of notation, we shall also useηX to denote the associated homology class inH2d(L(X ,d)).) Moreover, as required by Bloch–Ogus (1.3.4),α∗(ηX) = ηX′ wheneverα :X ′→X is an etale morphism because the flat pull-back of thed-cycleX onX is thed-cycleX ′ onX ′.

Observe that ifX is an irreducible variety of pure dimensiond, then the pairing of (5.2)restricted toηX ∈ (L(X ,d))2d,

−∩ ηX :M(−,s)→L(−,d− s)[−2d],

is induced by sending a continuous algebraic mapf :U →C0(Ps) to its graphΓf ∈ Cd(U × Ps)for anyU ∈ ZarX . Hence, wheneverX satisfies the condition that topological cycle cohomologyequals morphic cohomology (e.g.,X smooth by Corollary 5.6 below), the induced map is theduality map of [6,9]

D :Hj(X ,s)→H2d−j(X ,d− s).(5.6)

PROPOSITION 5.3. –Consider the following Cartesian square of varieties

Y ′

β

X ′

α

Y X

whose horizontal arrows are closed immersions and whose vertical arrows are etale. Then thefollowing square commutes

HjY (X ,s)⊗Hn(X ,r)

α∗⊗α∗

Hn−j(Y ,r− s)

β∗

HjY ′ (X ′,s)⊗Hn(X ′,r) Hn−j(Y ′,r− s)

for all j,n ∈ Z, r > s> 0.

4e SÉRIE– TOME 33 – 2000 –N◦ 1


Proof. –Becauseα∗ is exact, preserves quasi-isomorphisms, and sendsZX ∈ SX to ZX′ ∈S ′X , there is a natural map HomDX (ZX [i],F )→HomDX′ (ZX′ [i],α

∗F ) for anyF ∈ PX . Thus,it suffices to prove the commutativity of the following square inDX′ :

α∗MY (−,s)⊗ α∗L(−,r)

α∗⊗α∗

∩α∗L(−∩ Y ,r)

β∗

MY ′ (−s)⊗L(−,r) ∩ L(−∩ Y ′,r)

(5.7)

Using diagram (5.4), we readily see that it suffices to takeY = X , Y ′ = X ′ and to provethe commutativity inPX′ of this special case of (5.7). To prove this, it suffices to prove thecommutativity of the following square of topological abelian monoids whose horizontal mapsare given by (5.2)

Mor(U ,C0(Ps))× Cr(U )

◦α◦j, α∗◦j∗

Γ()|− Cr(U ×Ps)

α∗◦j∗

Mor(U ′,C0(Ps))× Cr(U ′)Γ()|− Cr(U ′ ×Ps)

(5.8)

wherej :U ′ ⊂ α−1(U ). Finally, the commutativity of (5.8) is verified by observing that the graphof f ◦α ◦ j :U ′→ U →C0(Ps) restricted toZ ′ ≡ U ′×X Z , Γf◦α◦j|Z′ ∈ Cr(U ′×Ps), equals thepull-back via (α ◦ j × 1) :U ′×Ps→ U ×Ps of Γf |Z ∈ Cr(U ×Ps). 2

The following “projection formula” is the Bloch–Ogus property (1.3.3).

PROPOSITION 5.4. –Consider the Cartesian square of varieties

Z

g

W

f

Y X

whose horizontal maps are closed immersions and whose vertical maps are proper. Then for anya ∈HjY (X ,s), z ∈Hn(W ,r),

a∩ f∗(z) = g∗(f∗(a)∩ z

).

Proof. –Observe that Theorem 3.1 implies that

HomDW(ZW [i], f∗L(−,r)

)= Hi

(f∗L(W ,r)

)= Hi

(L(X ,r)

)=Hi(X ,r)

sincef∗L(−,r)∈ PW is pseudo-flasque. Hence, as in the proof of Proposition 5.3, it suffices toconsider the diagram inPW of natural maps

M(−,s)⊗ L(−,r)

f∗

∩ L(−×As,r)[−2s]

f∗

f∗M(−,s)⊗

f∗

f∗L(−,r) ∩f∗L(−×As,r)[−2s]

(5.9)


72 E.M. FRIEDLANDER

HomDW (ZW [i],M(−,s))⊗ HomDW (ZW [j],L(−,r)) HomDW (ZW [i+ j],L(−,r))

HomDW (ZW [i], f∗M(−,s))⊗HomDW (ZW [j], f∗L(−,r)) HomDW (ZW [i+ j], f∗L(−,r))

HomDX (ZX [i],M(−,s))⊗ HomDX (ZX [j],L(−,r))

=

HomDX (ZX [i+ j],L(−,r))

=

DIAGRAM D.

and to show for anyV ∈ ZarW , anyα ∈ f∗M(V ,s), anys ∈ L(V ,r) that

α ∩ f∗(z) = f∗(f∗α ∩ z).

This will provide the “commutativity” of Diagram (D). To analyze (5.9), it suffices to considerthe following diagram of topological abelian monoids for anyU ∈ ZarX , V ∈ ZarW withi :V ⊂ f−1(U ):

Mor(V ,C0(Ps))×Cr(V )

f∗◦i∗

Cr(V ×Ps)

(f ×1)∗◦(i×1)∗

Mor(U ,C0(Ps))×

◦f◦i

Cr(U ) Cr(U ×Ps)

(5.10)

We readily verify for anyZ ∈ Cr(V ), φ ∈Mor(U ,C0(Ps)) that

(f × 1)∗ ◦ (i× 1)∗(Γφ◦f◦i|Z) = Γφ|f∗i∗(Z).

We interpret this equality as asserting the “commutativity” of (5.10) and thus the analogous“commutativity” of (5.9). 2

The next Bloch–Ogus property is the following assertion of Poincaré duality.

THEOREM 5.5. –Assume thatX is a smooth variety of dimensiond andY ⊂X is a closedsubvariety. Then cap product with the fundamental classηX ∈H2d(L(X ,d))

−∩ ηX :HjY (X ,s)→H2d−j(Y ,d− s)

is an isomorphism.

Proof. –The duality theorem of [6,9] implies that

−∩ ηX :M(−,s)[2s]→L(−×As,d)[−2d]

is a quasi-isomorphism. Applying homotopy invariance (cf. (5.3)), we conclude

−∩ ηX :M(−,s)→L(−,d− s)[−2d](5.11)

4e SÉRIE– TOME 33 – 2000 –N◦ 1


is a quasi-isomorphism. This extends to a map of distinguished triangles

MY (−,s) L(−∩ Y ,d− s)[−2d]

M(−,s) L(−,d− s)[−2d]

M(−∩U ,s) L(−∩U ,d− s)[−2d].

whose horizontal maps are quasi-isomorphisms, whereU =X − Y . 2The quasi-isomorphism (5.11) in conjunction with Corollary 3.2 immediately implies the

following equality of morphic cohomology and topological cohomology for smooth varieties.

COROLLARY 5.6. –If X is a smooth variety andY a closed subvariety, then the presheafMorY (−,s) in SX is pseudo-flasque. In particular, ifX is smooth, then

Hj(X ,s)' LsHj(X),

where the right hand side is morphic cohomology as defined in earlier papers.

The final Bloch–Ogus property [3, 1.5] verifies the triviality of the cycle class of a principaldivisor.

PROPOSITION 5.7. –If X is a smooth variety of pure dimensiond and if i :W → X isa principal divisor, then the imagei∗(ηW ) ∈ Zd−1(X) of the fundamental class ofW ishomologically trivial:

i∗(ηW ) = 0 ∈ H2d−2(X ,d− 1).

Proof. –We recall the existence of a Gysin mapi!W :Zr(X)→ Zr−1(W ) whose compositionwith i∗ :Zr(W )→ Zr(X) depends (in the derived category) only upon the line bundle of whichW is the zero locus of a global section [7, 2.4]. IfW is principal, thenL is the trivial line bundleso that this composition is 0. On the other hand,i!W (ηX ) = ηW [7, 2.4]. 2

6. Relationship to singular cohomology/homology

As observed in previous papers (esp., [8,12]), there are natural maps from morphiccohomology on normal varieties to singular cohomology and from Lawson homology to Borel–Moore homology. In this section, we confirm that these maps are naturally formulated in ourcontext of derived categories of presheaves and thereby extend to our more general setting. Aswe see, these maps are often compatible with cap product in singular homology/cohomologytheory (a result generalizing the main result of [11]).

We refer the reader to [11] for a quick sketch of integral cycles, Lipschitz neighborhoodretracts, and related matters.

DEFINITION. – For any varietyX , we denote by

ε : AnX → ZarX


74 E.M. FRIEDLANDER

the natural morphism of sites, where AnX is the site whose objects are analytic open subsets ofX and whose morphisms are inclusions.(The functorε is associated to the fact that every Zariskiopen sbset ofX is also an analytic open subset.)

We letPanX denote the triangluated category of presheaves on AnX with values in the abelian

category of chain complexes bounded below of abelian groups. We denote byDanX the localization

ofPanX by the thick subcategory of thoseP ∈ Pan

X each fibre of which is acyclic.

Observe that

ε∗ :PX →PanX

is an exact functor which induces

ε∗ :DX →DanX

since each fibre ofε∗P ∈ PanX is acyclic whenever each fibre ofP ∈ PX is acyclic.

DEFINITION. – We define the presheaf

Hom(−, 2s) =: U 7→ cone{

Homcont(U ,Z0(Ps−1)

)̃→Homcont

(U ,Z0(Ps)

)̃ }[−2s]

whereHomcont(U ,P ) denotes the mapping space of continuous maps fromU to P (each withthe analytic topology) given the compact-open topology and whereZ0(P ) = C0(P )+ denotes thenaïve topological group completion of the topological monoidC0(P ).

We define the presheaf

Z(−,m) =:U 7→(Zm(X )̃ /Zm(X −U )

)̃[m]

whereX ⊂X is a projective closure and whereZm(Y ) denotes the topological abelian groupof integralm-cycles onY (in the sense of geometric measure theory) provided with the flat-normtopology.

The following proposition interprets singular cohomology/homology in terms similar to ourformulation of topological cycle theory.

PROPOSITION 6.1. –For any varietyX , the presheaves

Hom(−, 2s), Z(−,m)∈ PanX

are pseudo-flasque. Moreover, for anyU ∈AnX ,

HomDanX

(ZX ,Hom(−, 2s)[j]

)= H−j

(Hom(U , 2s)

)= Hj(X), j 6 2s,(6.1)

HomDanX

(ZX [i],Z(−,m)

)= Hi

(Z(U ,m)

)= HBMi (U ), i>m,(6.2)

whereH∗(U ) andHBM∗ (U ) denote the singular cohomology and Borel–Moore homology ofU(considered with its analytic topology).

Proof. –Let K(Z,n) denote the Eilenberg–MacLane space with unique non-vanishinghomotopy groupZ in degreen, represented for example by the infinite symmetric powerSP (Sn)of then-sphere. SinceZ0(Ps−1)→Z0(Ps) can be identified with the natural embedding

s−1∏i=0

K(Z, 2i)→s∏i=0

K(Z, 2i),

4e SÉRIE– TOME 33 – 2000 –N◦ 1


Hom(−, 2s)[2s] is isomorphic inDanX to the presheafU 7→Homcont(U ,K(Z, 2s)). As in the proof

of Proposition 3.2, in order to prove (6.1) it suffices to show that

Homcont(U ,K(Z, 2s)) Homcont(U1,K(Z, 2s))

Homcont(U2,K(Z, 2s)) Homcont(U3,K(Z, 2s))

(6.3)

is homotopy Cartesian, wheneverU1,U2⊂ U in AnX with U = U1 ∪U2,U3 = U1 ∩U2. Since

πj(Homcont

(U ,K(Z, 2s)

))= H2s−j(U ),

the fact that (6.3) is homotopy Cartesian (and the validity of (6.1)) follows immediately fromMayer–Vietoris for singular cohomology.

A well known theorem of F. Almgren’s [1] implies that

πi(Zm(X )

)= Hi+m(X ).

As above, Mayer–Vietoris (this time for singular homology) implies thatZ(−,m) is pseudo-flasque and provides the equalities of (6.2).2

PROPOSITION 6.2. –For any varietyX , there are naturally constructed maps inPanX

ε∗M(−,s)→Hom(−, 2s)

ε∗L(−,r)→Z(−, 2r)

which determine natural maps

Φ∗ :Hj(X ,s)→Hj(X), Φ∗ :Hi(X ,r)→HBMi (X).

Proof. –By Proposition 6.1 and the fact thatε∗ is exact and preserves weak equivalences, inorder to constructΦ∗ andΦ∗ it suffices to exhibit the asserted maps inPan

X . For this, it suffices toexhibit natural maps for anyU ∈ ZarX

M(U ,s)→Hom(U , 2s), L(U ,r)→Z(U , 2r).

The first is induced by the natural inclusionMor(U ,C0(P ))→ Homcont(U ,Z0(P )) (induced byC0(P )→C0(P )+ = Z0(P )) and the second by the natural inclusionZr(X )⊂Z2r(X ). 2

The “duality map” relating morphic cohomology and Lawson homology is induced by theconstruction which sends a continuous algebraic map to its graph (cf. [9]). In view of (5.6), thiscan be interpreted as cap product in topological cycle theory with the fundamental class ofX .The basic result of [11] (namely, [11, 5.5]) is the assertion that ifX is a projective, normal varietythen this duality map is compatible with cap product with the fundamental class from singularcohomolgy to singular homology. The following compatibility theorem (more general than thatof [11]) is an easy consequence of [11, 5.5] in conjunction with Proposition 5.4.

PROPOSITION 6.3. –Let X be a projective variety. Then the natural maps of(6.2) arecompatible with cap product with fundamental classes of algebraic cycles in the sense that the


76 E.M. FRIEDLANDER

following square commutes

Hj(X ,s)⊗H2r(X ,r)

Φ∗⊗Φ∗

∩ H2r−2s+j(X ,r− s)

Φ∗

Hj(X)⊗H2r(X)∩ H2r−2s+j(X)

(6.4)

for r > s> 0, j 6 2s.

Proof. –Let j :Y ⊂ X be the closed embedding of an irreducible subvariety of dimensionr, let clX (Y ) ∈ H2r(L(X ,r)) denote its associated cycle class, and let [Y ] denoteclY (Y ).Let p :Y˜ → Y denote the normalization ofY and observe thati∗([Y˜ ]) = clX (Y ), wherei = j ◦ p :Y˜ → X . Then the commutativity of (6.4) follows from the following chain ofequalities for anyα ∈Hj(X ,s):

Φ∗(α ∩ clX (Y )

)= Φ∗

(i∗(i∗α ∩ [Y˜]

))= i∗

(Φ∗(i∗α)∩Φ∗([Y˜])

)= Φ∗(α)∩Φ∗

(clX (Y )

)where the first equality follows from Proposition 5.4, the second from [11, 5.5], and the thirdfrom the analogue in singular theory of Proposition 5.4.2

7. Local to global spectral sequence

Since we have shown that topological cycle theory satisfies all of the properties Bloch andOgus require of a “Poincaré duality theory with supports”, their analysis implies the validityof “Gersten’s Conjecture” for topological cycle cohomology (Theorem 7.1) and thus a local toglobal spectral sequence (Theorem 7.2).

We denote byHj(s) the sheaf associated to the presheaf onZarX sendingU toHj(U ,s).

THEOREM 7.1 (cf. [3, 4.2.2]). –For any smooth varietyX , there is a resolution of sheaves onZarX

0→Hj(s)→⊕

codim(x)=0

ixHj(Speck(x),s

)→

⊕codim(x)=1

ixHj−1(Speck(x),s− 1

)→ · · ·→

⊕codim(x)=j

ixH0(Speck(x),s− j

)→ 0,

whereixA denotes the extension by zero from{x} to X of the constant sheaf with value theabelian groupA on the closure{x} of the(Zariski) pointx ∈X .

THEOREM 7.2 (cf. [3, 6.2]). –Let X be a smooth variety. Then there is a natural spectralsequence

Ep,q2 = HpZar

(X ,Hq(s)

)⇒Hp+q(X ,s).

Moreover, the filtration{F pHn(X ,s)} onHn(X ,s) associated to this spectral sequence is thearithmetic filtration given by

F pHn(X ,s) =⋃Z

Ker{Hn(X ,s)→Hn(X −Z,s)

},

where the union is taken over all closed subvarietiesZ ⊂X of codimensionp.

4e SÉRIE– TOME 33 – 2000 –N◦ 1


We give one corollary of this theorem, a vanishing result confirming in this special case theexpected vanishing range of Lawson homology. Namely, one expects that ifX is a smooth varietyof dimensiond, thenHi(X ,∗) = 0 for i > 2d. By duality, this is equivalent to the expectationthat topological cycle cohomology should vanish in negative degrees.

COROLLARY 7.3. –LetX be a smooth, rational3-fold. Then

Hj(X , 2)= 0, j < 0.

Proof. –By Theorem 7.2, it suffices to prove thatHj(U , 2)= 0, j < 0, for all sufficiently smallU ∈ ZarX . SinceX is rational, this is implied by the assertion thatHj(V , 2)= 0, j < 0, for allV ⊂ P3. By duality (i.e., Theorem 5.5), it suffices to verify that Hi(L(V , 1))= 0, i > 6. Usingthe localization sequence (associated to the distinguished triangle of Theorem 3.2) in conjunctionwith the vanishing Hi(L(P3, 1))= 0, i > 6, as shown in [16], we conclude that it suffices to provethat Hi(L(Y , 1))= 0, i > 6, for any closed proper subvarietyY ⊂ P3.

Indeed, we show that Hi(L(Y , 1)) = 0, i > 4, for any closed proper subvarietyY ⊂ P3.Applying localization to the closed embedding of the singular locusYsing⊂ Y with Zariskiopen complementYns and observing thatZ1(Ysing) is discrete, we conclude that it suffices toassume thatY is smooth and possibly quasi-projective. Applying resolution of singularities tothe projective closureY of Y and localization once again, we conclude that it suffices to recallthe computation of Lawson homology for divisors on a smooth, projective variety given in [6,4.6]. 2

The proof of Corollary 7.3 shows somewhat more. Namely, the vanishing of topological cyclecohomology in negative dimensions and weight two less than the dimension ofX depends onlyupon the birational class ofX .

8. Gillet’s axioms

In this final section, we sketch how our topological cycle theory satisfies the axioms of H.Gillet [15], thereby presumably implying the existence of chern classes with values in topologicalcycle cohomology theory and a Riemman–Roch formula.

Gillet requires a graded complexΓ ∗(∗) of sheaves on the big Zariski siteZar. We take

Γ ∗(s) =M(−,s)Zar

the sheafification of the complexM(−,s) of presheaves onZar. Gillet further requires theexistence of an associative, (graded) commutative pairing

Γ ∗(∗)L⊗ Γ ∗(∗)→ Γ ∗(∗).

Such a pairing is established in [14, 3.1], using the join pairing of [8].Gillet then postulates a list of axioms [15, 1.2]. The first is given by Proposition 3.3, the second

by Proposition 3.4, and the third by Proposition 5.4. The existence of a functorial fundamentalclassηX is stated in (5.3), whereas Theorem 5.5 provides the duality isomorphism required inGillet’s fifth axiom.

Gillet’s sixth axiom is the condition that the duality isomorphism of Theorem 5.5 arises froma map of complexes (in the derived category); indeed, this is shown in the proof of Theorem 5.5.


78 E.M. FRIEDLANDER

Similarly, his seventh axiom is the condition that the projection formula arise from a diagram ofcomplexes as shown in the proof of Theorem 5.5. There is no requirement given in axiom eight.

Gillet’s ninth axiom, (1.2.ix), is the condition of homotopy invariance for cohomology. SinceLawson homology is homotopy invariant [7, 2.3], this homotopy invariance for topological cyclecohomology of smooth varieties is valid thanks to duality. More generally, this axiom does notappear to hold.

Axiom ten follows from the projective bundle theorem of [7, 2.5]. The final Gillet axiom,(1.2.xi) is essentially the existence of a first Chern classPic(−)→H2(−, 1). For a line bundleL generated by its global sections,c1(L) was exhibited in [8]; more generally, any line bundleLon a quasi-projective varietyX locally closed in some projective spacePN has the property thatL⊗O(m) is generated by its global sections form sufficiently large, so that we may define

c1(L) = c1(L⊗O(m)

)− c1

(O(m)

).

A more precise version of Gillet’s first Chern class axiom is formulated in terms of theexistence of a morphismO∗[−1]→ Γ ∗(1) in DX such that [OP1(1)] ∈ H2(P1,O∗[−1]) is sentto c1(OP1). Using [13], we observe thatO∗[−1] can be realized inDX as the sheaf of chaincomplexes associated to the group completion of the sheaf of simplicial abelian moniods

U 7→Mor(U × ∆•,C0(P1)

)/Mor

(U × ∆•,C0(P0)

).

Thus, the natural map from the algebraic singular complex to the topological singular complexof the topological abelian monoidMor(U ,C0(P1)) determines a natural map fromO∗[−1] toΓ ∗(1).

REFERENCES

[1] F.J. ALMGREN, JR., Homotopy groups of the integral cyclic groups,Topology1 (1962) 257–299.[2] D. BARLET, Espace Analytique Réduit des Cycles Analytiques Complexes Compacts d’un Espace

Analytique Complexe de Dimension Finite, Fonctions de Plusieurs Variables, II, Lecture Notes inMath., Vol. 482, Springer, Berlin, 1975, pp. 1–158.

[3] S. BLOCH and A. OGUS, Gersten’s conjecture and the homology of schemes,Ann. Éc. Norm. Sup.7(1974) 181–202.

[4] K. B ROWN and S. GERSTEN, AlgebraicK-Theory as Generalized Sheaf Cohomology, AlgebraicK-Theory, I, Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 266–292.

[5] E. FRIEDLANDER, Algebraic cocycles, Chow varieties, and Lawson homology,Compositio Math.77(1991) 55–93.

[6] E. FRIEDLANDER, Algebraic cocycles on normal, quasi-projective varieties,Compositio Math.110(1998) 127–162.

[7] E. FRIEDLANDER and O. GABBER, Cycles spaces and intersection theory,Topological Methods inModern Mathematics(1993) 325–370.

[8] E. FRIEDLANDER and H.B. LAWSON, A theory of algebraic cocycles,Annals of Math.136 (1992)361–428.

[9] E. FRIEDLANDER and H.B. LAWSON, Duality relating spaces of algebraic cocycles and cycles,Topology36 (1997) 533–565.

[10] E. FRIEDLANDER and H.B. LAWSON, Moving algebraic cycles of bounded degree,Inventiones Math.132 (1998) 91–119.

[11] E. FRIEDLANDER and H.B. LAWSON, Graph mappings and Poincaré duality.[12] E. FRIEDLANDER and B. MAZUR, Filtration on the Homology of Algebraic Varieties, Memoirs of

AMS, Vol. 529, 1994.

4e SÉRIE– TOME 33 – 2000 –N◦ 1


[13] E. FRIEDLANDER and V. VOEVODSKY, Bivariant cycle cohomology, in: V. Voevodsky, A. Suslin andE. Friedlander, eds.,Cycles, Transfers, and Motivic Homology Theories, Annals of Math. Studies,1999.

[14] E. FRIEDLANDER and M. WALKER, Function spaces and continuous algebraic pairings for varieties,Compositio Math.(to appear).

[15] H. GILLET ,Riemann–Roch theorems for higher algebraicK-theory,Adv. in Math.40 (1981) 203–289.[16] H.B. LAWSON, Algebraic cycles and homotopy theory,Annals of Math.129 (1989) 253–291.[17] P. LIMA -FILHO, Completions and fibrations for topological monoids,Trans. Amer. Math. Soc.340

(1993) 127–147.[18] A. SUSLIN and V. VOEVODSKY, Chow sheaves, in: V. Voevodsky, A. Suslin and E. Friedlander, eds.,

Cycles, Transfers, and Motivic Homology Theories, Annals of Math. Studies, 1999.

(Manuscript received May 11, 1998.)

Eric M. FRIEDLANDER

Department of Mathematics,Northwestern University, Evanston,

IL 60208-2730, USAe-mail: [email protected]


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Bloch–Ogus properties for topological cycle theory