Motives Over Simplicial Schemes

Home , Conservative functor, Smooth scheme

Motives over simplicial schemes

VLADIMIR VOEVODSKY

Abstract

This paper was written as a part of [8] and is intended primarily to provide the deﬁnitions and results concerning motives over simplicial schemes, which are used in the proof of the Bloch-Kato conjecture.

Key Words: relative motives, motivic cohomology, embedded simplicial schemes Mathematics Subject Classiﬁcation 2010: 14F42, 19E15

Contents

1 Introduction 1

2 Presheaves with transfers 3

3 Tensor structure 9

4 Relative motives 13

5 Relative Tate motives 16

6Embeddedsimplicialschemes 25

7 Coefﬁcients 33

8 Appendix: Internal Hom-objects 33

References 37

1. Introduction

For the purpose of this paper a scheme means a possibly inﬁnite disjoint union of separated noetherian schemes of ﬁnite dimension. A smooth scheme over a scheme 2V. VOEVODSKY

S is a disjoint union of smooth separated schemes of finite type over S.Asmooth simplicial scheme over S is a simplicial scheme such that all terms of are X X smooth schemes over S and all morphisms are over S. If is a smooth simplicial scheme over a field k then the complex of presheaves X with transfers defined by the simplicial presheaf with transfers Ztr. / gives an X object M. / in the triangulated category of motives DM eff .k/ over k. The X motivic cohomology of this object is called the motivic cohomology! of and we X denote its cohomology groups by p;q H . ;A/ HomDM .M. /;A.q/Œp/ X WD X where A is an abelian group of coefficients. The main goal of this paper is to define for any smooth simplicial scheme X over a perfect field k,atensortriangulatedcategoryDM eff . / such that ! X p;q H . ;A/ HomDM eff . /.Z;A.p/Œq/: (1.1) X D X For completeness we make our construction of DM eff . / for the case of a X general simplicial scheme and in particular we provide! a definition of “motivic cohomology” of simplicial schemes based on (1.1). If the terms of are not regular, X there are examples which show that the motivic cohomology defined by (1.1) does not satisfy the suspension isomorphism with respect to the T -suspension (which implies that they do not satisfy the projective bundle formula and do not have the Gysin long exact sequence). Therefore in the general case we have to distinguish the “effective” motivic cohomology groups given by (1.1) and the stable motivic cohomology groups given by p;q Hstable. ;A/ limHomDM eff . /.Z.n/;A.n q/Œp/: (1.2) X WD n X C The stable motivic cohomology groups should also have a description as morphisms between Tate objects in the properly defined T -stable version of DM and should have many good properties including the long exact sequence for blow-ups, which the unstable groups in the non-regular case do not have. If the terms of are regular schemes of equal characteristic then the can- X cellation theorem over perfect fields implies that this problem does not arise and the stable groups are same as the effective ones (see Corollary 5.5). Since in applications to the Bloch-Kato conjecture we need only the case of smooth schemes over a perfect field, we shall not consider stable motivic cohomology in this paper. Note also that while we use schemes which are smooth over a base as the basic building blocks of motives over a base, one can also consider all (separated) schemes instead, as done in [7] and [4]. As far as the constructions of this paper are concerned, this make no difference except that the resulting motivic category gets bigger. Motives over simplicial schemes 3

2. Presheaves with transfers

Let be a simplicial scheme with terms Xi , i 0.Foramorphism" Œj Œi in X ! W ! we let ! denote the corresponding morphism i j .DenotebySm= the X X ! X X category deﬁned as follows: 1. An object of Sm= is a pair of the form .Y;i/ where i is a non-negative X integer and Y i is a smooth scheme over i . ! X X 2. A morphism from .Y;i/ to .Z;j / is a pair .u;"/ where " Œj Œi is a W ! morphism in and u Y Z is a morphism of schemes such that the square W ! u Y Z """"! (2.1)

! ? X ? ?i ?j Xy """"! Xy commutes. A presheaf of sets on Sm= is a contravariant functor from Sm= to sets. Each X X presheaf F on Sm= deﬁnes in the obvious way a famlily of presheaves Fi on X Sm= i together with natural transformations F! .Fj / Fi for all morphisms X W X!" ! " Œj Œi in . W ! One can easily see that this construction provides a bijection between presheaves on Sm= and families .Fi ;F!/ such that FId Id and the obvious compatibility X D condition holds for composable pairs of morphisms in .Underthisbijectionthe presheaf h.Y;i/ represented by .Y;i/ has as its j -th component the presheaf

.h.Y;i//j hY ! j D # X ! a where " runs through the morphisms Œi Œj in . ! Our first goal is to develop an analog of this picture where the presheaves of sets are replaced with presheaves with transfers. Let us recall first the basic notions for presheaves with transfers over usual schemes. For a scheme X denote by SmCor.X/ the category whose objects are smooth schemes over X and whose morphisms are finite correspondences over X (in the case of a non-smooth see X [7] for a detailed definition of finite correspondences and their compositions). Note that we allow schemes which are infinite disjoint unions of smooth schemes of finite type to be objects of SmCor.X/.InparticularourSmCor.X/ has infinite direct sums. A presheaf with transfers on Sm=X is an additive contravariant functor from SmCor.X/ to abelian groups which takes infinite direct sums to products. 4V. VOEVODSKY

Presheaves with transfers form an abelian category PST .X/. The forgetful functor from presheaves with transfers to presheaves of sets has a left adjoint which we denote by Ztr. /.IfY is a smooth scheme over X and hY is the presheaf of " sets represented by Y then Ztr.hY / coincides with the presheaf with transfers represented by Y on SmCor.X/ and we denote this object by Ztr.Y /. It will be convenient for us to identify SmCor.X/ with its image in PST .X/ and to denote the object of SmCor.X/ corresponding to a smooth scheme Y over X by Ztr.Y /. A morphism of schemes f Y X deﬁnes the pull-back functor W !

Ztr.U / Ztr.U X Y/ 7! # from SmCor.X/ to SmCor.Y/ and therefore a pair of adjoint functors f ;f " " between the corresponding categories of presheaves with transfers. Since f " commutes with the forgetful functor, we conclude by adjunction that for a presheaf of sets F over X one has

Ztr.f ".F // f ".Ztr.F //: (2.2) D It is not necessarily true that pull-back functors on presheaves of sets and on presheaves with transfers commute with the forgetful functor. Deﬁnition 2.1 Let be a simplicial scheme. A presheaf with transfers on is the X X following collection of data:

1. For each i 0, a presheaf with transfers Fi on Sm= i . ! X 2. For each morphism " Œj Œi in the simplicial category ,amorphismof W ! presheaves with transfers

F! ".Fj / Fi : W X! !

These data should satisfy the condition that Fid Id and for a composable pair of D morphisms " Œj Œi, Œk Œj in , the obvious diagram of morphisms of W ! W ! presheaves commutes. We let PST. / denote the category of presheaves with transfers on .Thisis X X an abelian category with kernels and cokernels computed termwise.

Example 2.2 Let X be a scheme and asimplicialschemesuchthat i X for all X X D i and all the structure morphisms are identities. Then a presheaf with transfers over is the same as a cosimplicial object in the category of presheaves with transfers X over X. Motives over simplicial schemes 5

Let F .Fi ;F!/ be a presheaf of sets on Sm= . In view of (2.2), the collection D X of presheaves with transfers Ztr.Fi / has the natural structure of a presheaf with transfers on Sm= ,whichwedenotebyZtr.F /. One observes easily that F X 7! Ztr.F / is left adjoint to the corresponding forgetful functor. If .Y;i/ is an object of Sm= and h.Y;i/ is the corresponding representable presheaf of sets, we let Ztr.Y;i/ X denote the presheaf with transfers Ztr.h.Y;i//. For any presheaf with transfers F , we have Hom.Ztr.Y;i/;F / Fi .Y /: (2.3) D By construction, the i-th component of Ztr.Z;j / is

Ztr..h.Z;j //i / Ztr. hZ ! i / !Ztr.Z ! i / (2.4) D # X D˚ # X ! a where " runs through all morphisms Œj Œi in .Togetherwith(2.3)thisshows ! that

Hom.Ztr.Y;i/;Ztr.Z;j // !HomSmCor. i /.Y;Z ! i /: D˚ X # X Denote by SmCor. / the full subcategory in PST. / generated by direct X X sums of objects of the form Ztr.Yi /. The following lemma is an immediate corollary of (2.3). Lemma 2.3 The category PST. / is naturally equivalent to the category of X additive contravariant functors from SmCor. / to the category of abelian groups, X which commute with . ˚ Lemma 2.3 implies in particular that we can apply in the context of PST. / X the usual construction of the canonical left resolution of a functor by direct sums of representable functors. It provides us with a functor Lres from PST. / to X complexes over SmCor. / together with a familiy of natural quasi-isomorphisms X Lres.F / F: ! We let D. / D .PST . // X WD ! X denote the derived category of complexes bounded from above over PST. /.In X view of Lemma 2.3, it can be identiﬁed with the homotopy category of complexes bounded from above over SmCor. / by means of the functor X K Tot.Lres.K// (2.5) 7! which we also denote by Lres. 6V. VOEVODSKY

For a morphism of simplicial schemes f , the direct and inverse image $ W X ! Y functors fi";fi; deﬁne in the obvious way functors "

f " PST. / PST. / $ W Y ! X

f ; PST. / PST. / $ " W X ! Y and the adjunction morphisms Id fi; fi", fi"f ;i Id deﬁne morphisms ! " " !

Id f ; f " ! $ " $

f ; f " Id $ " $ ! which automatically satisfy the adjunction axioms and therefore make f ; into a $ " right adjoint to f ". $ The functor f " takes Ztr.Z;i/ to Ztr.Z i i ;i/ and commutes with direct #Y X sums. Therefore it$ restricts to a functor

1 f ! SmCor. / SmCor. /: $ W Y ! X

Using the equivalence of Lemma 2.3, we can now recover the functors f " and f ; 1 $ " $ as the direct and inverse image functors defined by f ! . $ The functors f ; are clearly exact and therefore define functors on the $ " corresponding derived categories. The functors f " for non-smooth f are in general only right exact but not left exact. To define the corresponding$ left adjoints one sets

Lf ".K/ f ".Lres.K// $ D $ where Lres is deﬁned on complexes by (2.5). The corresponding functor on the derived categories, which we continue to denote by Lf ", is then a left adjoint to $ f ; . $ " A group of functors relates the presheaves with transfers over with the X presheaves with transfers over the terms of .Foranyi 0,let X !

ri SmCor. i / SmCor. / W X ! X be the functor which takes a smooth scheme Y over i to Ztr.Y;i/.Thisfunctor X deﬁnes in the usual way a pair of adjoints

ri; PST. i / PST. / # W X ! X and r" PST. / PST. i / i W X ! X Motives over simplicial schemes 7

where ri" is the right adjoint and ri;# the left adjoint. Equation (2.3) implies that for a presheaf with transfers F on , r .F / is the i-th component of F .Tocompute X i" ri;#,notethat ri; .Ztr.Y // Ztr.Y;i/ (2.6) # D and ri; is right exact. Therefore for a presheaf with transfers F over i ,onehas # X

ri; .F / h0.ri; .Lres.F /// (2.7) # D # where Lres is the canonical left resolution by representable presheaves with transfers and the right hand side of (2.7) is deﬁned by (2.6).

The functors ri" are exact and therefore deﬁne functors between the corresponding derived categories which we again denote by ri". We do not know if the functors ri;# are exact, but in any event one can deﬁne the left derived functor

Lri; ri; Lres: # WD # ı This functor respects quasi-isomorphisms and the corresponding functor between the derived categories, which we continue to denote by Lri;#, is the left adjoint to ri". Lemma 2.4 The family of functors

r" D. / D. i / i W X ! X is conservative i.e. if r .K/ 0 for all i then K 0. i" Š Š Proof: Let K be an object such that r .K/ 0 for all i.Thenbyadjunction i" Š

Hom.Ztr.Y;i/;KŒn/ Hom.Lri; .Ztr.Y //;KŒn/ D # D

Hom.Ztr.Y /;r".K/Œn/ 0: D i D Since objects of the form Ztr.Y;i/ generate D. /,weconcludethatK 0. X Š Consider the composition

r"rj; PST. j / PST. i /: i # W X ! X

By (2.4) it takes Ztr.Y / to !Ztr.Y ! i / where " runs through the morphisms ˚ # X Œj Œi in . Therefore we have !

r"Lrj; !L " (2.8) i # D˚ X! and passing to h0. / we get " r"rj; ! ": i # D˚ X! 8V. VOEVODSKY

Remark 2.5 The functors ri behave as if the terms i formed a covering of the X simplicial scheme where the r were the inverse image functors for this covering X i" and the ri; were the functors which in the case of an open covering ji Ui X # W ! would be denoted by .ji /Š.

The functors ri" commute in the obvious sense with the functors f " for morphisms f of simplicial schemes. W X ! Y Let now be a simplicial scheme over a scheme S. We have a functor X

c" PST .S/ PST. / W ! X which sends a presheaf with transfers F over S to the collection

c".F / .. i S/".F //i 0 D X ! % with the obvious structure morphisms. This functor is clearly right exact and using the representable resolution Lres over S we may deﬁne a functor Lc" from complexes over PST .S/ to complexes over PST. /.ThenLc respects quasi- X " isomorphisms and therefore deﬁnes a triangulated functor

Lc" D.S/ D. /: W ! X The functors c are compatible with the pull-back functors f such that for f " " W , we have a natural isomorphism X ! Y

c" f "c" D and for the functors on the derived categories, we have natural isomorphisms

Lc" Lf "Lc": D

They are also compatible with the functors ri" such that one has

r"c" p" i D i and r"Lc" Lp " i D i where pi is the morphism i S. X ! If is a smooth simplicial scheme over S then the functor c has a left adjoint X " c# which takes Ztr.Y;i/ to the presheaf with transfers Ztr.Y=S/ on SmCor.S/.In particular in this case c" is exact. The functor c# being a left adjoint is right exact and we use representable resolutions to deﬁne the left derived functor

Lc c Lres: # WD # ı Motives over simplicial schemes 9

The functor Lc# respects quasi-isomorphisms and the corresponding functor on the derived categories is a left adjoint to c Lc . " D " Functors c# are compatible with the functors ri;# such that one has

ri; c pi; # # D # where pi is the smooth morphism i S,andonthelevelofthederivedcategories X ! one has Lri; Lc Lpi; : # # D #

3. Tensor structure

Recall that for a scheme X one uses the fiber product of smooth schemes over X and the corresponding external product of finite correspondences to define the tensor structure on SmCor.X/ (see e.g. [7]). One then defines a tensor structure on PST .X/ setting

F G h0.Lres.F / Lres.G// ˝ WD ˝ where the tensor product on the right is deﬁned by the tensor product on SmCor.X/. If f X X is a morphism of schemes then there are natural isomorphisms W 0 !

f ".F G/ f ".F / f ".G/ (3.1) ˝ D ˝ which are compatible on representable presheaves with transfers with the isomorphisms .Y X X 0/ X .Z X X 0/ .Y X Z/ X X 0: # # 0 # D # # Let now be a simplicial scheme. For presheaves with transfers F , G over the X X collection of presheaves with transfers Fi Gi over i has the natural structure of ˝ X a presheaf with transfers over deﬁned by the isomorphisms (3.1). This structure X is natural in F and G and one can easily see that the pairing

.F;G/ F G 7! ˝ extends to a tensor structure on presheaves with transfers over .Theunitof X this tensor structure is the constant presheaf with transfers Z which has as its components the constant presheaves with transfers over i . The following lemma X is straightforward. Lemma 3.1 Let F , G be presheaves of sets over .Thenthereisanatural X isomorphism Ztr.F G/ Ztr.F / Ztr.G/: # D ˝ 10 V. VOEVODSKY

A major difference between the categories of presheaves with transfers over a scheme and over a simplicial scheme lies in the fact that the tensor structure on PST. / does not come from a tensor structure on SmCor. /.Inparticular,for X X ageneral , Z is not representable and the tensor product of two representable X presheaves with transfers is not representable. Let us say that a presheaf with transfers F is admissible,ifitscomponents Fi are direct sums of representable presheaves with transfers over i .The X class of admissible presheaves contains Z and is closed under tensor products. The following straightforward lemma implies that any representable presheaf with transfers is admissible and in particular that Lres provides a resolution by admissible presheaves.

Lemma 3.2 A presheaf with transfers of the form Ztr.Y;i/ is admissible. Proof: Follows immediately from (2.4).

Lemma 3.3 Let K;K0;L be complexes of admissible presheaves with transfers and K K be a quasi-isomorphism. Then K L K L is a quasi-isomorphism. ! 0 ˝ ! 0 ˝ Proof: The analog of this proposition for presheaves with transfers over each i X holds since free presheaves with transfers are projective objects in PST. i /.Since X both quasi-isomorphisms and tensor products in PST. / are deﬁned term-wise, X the proposition follows. In view of Lemmas 3.2 and 3.3 the functor

L K L Lres.K/ Lres.L/ ˝ WD ˝ respects quasi-isomorphisms in K and L and therefore defines a functor on the L derived categories, which we also denote by . ˝ To see that this functor is a part of a good tensor triangulated structure on D. /, X we may use the following equivalent definition. Let A be the additive category of admissible presheaves with transfers over and H .A/ the homotopy category X ! of complexes bounded from above over A. The tensor product of presheaves with transfers makes A into a tensor additive category and we may consider the corresponding structure of the tensor triangulated category on H .A/.Observe ! now that the natural functor H .A/ D. / ! ! X is the localization with respect to the class of quasi-isomorphisms and that the tensor L product on D. / is the localization of the tensor product on H .A/.Since ˝ X ! atensortrinagulatedstructurelocalizeswellweconcludethatD. / is a tensor L X trinagulated category with respect to . To formulate this statement more precisely ˝ Motives over simplicial schemes 11 we will use the axioms connecting tensor and triangulated structures, which were introduced in [2]. Since we often work with categories which do not have internal Hom-objects we will write (TC2a) for that part of the axiom (TC2) of [2, Def. 4.1, p. 47], which refers to the tensor product, and (TC2b) for that part of the axiom which refers to the internal Hom-objects. As we show in the appendix, axiom (TC2b) follows from the axioms (TC1), (TC2a) and (TC3) whenever the internal Hom- objects exist. Proposition 3.4 The category D. / is symmetric monoidal with respect to the X tensor product introduced above and this symmetric monoidal structure satisfies axioms (TC1), (TC2a) and (TC3) with respect to the standard triangulated structure. The interaction between the tensor structure and the standard functors introduced above are given by the following lemmas. Lemma 3.5 For a morphism of simplicial schemes f one has canonical W X ! Y isomorphisms in PST. / of the form X

f ".F G/ f ".F / f ".G/ (3.2) ˝ D ˝ and canonical isomorphisms in D. / of the form X L L Lf ".K L/ Lf ".K/ Lf ".L/: (3.3) ˝ D ˝ Proof: The ﬁrst statement follows immediately from (3.1). The second follows from the ﬁrst and the fact that f " takes admissible objects to admissible objects.

Lemma 3.6 For a simplicial scheme one has canonical isomorphisms in X PST. i / of the form X r".F G/ r".F / r".G/ i ˝ D i ˝ i and canonical isomorphisms in D. i / of the form X L L Lr".K L/ Lr".K/ Lr".L/: i ˝ D i ˝ i Lemma 3.7 For a simplicial scheme over a scheme S one has canonical X isomorphisms in PST. / of the form X

c".F G/ c".F / c".G/ ˝ D ˝ and canonical isomorphisms in D. / of the form X L L Lc".K L/ Lc".K/ Lc".L/: ˝ D ˝ 12 V. VOEVODSKY

Proof: The ﬁrst statement follows immediately from (3.1). The second from the ﬁrst and the fact that f " takes representable presheaves with transfers over S to admissible presheaves with transfers over . X Lemma 3.8 For a simplicial scheme over a scheme S such that all i are smooth X X over S, one has canonical isomorphisms in PST. / of the form X

c .F c".G// c .F / G # ˝ D # ˝ and canonical isomorphisms in D. / of the form X L L Lc .F Lc".G// Lc .F / G: # ˝ D # ˝

Proof: Since (3.2) holds and c# is the left adjoint to c" there is a natural map

c .F c".G// c .F / G: # ˝ ! # ˝ Since all the functors here are right exact and every presheaf with transfers is the colimit of a diagram of representable presheaves with transfers, it is sufﬁcient to check that this map is an isomorphism for representable F and G.Thisfollows immediately from the isomorphisms

Ztr.Y;i/ c".Ztr.Z// Ztr.Y S Z;i/ (3.4) ˝ D # and c .Ztr.Y;i// Ztr.Y=S/: # D The isomorphism (3.4) implies also that for a representable G and a representable F , F c .G/ is representable. Therefore the ﬁrst statement of the lemma implies ˝ " the second.

To compute Lc# on the constant sheaf, we need the following result. Lemma 3.9 Consider the simplicial object LZ in SmCor. / with terms $ X

LZi Ztr. i ;i/ D X and the obvious structure morphisms. Let LZ be the corresponding complex. Then " there is a natural quasi-isomorphism

LZ Z: " ! Motives over simplicial schemes 13

Proof: We have to show that for any .Y;j / the simplicial abelian group LZ .Y;j / $ is a resolution for the abelian group

Z.Y / H 0.Y;Z/: D Indeed one veriﬁes easily that

LZ .Y;j / Z.j / Z.Y / $ D ˝ and since j is contractible, the projection j pt deﬁnes a natural quasi- ! isomorphism LZ .Y;j / Z.Y /. $ ! If is such that all its terms are disjoint unions of smooth schemes over S then X we may consider the complex Ztr. / deﬁned by the simplicial object represented X " by in the derived categories of presheaves with transfers over S.Notethat X

Ztr. / c#.LZ / X D $ and therefore Lemmas 3.8 and 3.9 imply the following formula. Proposition 3.10 For a complex of presheaves with transfers K over S one has

L Lc#Lc".K/ Ztr. / K: Š X " ˝

Remark 3.11 It is easy to see that the functors Lf " can be computed using more general admissible resolutions instead of the representable resolutions. But we cannot use admissible resolutions to compute Lc#, since for the (admissible) constant presheaf with transfers

Z Lc".Z/ D we have by 3.10:

c#.Z/ h0.Lc#.Z// h0.Ztr. / / Ztr. / Lc#.Z/: D D X " ¤ X" D Remark 3.12 It would be interesting to ﬁnd a nice explicit description of the complex Ztr. i ;i/ Ztr. j ;j/ or, equivalently, a nice simplicial resolution of X ˝ X h. i ;i/ h. j ;j / by representable presheaves (of sets). X # X

4. Relative motives

For a scheme X,letW el.X/ be the class of complexes over PST .X/ deﬁned as follows: 14 V. VOEVODSKY

1. For any pull-back square W V """"! p (4.1) ? j ? U? Y? y """"! y in Sm=X such that p is etale, j an open embedding and p 1.Y U/ Y U ! n ! n is an isomorphism, the corresponding Mayer-Vietoris complex

Ztr.W / Ztr.U / Ztr.V / Ztr.Y / ! ˚ ! is in W el.X/:

1 el 2. For any Y in Sm=X,thecomplexZtr.Y A / Ztr.Y / is in W .X/. # ! Let further W.X/ be the smallest class in D.X/ which contains W el.X/ and is closed under triangles, direct sums and direct summands. One says that a morphism in D.X/ is an A1-equivalence,ifitsconeliesinW.X/ and defines the triangulated category DM eff .X/ of (effective, connective) motives over X as the localization of D.X/ with! respect to A1-equivalences. For a simplicial , consider X el el W . / ri; .W . i // i X WD # X as classes of complexes in PST. /.LetW. / be the smallest class in D. / which X X X contains all W el. / and is closed under triangles, direct sums and direct summands. i X Definition 4.1 Amorphismu in D. / is called an A1-equivalence,ifitsconelies X in W. /. X Definition 4.2 Let be a simplicial scheme. The triangulated category DM eff . / X X of (effective, connective) motives over is the localization of D. / with! respect X X to A1-equivalences.

Lemma 4.3 1. For any morphism f of simplicial schemes the functor Lf " takes A1-equivalences to A1-equivalences,

1 1 2. for any simplicial scheme the functors ri" take A -equivalences to A - equivalences,

1 1 3. for any simplicial scheme the functors Lri;# take A -equivalences to A - equivalences,

1 4. for any simplicial scheme over S the functor Lc" takes A -equivalences to A1-equivalences, Motives over simplicial schemes 15

1 5. for any smooth simplicial scheme over S the functor Lc# takes A -equivalences to A1-equivalences.

Proof: It follows immediately from the deﬁnitions that the functors Lf " and Lc# 1 1 and ri;# take A -equivalences to A -equivalences. 1 1 The functor ri" takes A -equivalences to A -equivalences by (2.8). 1 1 To see that Lc" takes A -equivalences to A -equivalences consider a complex L over S which consists of representable presheaves with transfers. Let further Li be the pull-back of L to i which we consider as a complex of representable X presheaves with transfers over .Onehas X

Lc".L/ Z Lc".L/ LZ Lc".L/ D ˝ D " ˝ where LZ is the complex of Lemma 3.9. By (3.4) we conclude that Lc".L/ " is quasi-isomorphic to the total complex of a bicomplex with terms of the form el el ri; .Li /. Since for L W .S/ we have Li W . i / for all i,thisimpliesthat # 2 2 X Lc takes W.S/ to W. /. " X We keep the notations Lf ", ri", Lri;#, Lc" and Lc# for the functors between eff the categories DM which are deﬁned by Lf ", Lc" and Lc# respectively. Note (cf. [1, Prop. 2.6.2]) that these functors have the same adjunction properties as the original functors. Lemma 4.4 The family of functors

eff eff ri" DM . / DM . i / W ! X ! ! X is conservative i.e. if r .K/ 0 for all i then K 0. i" Š Š Proof: Same as in Lemma 2.4. L Proposition 4.5 The tensor product respects A1-equivalences. ˝ el Proof: It is enough to show that for K ri;#.W . i // and any L,theobject L 2 X K L is zero in DM eff . /.ByLemma4.4itissufficienttoshowthatr .K/ 0 ˝ X j" Š for all j . This follows! immediately from Lemma 3.6 and (2.8). By Proposition 4.5, the tensor structure on D. / defines a tensor structure X on DM eff . /. Since any distinguished triangle in DM eff . / is, by definition, X X isomorphic! to the image of a distinguished triangle in D. !/, Proposition 3.4 implies X immediately the following result. Proposition 4.6 The axioms (TC1)-(TC3) of [2] hold for DM eff . /. ! X Proposition 4.7 The category DM eff . / is Karoubian, i.e. projectors in this X category have kernels and images. ! 16 V. VOEVODSKY

Proof: For any K in DM eff . / the countable direct sum K exists in X ˚i1 1 DM eff . / for obvious reasons.! This implies the statement of theD proposition in X view! of the following easy generalization of [3, Prop.1.6.8 p.65]. Lemma 4.8 Let D be a triangulated category such that for any object K in D the countable direct sum i1 1K exists. Then D is Karoubian. ˚ D Proof: Same as the proof of [3, Prop.1.6.8 p.65].

5. Relative Tate motives

For any S we may define the elementary Tate objects Z.p/Œq in DM eff .S/ in the same way they were defined in [9, p.192] for S Spec.k/.For! over S D X we define the Tate objects Z.p/Œq in DM eff . / as Lc .Z.p/Œq/.Notethatthis X " definition does not depend on S -onemayalwaysconsider! as a simplicial scheme X over Spec.Z/ and lift the Tate objects from Spec.Z/.WedenotebyDT. / (resp. X DT . /) the thick (resp. localizing) subcategory in DM eff . / generated by Tate X X objects, i.e. the smallest subcategory which is closed under! shifts, triangles, direct summands (resp. and direct sums) and contains Z.i/ for all i 0.Wewillcall ! these categories the category of (effective) Tate objects over and the category of X effective Tate objects of finite type over ,respectively.When is clear from the X X context, we will write DT and DT instead of DT. / and DT . /,respectively, X X and since we never work with non-effective objects in this paper we will often omit the word "effective" . Both subcategories DT. / and DT . / are clearly closed under tensor product X X and Proposition 4.6 implies that they are tensor triangulated categories satisfying May’s axiom TC3. Remark 5.1 The category DT. / does not coincide in general with the triangulated X subcategory generated in DM eff . / by Tate objects. Consider for example the ! X case when 1 2 and both 1 and 2 are non-empty. Then the constant X D X q X X X presheaf with transfers Z is a direct sum of Z1 and Z2 where Zi is the constant presheaf with transfers on i .OnecaneasilyshowthattheZi ’s are not in the X triangulated subcategory generated by Tate objects. However, one can show that the problem demonstrated by this example is the only possible one - if H 0. ;Z/ is Z then the triangulated subcategory in X DM eff . / generated by Tate objects is closed under directs summands and X therefore! coincides with DT. /. eff X For M in DM ,wedenoteasusualbyH ; .M / the groups ! " " Hom.Z;M. q/Œ p/ for q 0 Hp;q.M / " " Ä D Hom.Z.q/Œp;M/ for q 0 ! ! Motives over simplicial schemes 17

; and by H " ".M / the groups

Hom.M;Z.q/Œp/ for q 0 H p;q.M / ! D 0 for q<0: ! Lemma 5.2 Let f M M be a morphism in DT which deﬁnes isomorphisms W ! 0 on the groups Hp;q. / for q 0.Thenf is an isomorphism. " ! Proof: For a given f , the class of all N such that the maps

Hom.N Œp;M/ Hom.N Œp;M 0/ ! are isomorphisms for all p is a thick subcategory of DT .Ourconditionmeansthat it contains all Z.q/. Therefore it coincides with the whole of DT and we conclude that f is an isomorphism by the Yoneda Lemma. Let be a smooth simplicial scheme over S.Forsucha we define M. / as X eff X X the object in DM .S/ given by the complex Ztr. / associated with the simplicial X object in SmCor.S/! .Notethatthisdefinitioniscompatiblewiththedefinition X of motives of smooth simplicial schemes given in [6]. Proposition 5.3 For as above, there are natural isomorphisms X

HomDM. /.Z.q0/Œp0;Z.q/Œp/ HomDM.S /.M. /.q0/Œp0;Z.q/Œp//: X D X Proof: We have by adjunction

HomDM. /.Z.p0/Œq0;Z.q/Œp/ HomDM. /.c"Z.q0/Œp0;c"Z.q/Œp// X D X D

HomDM.S /.Lc c"Z.q0/Œp0;Z.q/Œp/ D # and Proposition 3.10 implies that for any M in DM eff .S/,onehas !

Lc c".M / M. / M: # D X ˝

Corollary 5.4 For as above and any i>0,onehas X

HomDM. /.Z;ZŒ i/ 0: X " D Combining Proposition 5.3 with the Cancellation Theorem [7] we get the following result. 18 V. VOEVODSKY

Corollary 5.5 Let now be a smooth simplicial scheme over a perfect ﬁeld k. X Then one has HomDM. /.Z.q0/Œp0;Z.q/Œp/ X D 0 for q

0 DT<0 DT<1 DT. / D ' '&&&' X and we have nDT

Lemma 5.8 Let M be such that H ;i .M / 0 for all i n.ThenM lies in DT

ev $ M Hom.M;Z.n 1// Z.n 1/; ı W ˝ " ! " where $ is the permutation of multiples, is a morphism

M ‰.M/ W ! which is natural in M . Using Proposition 8.5 and Corollary 5.5 one veriﬁes immediately that ‰.M/ lies in DT

H ;i .M / H ;i .‰.M //: " ! " For i

… nM M …

Remark 5.10 Note that the proof of Lemma 5.8 shows that in the distinguished triangle of Lemma 5.9, one may choose M …

Lemma 5.11 Let f M1 M2 be a morphism in DT and let W !

… nM1 M1 …

… nM2 M2 …

… nM1 M1 …

Hom.… nM1Œ ;…

Hom.M1;…

1. For any N in DT

Hom.…

Hom.… nM;N/ 0: % D

2. For any N in DT n one has %

Hom.N;… nM/ Hom.N;M/ % D

Hom.N;…

Remark 5.13 The major difference between the slice ﬁltrations in the triangulated category of motives and in the motivic stable homotopy category is that in the later case, Lemma 5.12 does not hold. For N in SH n and M in SH

… n DT DT n % W ! % …

… n.M / Hom.Z.n/;M /.n/ % D …

… .n 1/M … nM sn.M / … .n 1/MŒ1 (5.3) % C ! % ! ! % C sn0 1.M / …

sn DT DTn: (5.5) W ! Since sn …

Proof: Follows easily by induction.

Lemma 5.15 Deﬁne a tensor product on nDTn by the formula ˚

.Mi /i 0 .Mj /j 0 . i j nMi Mj /n 0: % ˝ % D ˚ C D ˝ % Then for any N;M there is a natural isomorphism

s .N M/ s .N / s .M /: " ˝ D " ˝ "

Proof: For any M and N , the morphisms … i M M and … j N N deﬁne a % ! % ! morphism

si j .… i M … j N/ si j .M N/: (5.7) C % ˝ % ! C ˝ On the other hand the the projections … i M si .M / and … j N sj .N / deﬁne % ! % ! amorphism

si j .… i M … j N/ si j .si .M / sj .N // si .M / sj .N /: (5.8) C % ˝ % ! C ˝ D ˝ One can easily see that (5.8) is an isomorphism. The inverse to (5.8) together with (5.7) deﬁnes a natural morphism

i j nsi .M / sj .N / sn.M N/: ˚ C D ˝ ! ˝ One veriﬁes easily that it is an isomorphism for M Z.q/Œp, N Z.q /Œp , D D 0 0 which implies by the Five Lemma that it is an isomorphism for all M and N .

Lemma 5.16 The functors … n, …

X DT n P1 DT n 2 Ä 2 %

Y DT m P2 DT m: 2 Ä 2 % Then

.Hom.X;P1/ Hom.Y;P2/;evX;P evY;P / ˝ 1 ˝ 2 is an internal Hom-object from X Y to P1 P2. ˝ ˝ Motives over simplicial schemes 23

Proof: We need to verify that for any M the homomorphism

Hom.M;X0 Y 0/ Hom.M X Y;P1 P2/ ˝ ! ˝ ˝ ˝ deﬁned by evX;P evY;P is a bijection. Using Proposition 8.5 and the Five Lemma 1 ˝ 2 we can reduce the problem to the case when M;X;Y;P1 and P2 are all motives of the form Z.q/Œp with the appropriate restrictions of q. In this case the statement follows from Corollary 5.5. Lemma 5.18 Let n 0 be an integer and ! a b M0 M1 M2 (5.9) ! ! a sequence of morphisms in DT such that the following conditions hold:

1. M0 is in DT n and si .a/ is an isomorphism for i n, % ! 2. M2 is in DT

Then there exists a unique morphism M2 M0Œ1 such that the sequence ! a b M0 M1 M2 M0Œ1 ! ! ! is a distinguished triangle. This distinguished triangle is then isomorphic to the triangle … n.M1/ M1 …

Since the functor X X.n/ from DT0 to DTn is an equivalence (by Corollary 7! 5.5), we may consider s as a functor with values in nDT0. To describe the " ˚ category DT0 consider the projection

D. / DM eff . / (5.10) X ! ! X from the derived category of presheaves with transfers over to DM .Letus X say that a presheaf with transfers .Fi / on is locally constant if for every i X the presheaf with transfers Fi on Sm= i is locally constant. Locally constant X presheaves with transfers clearly form an abelian subcategory LC in the abelian category of presheaves with transfers. 24 V. VOEVODSKY

Remark 5.19 Let X CC.X/ be the functor which commutes with coproducts 7! and takes a connected scheme to the point. Applying CC to a simplicial scheme X we get a simplicial set CC. /.IfCC. / is a connected simplicial set then LC. / X X X is equivalent to the category of modules over %1.CC. //. X Let DLC be full the subcategory in D. / which consists of complexes of X presheaves with transfers with locally constant cohomology presheaves. Note that

DLC is a thick subcategory. Let further DT00 be the thick subcategory in DLC generated by the constant sheaf Z.NotethatthecategoryDLC is Karoubian and therefore the same holds for DT00.

Proposition 5.20 The projection (5.10) deﬁnes an equivalence between DT00 and DT0.

Proof: Let us show ﬁrst that the restriction of (5.10) to DT00 is a full embedding. In order to do this, we have to show that objects of DT00 are orthogonal to objects el of ri .W . i //. In order to do this, it is enough to show that for a smooth scheme X X the constant presheaf with transfers is orthogonal to complexes lying in W el.X/, i.e. that for any such presheaf F and such a complex K one has

HomD.K;F / 0: D This follows immediately from the fact that for a smooth X and constant F ,onehas

H i .X;F / 0 for i > 0 Nis D and F.X A1/ F.X/: # D To finish the proof of the proposition, it remains to check that the image of DT00 lies in DT and that any object of DT0 is isomorphic to the image of an object in DT00. The first statement is obvious from definitions. To see the second one, observe that since our functor is a full embedding and the source is Karoubian, its image is a thick subcategory. Since it contains Z it coincides with DT0. Remark 5.21 The category DLC has a t-structure whose heart is the category LC. / of locally constant presheaves with transfers. It is equivalent to the derived X category of LC. / if and only if CC. / is a K.%;1/. X X Remark 5.22 The condition that the terms of are disjoint unions of smooth X schemes over a field is important for Proposition 5.20. More precisely what is required is that the terms of are disjoint unions of geometrically unibranch X (e.g.normal) schemes. If this condition does not hold the Nisnevich cohomology of the terms with the coefficients in constant sheaves may be non-zero. Motives over simplicial schemes 25

6. Embedded simplicial schemes

In this section we consider a special case of the general theory developed above. Definition 6.1 Asmoothsimplicialscheme over k is called embedded (over k) X if the morphisms M. / M. / X # X ! X defined by the projections are isomorphisms. In the following lemma we consider as a simplicial presheaf on Sm=S and X denote by %0. / the Nisnevich sheaf associated with the presheaf U %0. .U //. X 7! X We also write pt for the final object in the category of sheaves, which is represented by S.

Lemma 6.2 Let be a smooth simplicial scheme over S such that %0. / is a X X ! X local equivalence in the Nisnevich topology, as a morphism of simplicial presheaves, and the morphism %0. / pt is a monomorphism. Then is embedded. X ! X Proof: Our conditions imply that pr is a local equivalence. W X # X ! X Therefore, M.pr/ is an isomorphism. X S C.X/ Example 6.3 Let be a smooth scheme over and L the Cech simplicial X C.X/ % .X/ scheme of (see [6, Sec.9]). Then L is embedded. The sheaf 0 takes a smooth connected scheme U to pt,ifforanypointp of U there exists a morphism Spec. h / X; OU;p ! and to otherwise. ; Example 6.4 For any subpresheaf F of the constant sheaf pt, the standard simplicial resolution G.F / of F is an embedded simplicial scheme. We will show below that for any embedded ,thereexistsF pt such that M. / M.G.F//. X ' X Š See Lemma 6.19. Lemma 6.5 Let be an embedded simplicial scheme over S and X a c"Lc c" c" W # ! the natural transformation defined by the adjunction. Then a is an isomorphism. Proof: By the definition of adjoint functors, the obvious map b c c Lc c is W " ! " # " asectionofa.Hence,itissufficienttoshowthat

b a c"Lc c" c"Lc c" ı W # ! # is the identity. This map is adjoint to the map

p1 Lc c"Lc c" Lc c" W # # ! # 26 V. VOEVODSKY

which collapses the second copy of the composition Lc#c" to the identity. On the other hand the identity on c"Lc#c" is adjoint in the same way to the map

p2 Lc c"Lc c" Lc c" W # # ! # which collapses the ﬁrst copy of the composition Lc#c" to the identity. It remains to show that p1 p2. By Proposition 3.10 we have D

Lc c".N / N M. / # D ˝ X and one can easily see that p1 and p2 can be identiﬁed with the morphisms

N M. / M. / N M. / ˝ X ˝ X ! ˝ X deﬁned by the two projections

M. / M. / M. / M. /: X ˝ X D X # X ! X These two projections are isomorphisms by our assumption on and since the X diagonal is their common section we conclude that they are equal. For any ,letDM denote the localizing subcategory in DM eff . /,which X X eff ! X is generated by objects of the form c".M / for M in DM .S/.NotethatDM X contains the category DT. / of Tate motives over . ! X X Lemma 6.6 If is embedded then Lc# deﬁnes a full embedding X eff Lc# DM DM .S/: W X ! !

eff Proof: It is sufﬁcient to verify that for M1;M2 DM .S/,onehas 2 !

HomDM. /.c"M1;c"M2/ HomDM.S /.Lc#c"M1;Lc#c"M2/ X D which immediately follows by adjunction from Lemma 6.5. Lemma 6.7 If is embedded and M;N are objects of DM then the canonical X X morphism Lc .M N/ Lc .M / Lc .N / (6.1) # ˝ ! # ˝ # is an isomorphism. Motives over simplicial schemes 27

Proof: Note ﬁrst that a natural morphism of the form (6.1) is deﬁned by adjunction, since c" commutes with the tensor products. Since both sides of (6.1) are triangulated functors in each of the arguments, the class of M and N such that (6.1) is an isomorphism is a localizing subcategory. It remains to check that it contains pairs of the form c"M.X/, c"M.Y / where X, Y are smooth schemes over S. With respect to isomorphisms

Lc c".M.X// M.X/ M. / # D ˝ X

Lc c".M.Y // M.Y / M. / # D ˝ X Lc c".M.X/ M.Y // M.X/ M.Y / M. / # ˝ D ˝ ˝ X and the morphism (6.1) coincides with the morphism

M.X/ M.Y / M. / M.X/ M. / M.Y / M. / ˝ ˝ X ! ˝ X ˝ ˝ X deﬁned by the diagonal of . This morphism is an isomorphism, since is X X embedded.

Lemma 6.7 shows that the restriction of Lc# to DM is almost a tensor functor. X Note that it is not really a tensor functor since

Lc .Z/ M. / Z: # D X ¤ We also have to distinguish the internal Hom-objects in DM and DM eff .S/. See X Example 6.23 below. ! From now on we assume that is embedded over S.WeuseLemma6.6 X to identify DM with a full subcategory in DM eff .S/. With respect to this X identiﬁcation the functor c takes M to M M. /!. " ˝ X Lemma 6.8 The subcategory DM coincides with the subcategory of objects M X such that the morphism M M. / M (6.2) ˝ X ! is an isomorphism. Proof: Let D be the subcategory of M such that (6.2) is an isomorphism. As was mentioned above the functor c takes a motive M to M M. /,soD is contained " ˝ X in DM .SinceD is a localizing subcategory and DM is generated by motives of X X the form M.X/ M. /, the opposite inclusion follows from the fact that for any ˝ X X the morphism

M.X/ M. / M. / M.X/ M. / ˝ X ˝ X ! ˝ X is an isomorphism. 28 V. VOEVODSKY

Remark 6.9 Lemma 6.8 show that DM is an ideal in DM eff .S/,i.e.foranyK X and any M in DM the tensor product K M is in DM . ! X ˝ X Lemma 6.10 For M in DM and N DM eff .S/,thenaturalmap X 2 ! Hom.M;N M. // Hom.M;N/ (6.3) ˝ X ! is an isomorphism. Proof: Consider the map Hom.M;N/ Hom.M;N M. // ! ˝ X 1 which takes f to .f IdM. // "! where " is the morphism of the form (6.2). ˝ X ı One can easily see that this map is both the right and the left inverse to (6.3). Lemma 6.10 has the following straightforward corollary. Lemma 6.11 Let M;N be objects of DM , P an object of DM eff .S/ and X ! e M N P W ˝ ! amorphismsuchthat.N;e/ is an internal Hom-object from M to P in DM eff .S/. Define ! e M N P M. / X W ˝ ! ˝ X as the morphism corresponding to e,byLemma6.10.Then.N;e / is an internal X Hom-object from M to P M. / in DM . ˝ X X Definition 6.12 An object M in DM is called restricted,ifforanyN in X DM eff .S/ the natural map ! Hom.N;M/ Hom.N M. /;M / (6.4) ! ˝ X is an isomorphism. For our next result, we need to recall the motivic duality theorem. For a smooth variety X over a field k and a smooth subvariety Z of X of pure codimension d, the Gysin distinguished triangle defines the motivic cohomology class of Z in X of the form M.X/ Z.d/Œ2d. In particular for X of pure dimension d,the ! diagonal gives a morphism M.X/ M.X/ Z.d/Œ2d which we denote by . ˝ ! " The following motivic duality theorem is proved in [5, Th. 4.3.2, p.234]. Theorem 6.13 For a smooth projective variety X of pure dimension d over a perfect field k,thepair.M.X/;"/ is the internal Hom-object from M.X/ to Z.d/Œ2d (see Appendix). Lemma 6.14 Let S Spec.k/ where k is a perfect field and let X be a smooth D projective variety such that M.X/ lies in DM .ThenM.X/ is restricted. X Motives over simplicial schemes 29

Proof: We may clearly assume that X has pure dimension d for some d 0. ! Theorem 6.13 implies that for M M.X/, the morphism (6.4) is isomorphic to the D morphism

Hom.N M.X/;Z.d/Œ2d/ Hom.N M. / M.X/;Z.d/Œ2d/ ˝ ! ˝ X ˝ which is an isomorphism by Lemma 6.8 and our assumption that M.X/ lies in DM . X Example 6.15 The unit object Z M. / of DM is usually not restricted. X D X X Consider for example the case when C.Spec.E// where E is a Galois X D L extension of k with Galois group G.Then

Hom.ZŒi;M. // Hi .G;Z/ X D and this group may be non-zero for i>0.IfM. / were restricted, this group X would be equal to

i;0 Hom.M. /Œi;M. // Hom.M. /Œi;Z/ H ! . ;Z/ X X D X D X which is zero for i>0. Lemma 6.16 Let M;N be objects of DM , P an object of DM eff .S/ and X ! e M N P M. / X W ˝ ! ˝ X amorphismsuchthat.N;e / is an internal Hom-object from M to P M. / in X ˝ X DM . Assume further that N is restricted. Then one has: X 1. .N;e / is an internal Hom-object from M to P M. / in DM eff .S/, X ˝ X ! 2. if e is the composition

M N P M. / P ˝ ! ˝ X ! then .N;e/ is an internal Hom-object from M to P in DM eff .S/. ! Proof: To prove the ﬁrst statement, we have to show that the map

Hom.K;N/ Hom.K M;P M. // (6.5) ! ˝ ˝ X is a bijection for any K in DM eff .S/.SinceN is restricted and M is in DM ,this X map is isomorphic to the map !

Hom.K M. /;N / Hom.K M M. /;P M. // ˝ X ! ˝ ˝ X ˝ X 30 V. VOEVODSKY which is a bijection since N is an internal Hom-object in DM . X To prove the second part, we have to show that the composition of (6.5) with the map Hom.K M;P M. // Hom.K M;P / (6.6) ˝ ˝ X ! ˝ is a bijection. This follows from the ﬁrst part and the fact that (6.6) is a bijection by Lemma 6.10. Lemma 6.17 Let X be a smooth scheme over S. Then M.X/ lies in DM if X and only if the canonical morphism u M.X/ Z factors through the canonical W ! morphism v M. / Z. W X ! Proof: If M.X/ is in DM then (6.2) is an isomorphism, which immediately X implies that u factors through v.Ontheotherhandifu v w where w is a D ı morphism M.X/ M. / then ! X .Id w/ M.X/ M.X/ M.X/ M.X/ M. / ˝ ı W ! ˝ ! ˝ X is a section of the projection (6.2). Therefore, M.X/ is a direct summand of an object of DM and since DM is closed under direct summands, we conclude that X X M.X/ is in DM . X Example 6.18 If C.X/and Y is any smooth scheme such that X D L Hom.Y;X/ ¤; then Lemma 6.17 shows that M.Y / lies in DM .InparticularM.X/ lies in DM . X X More generally, for any over a perfect ﬁeld one can deduce from Lemma 6.17 X that M.Y / lies in DM if and only if for any point y of Y there exists a morphism h X Spec. / Ztr. 0/ such that the composition OY;y ! X

h Spec. / Ztr. 0/ Z OY;y ! X ! equals 1. For an embedded ,let X C. 0/ XL D L X where 0 is the zero term of . X X Lemma 6.19 There is an isomorphism M. / M. /. X ! XL Motives over simplicial schemes 31

Proof: Let us show that both projections

M. / M. / M. / X ˝ XL ! X and M. / M. / M. / X ˝ XL ! XL are isomorphisms. Since both and are embedded, it is sufficient by Lemma 6.8 X XL to show that one has M. / DM X 2 XL and M. / DM : XL 2 X The terms of are smooth schemes i and for each i we have X X Hom. i ; 0/ : X X ¤; We conclude by Example 6.18 that M. i / are in DM and therefore M. / is in X XL X DM .Toseethesecondinclusion,itissufficienttoshowthatM. 0/ is in DM. /. XL X X This follows from Lemma 6.17, since the morphism 0 S factors through the X ! morphism S in the obvious way. X ! Remark 6.20 Let S Spec.k/ where k is a field. Recall from [6] that for X such D that X.k/ the projection C.X/ Spec.k/ is a local equivalence. Since a ¤; L ! non-empty smooth scheme over a field always has a point over a finite separable extension of this field, we conclude from Lemma 6.19 that for any embedded X such that M. / 0 there exists a finite separable field extension E=k such that the X ¤ pull-back of M. / Z to E is an isomorphism. X ! Lemma 6.21 Let be an embedded simplicial scheme and X asmoothscheme X over S.Assumethatthefollowingconditionshold: 1. M.X/ DM , 2 X 2. for any Y such that M.Y / DM , there exists a Nisnevich covering U X 2 X ! of X and a morphism M.U/ M.Y / such that the square ! M.U/ M.Y / """"!

? Id ? Z? Z? y """"! y commutes. Then M.C.X// M. /. L Š X 32 V. VOEVODSKY

Proof: We need to verify that the projections M. C.X// M. / X # L ! X and M. C.X// M.C.X// X # L ! L are isomorphisms. The second one is an isomorphism by Lemma 6.8, since M.X/ 2 DM and therefore M.C.X// DM .Tocheckthatthefirstprojectiondefinesan X L 2 X isomorphism, it is sufficient by the same lemma to verify that M. / is in DMC .X/. X L In view of Lemma 6.19 it is sufficient to check that M. 0/ is in DMC.X/.Since 0 X L X is a disjoint union of smooth varieties of finite type Y such that M.Y / is in DM , X it remains to check that for such Y one has

M.Y / DMC.X/: (6.7) 2 L One can easily see (cf. [6]) that for a Nisnevich covering U Y the corresponding ! map C.U/ C.Y/is a local equivalence. Hence we may assume that U Y ,i.e. L ! L D that there is a morphism M.Y / M.X/ over Z. Then the morphism M.Y / Z ! ! M.C.X// factors throught L and we conclude by Lemma 6.17 that (6.7) holds. Remark 6.22 One can show that (at least over a perfect ﬁeld) the conditions of Lemma 6.21 are in fact equivalent to the condition that M.C.X// M. /. L Š X Example 6.23 In the notation of Example 6.15, consider the pair .M. /;e/ where X e is the canonical morphism M. / M. / M. /: X ˝ X ! X Since M. / is the unit of DM , this pair is an internal Hom-object from M. / X X X to itself in DM .HoweveritisnotaninternalHom-objectfromM. / to itself in X X DM eff .k/, since if it were we would have ! Hom.M;M. // Hom.M M. /;M. // X D ˝ X X for all M in DM eff .k/ and we know that this equality does not hold for M Z. ! D Example 6.24 Since for the terms i of we have X X M. i / DM ; X 2 X the motive M. / lies in the localizing subcategory generated by motives of i . X X If all i are smooth projective varieties, this implies by Lemma 6.14 that M. / X X lies in the localizing subcategory generated by restricted motives. Together with the previous example this shows that the category of restricted motives is not localizing. Indeed, one can easily see that it is closed under triangles and direct summands but not necessarily under inﬁnite direct sums. Motives over simplicial schemes 33

7. Coefﬁcients

All the results of Sections 2-6 can be immediately reformulated in the R-linear context where R is any commutative ring with unit. Note that the notion of embedded simplicial scheme depends on the choice of coefficients. If we consider motives with coefficients in R where R is of characteristic zero then the pull-back with respect to a finite separable field is a conservative functor (i.e. it reflects isomorphisms). Therefore Remark 6.20 implies that for S Spec.k/ D and motives with coefficients in a ring R of characteristic zero, one has

M. / R X Š for any non-empty embedded simplicial scheme .Thismeansthatinthecaseof X motives over a ﬁeld, the theory of Section 6 is interesting only if we consider torsion phenomena.

8. Appendix: Internal Hom-objects

Recall that for two objects X;S in a tensor category, an internal Hom-object from X to S is a pair .X ;e/ where X is an object and e X X S amorphismsuch 0 0 W 0 ˝ ! that for any Q the map

Hom.Q;X0/ Hom.Q X;S/ (8.1) ! ˝ given by f e .f IdX / is a bijection. 7! ı ˝ If .X 0;eX / is an internal Hom-object from X to S and .Y 0;eY / an internal Hom- object from Y to S and if we have a morphism f X Y then the composition W ! eY Y 0 X Y 0 Y S ˝ ! ˝ ! is the image under (8.1) of a unque morphism Y X which we denote by 0 ! 0 DS;eX ;eY f or simply Df ,ifS, eX and eY are clear from the context. One veriﬁes easily that D.gf / Df D g , if all the required morphisms are deﬁned. The same is D true with respect to the functoriality of internal Hom-objects in S. The internal Hom-objects are unique up to a canonical isomorphism in the following sense.

Lemma 8.1 Let .X 0;e0/, .X 00;e00/ be internal Hom-objects from X to S.Thenthere is a unique isomorphism " X X such that e e ." IdX /. W 0 ! 00 0 D 00 ı ˝ If .Y ;eY / is an internal Hom-object from Y to S and f Y X amorphism 0 W ! then ! D00f .D0f X 0 Y 0/ .X 0 X 00 Y 0/ W ! D ! ! 34 V. VOEVODSKY

where D0 is the dual with respect to e0 and eY and D00 the dual with respect to e00 and eY . A similar property holds for morphisms X Y and for morphisms in S. ! A specification of internal Hom-objects for a tensor category is a choice of C one internal Hom-object for each pair .X;S/ such that there exists an internal Hom- object from X to S. We will always assume below that a specification of internal Hom-objects is fixed. The distinguished internal Hom-object from X to S with respect to this specification will be denoted by .Hom.X;S/;evX;S /. The construction of DS f described above shows that for each S,

X Hom.X;S/ (8.2) 7! is a contravariant functor from the full subcategory of consisting of X such that C Hom.X;S/ exists to .Lemma8.1showsthatdifferentchoicesofspeciﬁcationsof C internal Hom-objects lead to isomorphic functors of the form (8.2). The same holds for functoriality in S. Consider now the case of a tensor triangulated category which satisﬁes the C obvious axioms (TC1), (TC2a) connecting the tensor and the triangulated structure. We want to investigate how internal Hom-objects behave with respect to the shift functor and distinguished triangles. Let X, S be a pair of objects of and .X ;e X X S/ an internal Hom- C 0 W 0 ˝ ! object from X to S.Considerthepair.X Œ 1;X Œ 1 XŒ1 S/ where the 0 " 0 " ˝ ! morphism is the composition

X 0Œ 1 XŒ1 X 0Œ 1Œ1 X X 0 X S: (8.3) " ˝ ! " ˝ ! ˝ ! One veriﬁes easily that this pair is an internal Hom-object from XŒ1 to S. Similar behavior exists with respect to shifts of S. Together with Lemma 8.1, this shows that for a given speciﬁcation of internal Hom-objects there are canonical isomorphisms

Hom.XŒ1;S/ Hom.X;S/Œ 1 (8.4) ! " Hom.X;SŒ1/ Hom.X;S/Œ1: (8.5) ! Remark 8.2 There is another possibility for the pairing (8.3), which one gets by moving Œ 1 to X instead of Œ1 to X . It differs from (8.3) by sign and also makes " 0 X Œ 1 into an internal Hom-object from XŒ1 to S. The isomorphisms (8.4), (8.5) 0 " constructed using different pairings (8.3) will differ by sign. If h Z XŒ1 is a morphism and Hom.Z;S/ and Hom.X;S/ exist then the W ! composition of Dh with (8.4) gives a morphism Hom.X;S/Œ 1 Hom.Z;S/ " ! which we will also denote by Dh.Thisdoesnotleadtoanyproblemssinceit is always possible to choose a speciﬁcation of internal Hom-object such that the morphisms (8.4) and (8.5) are identities. Motives over simplicial schemes 35

Theorem 8.3 Let f g h X Y Z XŒ1 (8.6) ! ! ! be a distinguished triangle in a tensor triangulated category satisfying axioms (TC1), (TC2a) and (TC3) of [2, p.47-49] and such that Hom.X;S/ and Hom.Z;S/ exist. Then for any distinguished triangle of the form

g0 f 0 DhŒ1 Hom.Z;S/ Y 0 Hom.X;S/ Hom.Z;S/Œ1 ! ! ! there exists a morphism eY Y Y S such that .Y ;eY / is an internal Hom- W 0 ˝ ! 0 object from Y to S and one has g Dg, f Df . 0 D 0 D Proof: To simplify notation, set

X 0 Hom.X;S/ eX evX;S D D

Z0 Hom.Z;S/ eZ evZ;S : D D We want to ﬁnd eY such that for any Q,themap

Hom.Q;Y 0/ Hom.Q Y;S/ (8.7) ! ˝ given by f eY .f IdY / is a bijection and the induced maps Dg and Df 7! ı ˝ coincide with g0 and f 0,respectively.Considerthediagram

Hom.Q;Z / Hom.Q;Y / Hom.Q;X / 0 """"! 0 """"! 0 (8.8) ? ? Hom.Q? Z;S/ Hom.Q Y;S/ Hom.Q? X;S/: y˝ """"! ˝ """"! y˝ If we can find eY such that the corresponding map (8.7) subdivides this diagram into two commutative squares then this map will be a bijection by the Five Lemma. In addition setting Q Z and using the commutativity of the left square on IdZ , D 0 0 we will get g Dg and setting Q Y and using the commutativity of the right 0 D D 0 square on IdY ,wewillgetf Df . It suffices therefore to find eY which for any 0 0 D Q splits (8.8) into two commutative squares. A simple diagram chase shows that the commutativity of the left square is equivalent to the commutativity of the square

Z Y Y Y 0 ˝ """"! 0 ˝

? eZ ? Z ? Z S? 0 ˝y """"! y 36 V. VOEVODSKY and the commutativity of the right square to the commutativity of the square

Y X Y Y 0 ˝ """"! 0 ˝

? eX ? X ? X S:? 0 ˝y """"! y Together, we may express our condition as the commutativity of the square

.Y X/ .Z Y/ Y Y 0 ˝ ˚ 0 ˝ """"! 0 ˝

? eX eZ ? .X X/ ? .Z Z/ C S:? 0 ˝ ˚y 0 ˝ """""! y Applying Axiom TC3’ ([2]) to our triangles, we see that there is an object W which ﬁts into a commutative diagram

.Y X/ .Z Y/ Y Y 0 ˝ ˚ 0 ˝ """"! 0 ˝

? k3 k1 ? .X X/ ? .Z Z/ C W:? 0 ˝ ˚y 0 ˝ """"! y

It remains to show that eX eZ factors through k3 k1. By [2, Lemma 4.9] the C C lower side of this square extends to an exact triangle of the form

k3 k1 .X 0 Z/Œ 1 .X 0 X/ .Z0 Z/ C W X 0 Z: ˝ " ! ˝ ˚ ˝ ! ! ˝ Therefore it is sufﬁcient to show that the diagram

.X Z/Œ 1 X X‘ 0 ˝ " """"! 0 ˝

? eZ ? Z ? Z S? 0 ˝y """"! y anticommutes. A diagram of this form can be deﬁned for any morphism of the form X ZŒ1 and its anticommutativity follows easily from the elementary axioms. !

Remark 8.4 Applying Theorem 8.3 to the category opposite to ,oneconcludesthat C a similar result holds for distinguished triangles with respect to the second argument of Hom. Motives over simplicial schemes 37

Theorem 8.3 together with the preceeding discussion of internal Hom-objects and the shift functor, implies in particular that for a given S (resp. given X) the subcategory . ;S/ (resp. .X; /), consisting of all X (resp. all S)suchthat C ! C ! Hom.X;S/ exists, is a triangulated subcategory. Proposition 8.5 The functors

Hom. ;S/ . ;S/ " W C ! ! C

Hom.X; / .X; / " W C ! ! C considered together with the canonical isomorphisms (8.4), (8.5) are triangulated functors. Proof: It clearly sufﬁces to prove the part of the proposition related to Hom. ;S/, " i.e. to show that this functor takes distinguished triangles to distinguished triangles. Consider a distinguished triangle of the form (8.6) and the resulting triangle

Dg Df DhŒ1 Hom.Z;S/ Hom.Y;S/ Hom.X;S/ Hom.Z;S/Œ1: (8.9) ! ! !

In view of Theorem 8.3, there exists an internal Hom-object .Y ;eY / from Y to Q 0 Q S Dg Df DhŒ1 such that the triangle formed by Q , Q and is distinguished. By Lemma 8.1, there is an isomorphism Y Hom.Y;S/ which extends to an Q 0 ! isomorphism of triangles. We conclude that (8.9) is isomorphic to a distinguished triangle and therefore is distinguished.

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7. Vladimir Voevodsky. Cancellation theorem. arXiv:math/0202012, To appear in Documenta Mathematica,2009. 8. Vladimir Voevodsky. Motivic cohomology with Z=l-coefﬁcients. arXiv:0805.4430, Submitted to Annals of Mathematics,2009. 9. Vladimir Voevodsky, Andrei Suslin, and Eric M. Friedlander. Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies 143 (2000), Princeton University Press, Princeton, NJ

VLADIMIR VOEVODSKY [email protected] Institute for Advanced Study Princeton, NJ 08540 Received: January 6, 2009