SOME REMARKS ON THE INTEGRAL HODGE REALIZATION OF VOEVODSKY S MOTIVES

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1 SOME REMRKS ON THE INTEGRL HODGE RELITION OF VOEVODSKY S MOTIVES VDIM VOLOGODSKY bstract. We construct a functor from the triangulated category of Voevodsky s motives to the derived category of mixed structures enriched with integral weight filtration and prove that our realization functor commutes with the lbanese functor. This extends the previous results of Barbieri-Viale, Kahn and the author ([BK], [Vol] for motives with rational coicients. 1. Introduction Let DM gm(c; be the triangulated category of Voevodsky s motives over C with integral coicients. We have the realization functor ([Hu1], [Hu2], [DG] (1.1 R : DMgm(C; D b (MHS to the bounded derived category of polarizable mixed - structures. Recall that, a mixed - structure consists of a finitely generated abelian group V together with a weight filtration on the rational vector space V Q = V Q and a filtration on the C-vector space V C satisfying certain compatibility conditions. On the other hand, it is shown by Gillet and Soulé ([GS] that the integral homology of every complex algebraic variety carries a canonical integral weight filtration. The goal of this paper is to construct a realization functor from the triangulated category of Voevodsky s motives to a certain derived category of mixed structures with the weight filtration defined integrally. n enriched mixed - structure V = (W V, F V C consists of a finitely generated abelian group V together with a weight filtration on V and a filtration on V C such that the triple (V, W Q, F constitutes a mixed structure in the sense of Deligne. n enriched structure is called ective if F 1 = 0, W 0 = V and GrW 0 V is torsion free; V is called polarizable if (V, W Q, F is polarizable. We denote the category of enriched ective polarizable mixed structures by MHS,en. The category MHS,en is not abelian, but it carries a natural exact structure. We postulate that a sequence 0 V 0 V 1 V 2 0 is exact if, for every i, the sequence of pure integral structures 0 Gr i W V 0 Gr i W V 1 Gr i W V 2 0 splits. Let D b (MHS,en be the corresponding bounded derived category ([N]. We lift the functor R to a triangulated functor (1.2 : DM gm(c; D b (MHS,en Mathematics Subject Classification. Primary 14F42, 14C25; Secondary 14C22, 14F05. Key words and phrases. lgebraic cycles, Voevodsky motives, theory. 1

2 2 VDIM VOLOGODSKY Composing with the functor D b (MHS,en Grn W K b (HSn to the homotopy category of complexes of pure weight n structures, we recover the weight complexes constructed earlier by Bondarko ([Bon]. The construction of the functor is very simple: the hard part of the job is already done in ([BV], [Bon], [Vol]. In fact, we show that if B is an abelian category of homological dimension 1 then every DG enriched functor from the category of Voevodsky s motives over a field k of characteristic 0 to the derived category of B canonically lifts to a functor from DMgm(k; to a certain filtered category D b (F B. In the second part of the paper we apply the above construction to study the motivic lbanese functor Llb : DM gm(k; D b (M 1 (k from the category of Voevodsky s motives over a subfield k C to the derived category of Deligne s 1-motives, introduced by Barbieri-Viale and Kahn ([BK]. Let MHS1 be the category of torsion free polarizable mixed - structures of type {(0, 0, (0, 1, ( 1, 0, ( 1, 1}. This category has two natural exact structures induced by the embeddings of MHS1 to the abelian category MHS and to the exact category MHS,en. busing notation, denote by Db (MHS1 and D b (MHS,en 1 the corresponding derived categories. We show that the functor D b (MHS,en 1 D b (MHS,en has a left adjoint and denote by Llb : D b (MHS,en Db (MHS 1 its composition with the projection D b (MHS,en 1 D b (MHS 1. We remark that the lbanese functor Llb does not factor through the category D b (MHS 1. Our main result, Theorem 1, is the commutativity of the following diagram (1.3 DM gm(k; Llb D b (M 1 (k T D b (MHS,en Llb D b (MHS 1, where T is the realization functor defined by Deligne ([D3]. The commutativity of (1.3 can be viewed as a derived version of Deligne s conjecture on 1-motives. Notation. For DG categories C, C we denote by T (C, C the category of DG quasi-functors ([Dri], If F : C C is a DG quasi-functor we will write Ho F : Ho C Ho C for the corresponding functor between the homotopy categories. We denote by C (resp. C the DG ind-completion (resp. the DG proj-completion of C ([Dri], 14 or [BV] 1.6. busing notation, we use the same letter F to denote the quasi-functors C C, C C induced by F T (C, C. We write F! : C C and F : C C for the right and left adjoint functors respectfully. cknowledgments. I would like to thank Leonid Positselski for helpful conversations related to the subject of this paper. 1 We denote by MHS a full subcategory of MHS formed by mixed structures with F 1 = 0.

3 SOME REMRKS ON THE INTEGRL HODGE RELITION OF VOEVODSKY S MOTIVES 3 2. Enriched structures. Let be a commutative ring, B an -linear abelian category, and let F B be the category of finitely filtered objects of B. That is, objects of F B are pairs (V, W V, where V B and W V is a increasing filtration on V, such that, for sufficiently large i, we have W i V = V, W i V = 0. Morphisms between (V, W V and (V, W V form a subgroup of Hom B (V, V consisting of those f : V V that carry W i V to W i V, for every i. We introduce an exact structure on F B postulating that a sequence 0 (V 0, W V 0 (V 1, W V 1 (V 2, W V 2 0 is exact if, for every i, the sequence 0 Gr i W V 0 Gr i W V 1 Gr i W V 2 0 splits. We denote by D b dg (F B the corresponding derived DG category and by Db (F B the triangulated derived category ([N]. For an object V of B, let V {i} be the object of F B with W j V {i} equal to V if j i and to 0 otherwise. The following result is due to Positselski ([P], Example in 8. Lemma 2.1. If n j i the forgetful functor D b (F B D b (B induces an isomorphism Hom Db (F B(V {i}, V {j}[n] Hom Db (B(V, V [n]. If n > j i then Hom D b (F B(V {i}, V {j}[n] is trivial. Let k be a field of characteristic 0. The following construction is inspired by the work of Bondarko ([Bon]. Let DM gm,et(k; (resp. DM gm(k; be the DG category of étale (resp. Nisnevich geometric Voevodsky s motives ([Vol], 2; [BV], and let R : DM gm,et(k; Ddg(B b be a DG quasi-functor ([Dri]. ssuming that the homological dimension of B is less or equal than 1 we shall construct a commutative diagram (2.1 DM α gm(k; DM gm,et(k; Ddg b (F B R Φ D b dg (B, where α is the change of topology functor, Φ is the forgetful functor, and is a DG quasi-functor defined below. Let P DM gm(k; be a full DG subcategory formed by those motives that are isomorphic in DMgm(k; to direct summands of motives of smooth projective varieties. Recall (see, e.g. [BV] that for every X, Y P the complex Hom(X, Y has trivial cohomology in positive degrees. It follows that the restriction of R to P factors as follows P P D b dg (B trunc Ddg(B. b Here Ddg b (B trunc is a DG category having the same objects as Ddg b (B and morphisms defined by the formula Hom D b dg (B trunc (V, V = τ 0 Hom D b dg (B(V, V, where τ i C denotes the canonical truncation of a complex C in degree i.

4 4 VDIM VOLOGODSKY Lemma 2.2. The pretriangulated completion ([BV], 1 of the category Ddg b (B trunc is homotopy equivalent to Ddg b (F B. Proof. Let Q be the full DG subcategory of Ddg b (F B formed by objects V {i}[i], with V B, i. Since, by Lemma 2.1, the complex Hom D b dg (F B(V {i}[i], V {j}[j] has trivial cohomology groups in positive degrees the restriction of the forgetful functor to Q factors as Q D b dg(b trunc D b dg(b. Since Ddg b (F B is strongly generated by Q we get a quasi-functor (2.2 D b dg(f B (D b dg(b trunc pretr, which is a homotopically fully faithful by the Lemma 2.1. s B has homological dimension 1, every complex in Ddg b (B trunc is homotopy equivalent to a finite direct sum of objects in image of Q. It follows that (2.2 is a homotopy equivalence. Using the above lemma we construct the functor in the diagram (2.1 as follows. ccording to ([Bon], 6 or [BV], the category DM gm(k; is strongly generated by P. Thus, the quasi-functor functor P extends uniquely to a functor DM gm(k; (D b dg(b trunc pretr D b dg(f B. The composition of the above quasi-functors is our. Observe that, for the motive M(X of a smooth projective variety, we have an isomorphism in the homotopy category of complexes over B (2.3 Gr i W ( (M(X H i R(M(X[ i]. Let us also remark that for every smooth projective varieties X and Y the forgetful functor Φ yields a quasi-isomorphism (2.4 Hom D b dg (F B( (M(X, (M(Y τ 0 Hom D b dg (B(R(M(X, R(M(Y. Throughout the rest of the paper denotes a subring of Q. We apply the above construction to the realization DG quasi-functor (2.5 R : DM gm,et(c; D b dg(mhs. constructed in ([Vol], 2. By Lemma 1.1 from [Bei] the category MHS of polarizable mixed - structures has cohomological dimension 1. Consider the full subcategory MHS,en of F (MHS whose objects (called enriched mixed structures are polarizable mixed - structures V equipped with a filtration W V such that GrW i V is pure of weight i (in particular, the filtration W V Q coincides with the weight filtration on V Q. The category MHS,en inherits an exact structure from F (MHS. n enriched mixed structure (V, W MHS,en is called ective if F 1 = 0, W 0 V = V and GrW 0 V is torsion free. Denote by MHS,en MHS,en the full subcategory of ective enriched mixed structures. Lemma 2.3. The functor D b (MHS,en Db (F (MHS is a fully faithful embedding.

5 SOME REMRKS ON THE INTEGRL HODGE RELITION OF VOEVODSKY S MOTIVES 5 Proof. s MHS has homological dimension 1, Lemma 2.1 implies that, for every objects V, V F (MHS and every i > 1, we have Hom F (MHS (V, V [i] = 0. Since the subcategory MHS,en F (MHS is closed under extensions the lemma follows from ([P], Proposition.7. The lemma together with formula (2.3 imply that the functor from the diagram (2.1 factors through Ddg b (MHS,en yielding the enriched realization functor DM gm(c; Ddg(MHS b,en. Notation: If k is a subfield of C, we will write, by abuse of notation, composition (2.6 where f is the base change functor. : DM gm (k; f DM gm (C; D b dg(mhs,en Remark 2.4. Our category MHS Q,en is just MHS Q and Q for the = R Q. On the does not even factor through the other hand, the integral realization functor category of étale motives. Indeed, the functor takes the motive /m(n = m m cone((n (n to the complex (n (n of enriched structures of weight 2n and, in particular, (/m(n is not isomorphic to (/m(n in D b (MHS,en, unless n = n. Remark 2.5. Composing Ho( with the functor D b (MHS,en Gr W i 0 K b (HS i, where K b (HSi is the homotopy category of complexes of pure weight i structures, we recover the realization of the Gillet-Soulé weight complexes ([GS]. Let (2.7 γ M : d 1 DM gm(k; DM gm(k; be the smallest full homotopy Karoubian DG subcategory of DM gm(k; that contains motives of smooth varieties of dimension 0 and 1. Lemma 2.6. For every M DM gm (C; and M d 1 DM induces a quasi-isomorphism Hom DM gm (C; (M, M Hom D b dg (MHS,en ( (M, gm (C; the functor (M. Proof. By formula (2.4 it suffices to prove that for every smooth projective variety X and a smooth projective curve C the canonical morphism Hom DM gm (C; (M(X, M(C τ 0 Hom D b dg (MHS (R (M(X, R (M(C is a quasi-isomorphism. In turn, the above is equivalent to showing that the morphism (2.8 Hom DM gm (C; (M(X C, (1 τ 2 Hom D b dg (MHS (R (M(X C, (1

6 6 VDIM VOLOGODSKY is a quasi-isomorphism. In fact, the complex at the left-hand side of (2.8 computes the groups H 1 ar (X C, O X C that are mapped isomorphically via (2.8 to Deligne s cohomology H (X C, (1 D for 2. Remark 2.7. Unless = Q, Lemma 2.6 does not hold with MHS,en MHS. replaced by Let MHS1 be the full subcategory of the category MHS that consists of torsionfree objects of type {(0, 0, (0, 1, ( 1, 0, ( 1, 1}. The category MHS1 has two natural exact structures induced by the embedding into the abelian category MHS and into the exact category MHS,en. Denote by Ddg b (MHS 1, Ddg b (MHS,en 1 the corresponding derived DG categories. Lemma 2.8. The functor induced by the embedding MHS1 a left adjoint DG quasi-functor D b dg(mhs,en 1 D b dg(mhs,en MHS,en (2.9 Ddg(MHS b,en Db dg(mhs,en 1. is homotopically fully faithful and has Proof. The first claim follows from the fact that the subcategory MHS1 is closed under extensions and from the vanishing of Hom Db (MHS,en MHS,en (V, V [i], for. For the existence of a left adjoint functor it suffices to such that, for every i > 1 and V, V MHS,en check that, there exists a set S of generators of D b (MHS,en V S the functor Φ V : D b (MHS,en 1 Mod(, Φ V (V = Hom Db (MHS,en (V, V is representable by an object of D b (MHS,en 1. We take for S the set of pure enriched structures. If V has weight < 2, then Φ V is 0. Let V be a pure polarizable structure of weight 2. There exists a unique decomposition V Q into the direct sum of a -Tate substructure and a substructure P V Q that has no -Tate subquotients. Then Φ V is representable by Im(V V Q/P. Next, every pure weight 1 structure V is the direct sum of a torsion free structure V f and structures of the form /m, m (with W 1 (/m = /m, W 2 (/m = 0. The functor Φ Vf is representable by V f and the functor Φ /m is representable by the complex (1 m (1. Finally, every ective weight 0 structure is already in MHS1. Remark 2.9. Unless = Q, the embedding D b dg (MHS 1 D b dg (MHS does not have a left adjoint functor. We define the lbanese functor Llb : D b dg(mhs,en D b dg(mhs 1. to be the composition of the adjoint functor (2.9 and the projection D b dg (MHS,en 1 D b dg (MHS 1.

7 SOME REMRKS ON THE INTEGRL HODGE RELITION OF VOEVODSKY S MOTIVES 7 3. The lbanese functor Let k be a field of finite étale homological dimension embedded into C and let M 1 (k; := M 1 (k be the category of 1-motives with coicients in. We consider the exact structure on M 1 (k; induced by the realization functor ([D3] : M 1 (k; MHS T and denote by Ddg b (M 1(k; the corresponding derived DG category. Barbieri-Viale and Kahn defined the motivic lbanese functor Llb : DM gm(k; d 1 DM gm,et(k; D b dg(m 1 (k; as follows. The composition D α In [BK], DM gm(k; DM gm(k; DM gm,et(k; of the duality, D(M = Hom(M, (1, and the change of topology functor factors through the embedding γ M et : d 1 DM gm,et(k; DM et (k;. The functor Llb is defined to be the composition DM gm(k; d 1 DM gm,et(k; Det d 1 DM gm,et(k;. Here is an equivalent more convenient for us construction of Llb. Lemma 3.1. The composition DM gm (k; γ M d 1 DM gm (k; d 1 DM gm,et(k; of the functor left adjoint to the embedding (2.7 with the change of topology functor is isomorphic to the motivic lbanese functor Llb. Proof. Consider the diagram α (3.1 DM gm(k; γ M d 1 DM gm(k; α d 1 DM gm,et(k; D D D et DM γ gm(k;! M d 1 DM gm(k; α d 1 DM gm,et(k; α γ! M,et DM gm,et(k; The rectangular part of the diagram is commutative. To prove the lemma we will show that the lower triangle is commutative: the morphism α γ M! γ! M,et α induced by γ et M α γ M! α γ M γ M! α is an isomorphism. Let X be an object of DM gm(k; and let X = cone(γ M γ M! (X X. We have γ! M (X = 0. Thus, it suffices to check that γ M,et! α(x = 0. In fact, if X is represented by a complex F of Nisnevich sheaves with transfers with homotopy invariant cohomology, then α(x is represented by its sheafification F et in étale topology. s γ M! (X = 0, for every

8 8 VDIM VOLOGODSKY curve C, the restriction of F to the small Nisnevich site of C is 0. Thus the restriction of F et to the small étale site of C is 0 i.e., γ! M,et α(x = 0. We will need a similar description of the lbanese functor. Denote by (3.2 γ H : d 1 Ddg(MHS b,en Ddg(MHS b,en, the smallest full homotopy Karoubian DG subcategory of Ddg b (MHS,en that contains structures (M(C of smooth varieties C of dimension 0 and 1. Equivalently, d 1 Ddg b (MHS,en is the derived category of the subcategory d 1 MHS,en MHS,en 1 formed by those V MHS,en 1 that the structure Gr 1 W V is a direct summand of the first homology of some smooth proper curve. Lemma 3.2. The composition Ddg(MHS b γ H d 1 Ddg(MHS b,en d 1Ddg(MHS b Db dg(mhs1 is isomorphic to the lbanese functor Llb. Proof. The lemma is implied by the following concrete statement: for every pure polarizable mixed structure V of weight 1 there exists a complex V D b (d 1 MHS,en and a morphism V V in D b (MHS,en that becomes an isomorphism in D b (MHS. Let X be an abelian variety such that V = H 1(X. We can find a direct summand J in the Jacobian of a smooth projective curve and a surjective morphism X J with finite kernel K. Pick an embedding of K into an algebraic torus T and let T = T/K. Consider the quotient G of X T by K diagonally embedded into X T. The algebraic group G is an extension of J by T. The exact sequence 0 X G T 0 yields a morphism of complexes of 1-motives X (G T which is an isomorphism in D b (M 1 (C;. pplying T By adjunction we get a morphism of quasi-functors (3.3 γ H (3.4 DM gm,et(k; D b dg (MHS,en γ M γ H γ M d 1 DM gm(k; d 1 Ddg b (MHS,en. Composing this morphism with the projection to Ddg b (MHS 1, we get (3.5 Llb R Llb. Theorem 1. The morphism (3.5 is an isomorphism of quasi-functors. we obtain the result.

9 SOME REMRKS ON THE INTEGRL HODGE RELITION OF VOEVODSKY S MOTIVES 9 Proof. We will prove that morphism (3.3 is an isomorphism. Since the motivic lbanese functor commutes with an arbitrary base field extension we may assume that k = C. By Lemma 2.6 the functor : d 1 DM gm(k; d 1 D b dg(mhs,en is a homotopy equivalence. Thus, the right vertical arrow in (3.4 is also a homotopy equivalence and the adjoint functor ( : d 1 Ddg(MHS b,en d 1DM gm(k; is its inverse. Hence, is suffices to check the commutativity of the diagram (3.6 DM gm,et(k; D b dg (MHS,en which is equivalent to Lemma 2.6. Using the commutative diagram (3.7 d 1 DM gm,et(k; γ M d 1 DM gm(k; (γ H D b dg (M 1(k; R T D b dg (MHS 1 from ([Vol], Theorem 3 we obtain our main result (1.3 stated in the introduction. References [BK] L. Barbieri-Viale, B. Kahn, On the derived category of 1-motives, arxiv: (2010. [Bei]. Beilinson, Notes on absolute cohomology, pplications of algebraic K-theory to algebraic geometry and number theory, Contemp. Math., 55, mer. Math. Soc. (1986. [BV]. Beilinson, and V. Vologodsky, guide to Voevodsky s motives. Geom. Funct. nal. 17 (2008, no. 6, [Bon] M. Bondarko, Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky versus Hanamura, J. Inst. Math. Jussieu 8 (2009, no. 1, [D3] P. Deligne, Théorie de 3, Publ. Math. IHES 44 (1974. [DG] P. Deligne,. Goncharov, Groupes fondamentaux motiviques de Tate mixte, nn. Sci. École Norm. Sup. (4 38 (2005, no. 1, [Dri] V. Drinfeld, DG quotients of DG categories, J. lgebra 272 (2004, no. 2, [GS] H. Gillet, C. Soulé, Descent, motives and K-theory, J. Reine ngew. Math. 478 (1996, [Hu1]. Huber, Realization of Voevodsky s motives, J. lgebraic Geom. 9 (2000, no. 4. [Hu2]. Huber, Corrigendum to: Realization of Voevodsky s motives, J. lgebraic Geom. 13 (2004, no. 1. [N]. Neeman, The derived category of an exact category. J. lgebra 135 (1990, no. 2, [P] L. Positselski, Mixed rtin-tate motives with finite coicients, arxiv: (2010. [Vol] V. Vologodsky, The lbanese functor commutes with the realization, arxiv: (2008.