MOTIVES WITH MODULUS

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1 MOTIVES WITH MODULUS BRUNO KAHN, SHUJI SAITO, AND TAKAO YAMAZAKI Abstract. We construct and study a triangulated category of motives with modulus that extends Voevodsky s category in such a way as to encompass non-homotopy invariant phenomena. It is intimately related to the theory of reciprocity sheaves, introduced by the authors in a previous paper. Contents 1. Modulus pairs and admissible correspondences 5 2. Presheaf theory Sheaf theory Motives with modulus Cubical objects and intervals Main results and computations Relationship with reciprocity sheaves 52 Appendix A. Categorical toolbox 62 References 71 In this paper, we construct triangulated categories of motives with modulus over a field k, in parallel with Voevodsky s construction of triangulated categories of motives in [32]. Our motivation comes from the reciprocity sheaves studied in [13]; the link between the present theory and [13] is explained in Section 7. In Voevodsky s construction, the models are smooth k-schemes. Here they are replaced by modulus pairs, which only played an auxiliary rôle in [13]. A modulus pair is a smooth separated k-scheme X provided with a partial compactification X and the datum of an effective, not Date: November 22, Mathematics Subject Classification. 19E15 (14F42, 19D45, 19F15). The first author acknowledges the support of Agence Nationale de la Recherche (ANR) under reference ANR-12-BL The second author is supported by JSPS KAKENHI Grant (15H03606). The third author is supported by JSPS KAK- ENHI Grant (15K04773). 1

2 2 BRUNO KAHN, SHUJI SAITO, AND TAKAO YAMAZAKI necessarily reduced, Cartier divisor X on X with support X X. We don t require X to be proper, although this provides the most important type of modulus pairs. Voevodsky s construction is based on the notion of homotopy invariant presheaf with transfer (we now abbreviate presheaf with transfer by PST). In [13] we introduced the notion of reciprocity on a PST, which generalizes homotopy invariance; to a modulus pair as above, with X proper and X quasi-affine, we associated a universal reciprocity PST h(x X ), such that h(x X )(k) = CH 0 (X X ), the Chow group of zero-cycles with modulus introduced in [18]. In this way we recovered some (but not all) cycle-theoretic invariants associated to (X, X ) in [3, 28, 17, 6]. Recall that when k is finite of characteristic 2, the groups CH 0 (X X ) control wild ramification along the boundary X X in coverings of X [18, Th. III]; wild ramification is non-homotopy invariant in nature, hence impossible to control by invariants of X arising from the theory of [32]. Thus, reciprocity PST s already enlarge the scope of applications of algebraic cycles to algebraic and arithmetic geometry. The presheaves h(x X ) of [13] are modulus analogues of the presheaves h 0 (X) associated to a smooth X by Voevodsky in [31, 3.2]; the latter yield the first nonzero homology sheaf of the motive of X. In the present paper, we take the next parallel step: associate a motive to any modulus pair. After a first unsuccessful attempt to use the category PST of presheaves with transfers, we were led to modify Voevodsky s paradigm: abandon PST and work with an analogous category directly built on modulus pairs. Recall that Voevodsky s construction of triangulated categories of motives in [32] consists of functors (0.1) Cor M DM eff ι gm DM eff, where Cor is the -category whose objects are smooth separated k- schemes and morphisms are finite correspondences, and DM eff gm (resp. DM eff ) is the category of effective geometric motives (resp. of motivic complexes). With this notation, PST is just the category of additive contravariant functors from Cor to abelian groups. The category DM eff gm is obtained by localizing the homotopy category K b (Cor) of bounded complexes in Cor by annulling complexes arising from Mayer-Vietoris and A 1 -invariance. The category DM eff is much bigger than DM eff gm; it is defined as a localization of the derived category of unbounded complexes of Nisnevich sheaves on Cor by A 1 - invariance, and serves as a computational tool. A fundamental result is the following.

3 MOTIVES WITH MODULUS 3 Theorem 1 ([32, 4.2.6] in char. 0, [4, 6.7.3] in general). Assume that k is perfect. For X, Y Cor, with X proper of dimension d and j Z, there is a canonical isomorphism Hom DM eff gm (M(Y ), M(X)[j]) H 2d+j (Y X, Z(d)) where the right hand side is Voevodsky s motivic cohomology. In particular, this group is 0 for j > 0 and isomorphic to CH d (Y X) for j = 0. Recall that, more generally, motivic cohomology is isomorphic to Bloch s higher Chow groups by [33]. We mimic Voevodsky s construction by introducing a -category MCor (see Definition 1.8), whose objects are modulus pairs and morphisms are finite correspondences satisfying some admissibility and properness conditions with respect to the compactifications and Cartier divisors at infinity. This is our analogue of the category Cor. To associate to MCor triangulated categories in the style of (0.1), one has to give analogues of Mayer-Vietoris and A 1 -invariance. The first notion is a bit tricky and is the main reason for allowing nonproper modulus pairs. It turns out that the right object replacing A 1 in this context is = (P 1, 1) the compactification of A 1 P 1 {1} with reduced divisor at infinity. We may then define -categories MDM eff gm and MDM eff, containing motives M(X ) of modulus pairs X = (X, X ) and fitting in a naturally commutative diagram when k is perfect 1 M MCor MDM eff ι gm MDM eff ω ω eff,gm ω eff Cor M DM eff gm ι DM eff where ι : MDM eff gm MDM eff is fully faithful and the vertical functors are induced by the natural functor ω : MCor Cor which forgets the compactifications. The functor ω eff has a right adjoint ω eff, and we have the following partial analogue of Theorem 1: Theorem 2 (See Th. 6.5 and Cor. 6.7). Let X be a smooth proper k-variety of dimension d. Then ω eff M(X) = M(X, ). If k is perfect 1 See 4.6 and Remark 4.9 for the situation in general.

4 4 BRUNO KAHN, SHUJI SAITO, AND TAKAO YAMAZAKI and j Z, we have a canonical isomorphism: Hom MDM eff gm (M(Y), M(X, )[j]) H 2d+j ((Y Y ) X, Z(d)) for any modulus pair Y = (Y, Y ), where the right hand side is Voevodsky s motivic cohomology. As a consequence, we find that ω eff sends DM eff gm[1/p] into MDM eff gm[1/p], where p is the exponential characteristic of k (Corollary 6.6). Another application is the motive of a projective bundle: Theorem Theorem 2 suggests that for any modulus pairs Y = (Y, Y ) and X = (X, X ) with X proper over k, one can relate Hom MDM eff gm (M(Y), M(X )[j]) to cycle-theoretic invariants such as additive higher Chow groups of Bloch-Esnault-Park [3, 28] and higher Chow groups of modulus pairs of Binda-Kerz-Saito [6]. Also, one may expect motivic cohomology of a modulus pair (X, X ) to be related to the relative algebraic K-group K (X, X ) by an Atiyah-Hirzebruch type spectral sequence. To achieve the first objective, it would be important to extend Theorem 2 to more general modulus pairs. As a first step, we have: Theorem 3 (see Corollary 6.3). For any X, Y MCor and i Z, we have an isomorphism Hom MDM eff gm (M(X ), M(Y)[i]) H i Nis(X, RC (Y) X ). Here RC (Y) is the derived Suslin complex of the modulus pair Y, see Subsection 6.1. Briefly, it is defined like the Suslin complex of a smooth variety X, with 3 differences: a) we use instead of A 1 ; b) we use a cubical version instead of Suslin-Voevodsky s simplicial version (see Remarks 5.9 and 6.4 for an important comment on this point); c) we use derived internal Homs instead of classical internal Homs. The latter point allows us to get Theorem 3 elementarily, without having to develop an analogue of Voevodsky s deep theory of homotopy invariant PST over a perfect field [31]. This will become essential, however, if we want to replace derived Suslin complexes by the naïve ones from Definition 7.3, and obtain a homotopy t-structure on MDM eff. It would be the first step towards a purely cycle-theoretic computation of Hom groups in MDM eff gm, in the style of [32]. Let us conclude by explaining the relationship of the above story with reciprocity sheaves. Let MCor be the full subcategory of MCor consisting of proper modulus pairs, and let MPST = Mod MCor be the category of modulus presheaves with transfers. An object F MPST is -invariant if F (M) F (M ) for any M

5 MOTIVES WITH MODULUS 5 MCor. The forgetful functor ω : MCor Cor induces a functor ω! : MPST PST, where PST is Voevodsky s category of presheaves with transfers. Theorem 4 (See Prop. 7.7 and Th. 7.10). If F MPST is - invariant, then ω! F PST has reciprocity in the sense of [13, Def. 2.3]. As an application, we get a -structure on (a slight strengthening of) the category of reciprocity sheaves, see Corollary Independently of our work, Moritz Kerz conjectured in [19] the existence of a category of motives with modulus and gave a list of expected properties. Our construction realizes part of his conjectures. We thank him for communicating his ideas, which were very helpful to us. Acknowledgements. Part of this work was done while the authors stayed at the university of Regensburg supported by the SFB grant Higher Invariants. Another part was done in a Research in trio in CIRM, Luminy. Yet another part was done while the third author was visiting IMJ-PRG supported by the Foundation Sciences Mathématiques de Paris. We are grateful to the support and hospitality received in all places. Notation and conventions. In the whole paper we fix a base field k. Let Sm be the category of separated smooth schemes of finite type over k, and let Sch be the category of separated schemes of finite type over k. We write Cor for Voevodsky s category of finite correspondences [32]. 1. Modulus pairs and admissible correspondences 1.1. Admissible correspondences. Definition 1.1. (1) A modulus pair M consists of M Sch and an effective Cartier divisor M M such that M is locally integral and the dense open subset M o := M M is smooth over k. (The case M = is allowed.) We say that M is proper if M is. We write M = (M, M ), since M is completely determined by the pair, although we regard M o as the main part of M. (2) Let M 1, M 2 be modulus pairs. Let Z Cor(M1 o, M2 o ) be an elementary correspondence. We write Z N for the normalization of the closure of Z in M 1 M 2 and p i : Z N M i for the canonical morphisms for i = 1, 2. We say Z is admissible for (M 1, M 2 ) if

6 6 BRUNO KAHN, SHUJI SAITO, AND TAKAO YAMAZAKI p 1M 1 p 2M 2. An element of Cor(M o 1, M o 2 ) is called admissible if all of its irreducible components are admissible. We write Cor adm (M 1, M 2 ) for the subgroup of Cor(M o 1, M o 2 ) consisting of all admissible correspondences. Remark 1.2. In [13, Def. 2.1], we used a different notion of modulus pair, where M is supposed proper, M o smooth quasi-affine and M is any closed subscheme of M. Definition 1.1 (1) is the right one for the present work. Definition 1.1 (2) is the same as [13, Def. 2.16], mutatis mutandis. The following lemma will play a key rôle: Lemma 1.3. Let X Sm and let X be a variety containing X as a dense open subset. Assume that X X is the support of a Cartier divisor. Then we have MCor adm (M, N) = Cor(X, N o ) {M Cor adm M=X and M o =X} for any modulus pair N. Proof. This is proven in [13, Lemma 2.17]. In loc. cit. X and N o are assumed to be quasi-affine, and X and N proper and normal (see Remark 1.2). But these assumptions are not used in the proof. (Nor is the assumption on Cartier divisors, but the latter is essential for the proof of Proposition 1.6 below.) 1.2. Composition. To discuss composability of admissible correspondences, we need the following generalization of a lemma of Krishna and Levine [17, Lemma 3.2]. It will also be used in the proof of Lemma Lemma 1.4. Let f : X Y be a surjective morphism of normal schemes, and let D, D be two Cartier divisors on Y. Assume either (i) f is flat or (ii) f is proper. If f D f D, then D D. Proof. We proceed as in loc. cit. but avoid the Bertini argument. We may reduce to D = 0 by replacing D by D D, and then to Y = Spec A for A a discrete valuation ring by localizing at generic points of D. Then D = (a) for a K and f (D) = div X (f (a)) 0, where K is the field of fractions of A. We ought to show that a A. In Case (i), we may further assume X = Spec B for B a discrete valuation ring by localizing at a generic point of f D that maps to the closed point of Y. Then div X (f (a)) 0 implies f (a) B and hence a K (f ) 1 (B) = A. In Case (ii), let X Y Y be the Stein factorization of f: then Y is normal, hence we are reduced to

7 MOTIVES WITH MODULUS 7 the two cases (ii-1) Γ(Y, O Y ) Γ(X, O X ) and (ii-2) f finite. In Case (ii-1), div X (a) 0 implies a Γ(X, O X ) A since X is normal. In Case (ii-2), f is flat because it is finite and surjective over a discrete valuation ring. Thus the claim follows from the case (i). Definition 1.5. Let M 1, M 2, M 3 be three modulus pairs, and let α Cor adm (M 1, M 2 ), β Cor adm (M 2, M 3 ). We say that α and β are composable if their composition βα is admissible. Proposition 1.6. With the above notation, assume α and β integral and let ᾱ and β be their closures in M 1 M 2 and M 2 M 3 respectively. Then α and β are composable provided the projection ᾱ M 2 β M 1 M 3 is proper. This happens in the following cases: (i) (ii) ᾱ M 1 is proper. β M 3 is proper. In particular, both (i) and (ii) hold if M 2 is proper. Proof. Note that α M o 2 β is a closed subscheme of (M1 o M2 o ) M o 2 (M2 o M3 o ) = M1 o M2 o M3 o ; we have βα = p (α M o 2 β) where p : M1 o M2 o M3 o M1 o M3 o is the projection. Let γ be a component of α M o 2 β. We have a commutative diagram γ α M o 2 β M1 o M2 o M3 o p M o 1 M o 3 δ γ ᾱ M 2 β M 1 M 2 M 3 M 1 M 3 δ where p 13 and p 13 are projections, δ = p 13 (γ), and denotes closure. The hypothesis implies that γ δ is proper sujrective. The same holds for π γδ appearing in the second part of two other commutative diagrams: ᾱ N ϕ α M 1 M 2 γ N ϕγ M 1 M 2 M 3 γ N π γα ϕ γ p 12 M 1 M 2 M 3 π γδ p 13 δ N ϕ δ M 1 M 3 π γβ p 23 β N ϕ β M 2 M 3

8 8 BRUNO KAHN, SHUJI SAITO, AND TAKAO YAMAZAKI where N means normalisation. We have the admissibility conditions for α and β: ϕ α(m 1 M 2 ) ϕ α(m 1 M 2 ) ϕ β(m 2 M 3 ) ϕ β(m 2 M 3 ). Applying πγα (resp. πγβ ) to the first (resp. second) inequality and using the left half of the above diagram, we get an inequality ϕ γ(m 1 M 2 M 3 ) ϕ γ(m 1 M 2 M 3 ) ϕ γ(m 1 M 2 M 3 ), which implies by the right half of the above diagram π γδϕ δ(m 1 M 3 ) π γδϕ δ(m 1 M 3 ) hence ϕ δ (M 1 M 3 ) ϕ δ (M 1 M 3 ) by Lemma 1.4. Finally, cases (i) and (ii) are trivially checked. Example 1.7. Let M 1 = M 3 = P 1, M 2 = A 1, M o i = A 1, M 1 =, M 2 =, M 3 = 2, α = β = graph of the identity. Then α and β are admissible but β α is not admissible The categories MSm, MSm, MCor and MCor. We now define 4 categories: Definition 1.8. An admissible correspondence α Cor adm (M, N) is left proper if the closures of all components of α are proper over M; this is automatic if N is proper. Modulus pairs and left proper admissible correrspondences define an additive category MCor by Proposition 1.6. We write MCor for the full subcategory of MCor whose objects are proper modulus pairs (see Definition 1.1(1)). These are indeed categories by Case (i) of Proposition 1.6. In the context of modulus pairs, the category Sm and the graph functor Sm Cor are replaced by the following: Definition 1.9. We write MSm for the category with same objects as MCor a morphism of MSm(M 1, M 2 ) being a (scheme-theoretic) k-morphism f : M o 1 M o 2 whose graph belongs to MCor(M 1, M 2 ). We write MSm for the full subcategory of MSm whose objects are proper modulus pairs Tensor structure. Definition For M, N MCor, we define L = M N by L = M N, L = M N + M N. This gives the categories of Definitions 1.8 and 1.9 symmetric monoidal structures with unit (Spec k, ). To see this, we have to check:

9 MOTIVES WITH MODULUS 9 Lemma Let f MCor(M 1, N 1 ) and g MCor(M 2, N 2 ). Consider the tensor product correspondence f g Cor(M o 1 M o 2, N o 1 N o 2 ). Then f g MCor(M 1 M 2, N 1 N 2 ). Proof. We may assume that f and g are given by integral cycles Z M o 1 N o 1 and T M o 2 N o 2. Then f g is given by the product cycle Z T. Let Z N Z be the normalizations of the closures Z of Z, and similarly for T N T. By hypothesis, we have (p Z 1 ) M 1 (p Z 2 ) N 1, (p T 1 ) M 2 (p T 2 ) N 2, where p Z 1 is the composition Z N Z M 1 M 2 M 1, and likewise for p Z 2, p T 1, p T 2. Hence: (p Z 1 p T 1 ) (M 1 M 2 + M 1 M 2 ) = (p Z 1 ) M 1 T + Z (p T 1 ) M 2 (p Z 2 ) N 1 T + Z (p T 2 ) N 2 = (p Z 2 p T 2 ) (N 1 N 2 + N 1 N 2 ) hence Z T MCor(M 1 M 2, N 1 N 2 ), because the projection (Z T ) N Z T factors through Z N T N. Finally, Z T is obviously proper over M 1 M 2. Warning Definition 1.10 does not have the universal property of products, even when restricted to MSm. Indeed, take M = N MSm. If M M represented the self-product of M in MSm, the diagonal M M M would have to be admissible; this happens if and only if M =. Most likely, products are not representable in MSm, starting with the self-product of = (P 1, ) The functors ( ) (n). Definition Let n 1 and M = (M, M ) MCor. We write M (n) = (M, nm ). This defines an endofunctor of MCor. Those come with natural transformations (1.1) M (n) M (m) if m n. Lemma The functor ( ) (n) is monoidal and fully faithful Changes of categories. We now have a basic diagram of additive categories and functors (1.2) MCor ω τ Cor λ ω MCor

10 10 BRUNO KAHN, SHUJI SAITO, AND TAKAO YAMAZAKI with τ(m) = M; ω(m) = M o ; ω(m) = M o ; λ(x) = (X, ). All these functors are monoidal and faithful, and τ is fully faithful; they restrict to analogous functors between MSm, MSm and Sm. Note that ω ( ) (n) = ω for any n. Moreover: Lemma We have ωτ = ω, and λ is left adjoint to ω. Moreover, the restriction of λ to Cor prop (finite correspondences on smooth proper varieties) is right adjoint to ω. Proof. The first identity is obvious. For the adjointness, let X Cor, M MCor and α Cor(X, M o ) be an integral finite correspondence. Then α is closed in X M, since it is finite on X and M is separated; it is evidently proper over X and q M = 0 where q is the composition α N α M o M. Therefore α MCor(λ(X), M). For the second statement, assume X proper and let β Cor(M o, X) be an integral finite correspondence. Then β is trivially admissible, and its closure in M X is proper over M, so β MCor(M, λ(x)). The following theorem is an important refinement of Lemma 1.15: it will be proven in the next subsections. Theorem The functors ω and τ have monoidal pro-left adjoints (see A.2). General definitions and results on pro-objects and pro-adjoints are gathered in Appendix. We shall freely use results from there Proof of Theorem 1.16: case of ω. We need a definition: Definition Let Σ be the class of all morphisms in MCor such that σ : M 1 M 2 belongs to Σ if and only if it restricts to the identity on M o 1 = M o 2. In view of Proposition A.15, the existence of the pro-left adjoint of ω is a consequence of the following more precise result: Proposition a) The class Σ enjoys a calculus of right fractions. b) The functor ω induces an equivalence of categories Σ 1 MCor Cor. Proof. a) We check the axioms of Definition A.8: (1) Identities, stability under composition: obvious.

11 (2) Given a diagram in MCor MOTIVES WITH MODULUS 11 α M 1 M 2 with M2 o = M 2 o, Lemma 1.3 provides a M 1 MCor such that M 1 o = M1 o and α MCor(M 1, M 2). We may choose M 1 such that M 1 = M 1. Then M 1 = (M 1, M 1 ) with any M 1 such that M 1 M 1, M 1 M 1 allows us to complete the square. (3) Given a diagram M 2 f s M 1 M 2 M g 2 with M 1, M 2, M 2 as in (2) and such that sf = sg, the underlying correspondences to f and g are equal since the one underlying s is 1 X2. Hence f = g. b) now follows from a), Lemma 1.3 and Corollary A.12, noting that ω is essentially surjective. Let ω! : Cor pro MCor be the pro-left adjoint of ω. It remains to show that ω! is monoidal (for the monoidal structure on pro MCor induced by the one on MCor, given by Definition 1.10). By Proposition A.15, we have for X Cor: ω! X = lim M. M Σ X Let us spell out the indexing set MSm(X) of this pro-object, and refine it: Definition (1) For X Sm, we define a subcategory MSm(X) of MSm as follows. The objects of MSm(X) are those M MSm such that M o = X. Given M 1, M 2 MSm(X), we define MSm(X)(M 1, M 2 ) to be {1 X } if 1 X is admissible for (M 1, M 2 ), and otherwise. (2) Let X Sm and fix a compactification X such that X X is the support of a Cartier divisor (for short, a Cartier compactification). Define MSm(X, X) to be the full subcategory of MSm(X) consisting of objects M MSm(X) such that M = X. Lemma a) For any X Sm and any Cartier compactification X, MSm(X) is a cofiltered ordered set, and MSm(X, X) is cofinal in

12 12 BRUNO KAHN, SHUJI SAITO, AND TAKAO YAMAZAKI MSm(X). b) Let X Cor, and let M MSm(X). Then (M (n) ) n 1 defines a cofinal subcategory of MSm(X). Proof. a) Ordered is obvious and cofiltered follows from Propositions 1.18 and A.9 a); the cofinality follows again from Lemma 1.3. b) Let M = (X, X ). By a) it suffices to show that (M (n) ) n 1 defines a cofinal subcategory of MSm(X, X). If (X, Y ) MSm(X, X), Y and X both have support X X, so there exists n > 0 such that nx Y. Let X, Y Cor and let M MSm(X), N MSm(Y ). M N MSm(X Y ). By Lemma 1.20, we have Then ω! X = lim M (n), ω! Y = lim N (n), ω! (X Y ) = lim (M N) (n). n 1 n 1 n 1 This shows the monoidality of ω!, since (M N) (n) = M (n) N (n). We also note: Proposition Define Σ Ar(MCor) as in Definition Then Proposition 1.18 extends to MCor, Σ and ω. Proof. Same as for Proposition 1.18, except for b): we must show for M, N MCor, the injection lim MCor(M, N) Cor(M o, N o ) M Σ M is surjective (Lemma 1.3 is not sufficient because of the properness condition). For this, we simply note that Σ M contains the initial object (M o, ) = λω(m), and apply Lemma Proof of Theorem 1.16: case of τ. We need a definition: Definition Take M = (M, M ) MCor. Let Comp(M) be the category whose objects are pairs (N, j) consisting of a modulus pair N = (N, N ) MCor equipped with a dense open immersion j : M N such that the schematic closure MN of M in N is a Cartier divisor and N = MN + Σ for an effective Cartier divisor Σ on N whose support is N M. Note that for N Comp(M) we have N o = M o and N is equipped with j N MCor(M, N) which is the identity on M o = N o. For N 1, N 2 Comp(M) we define Comp(M)(N 1, N 2 ) = {γ MCor(N 1, N 2 ) γ j N1 = j N2 }.

13 MOTIVES WITH MODULUS 13 For M MCor and L MCor we have a natural map Φ : lim MCor(N, L) MCor(M, τl), N Comp(M) which maps a system (α N ) N to α N j N which is independent of N by definition. We also have a natural map Ψ : MCor(τL, M) which maps a morphism β to (j N β) N. The following is analogue to Lemma 1.3: Lemma Φ and Ψ are isomorphisms. lim MCor(L, N), N Comp(M) Proof. We start with Φ. The injectivity is obvious since both sides are subgroups of Cor(M o, L o ). We prove the surjectivity. Choose a dense open immersion j 1 : M N 1 with N 1 integral proper such that N 1 M is the support of an effective Cartier divisor Σ 1. Let M1 be the schematic closure of M in N 1. Let π : N 2 N 1 be the blowup with center in M1 and put M2 = π 1 (M1 ) and Σ 2 = π 1 (Σ 1 ). By the universal property of the blowup [11, Ch. II, Prop. 7.14], j 1 extends to an open immersion j 2 : M N 2 so that j 1 = πj 2. Then N 2 M o is the support of the Cartier divisor M2 + Σ 2 so that It suffices to show the following: (N 2, M 2 + Σ 2 ) Comp(M). Claim For any α MCor(M, L), there exists an integer n > 0 such that α MCor((N 2, M2 + nσ 2 ), L). Indeed we may assume α is an integral closed subscheme of M o L o. We have a commutative diagram α N j 1 α N 1 π α N 2 ϕ α ϕ α1 ϕ α2 M L j 1 N 1 L N 2 L where α N (resp. α N 1, resp. α N 2 ) is the normalization of the closure of α M o L 0 in M L (resp. N 1 L, resp. N 2 L), and j 1 and π are iduced by j 1 : M N 1 and π : N 2 N 1 respectively. Now the admissibility of α MCor(M, L) implies ϕ α(m L ) ϕ α(m L). π

14 14 BRUNO KAHN, SHUJI SAITO, AND TAKAO YAMAZAKI Since α N 1 j 1 (α N ) is supported on ϕ 1 α 1 (Σ 1 L), this implies ϕ α 1 (N 1 L ) ϕ α 1 ((M 1 + nσ 1 ) L) for a sufficiently large n > 0. Appllying π to this inequality, we get ϕ α 2 (N 2 L ) ϕ α 2 ((M 2 + nσ 2 ) L) which proves the claim. Next we prove that Ψ is an isomorphism. The injectivity is obvious since both sides are subgroups of Cor(L o, M o ). We prove the surjectivity. Take γ lim MCor(L, N). Then γ Cor(L o, M o ) is such N Comp(M) that any component δ L o M o of γ satisfies the following condition: Take any open immersion j : M N with N integral proper such that the schematic closure MN of M in N is a Cartier divisor and N M is the support of an effective Cartier divisor Σ on N. Let δ N be the normalization of the closure of δ in L N with the natural map ϕ δ : δ N L N. Then we have ϕ δ(l (M N + nσ)) ϕ δ(l N) for any integer n > 0. Clearly this implies that δ does not intersect with Σ so that δ L M. Noting δ is proper over L since N is proper, this implies δ MCor(L, M) which proves the surjectivity of Ψ as desired. Corollary The category Comp(M) is a cofiltering ordered set. Proof. Ordered is obvious as Comp(M)(N 1, N 2 ) has at most 1 element for any (N 1, N 2 ). Cofiltering then follows from Lemma 1.23 applied with L Comp(M). Lemma 1.23 implies the existence of the pro-left adjoint τ! to τ, with the formula τ! M = lim N. N Comp(M) To show the monoidality of τ!, we argue as in the case of ω! (although we cannot quite use the functors ( ) (n) here): let M MCor. if N Comp(M), define Comp(N, M) as the full subcategory of Comp(M) consisting of those P such that P = N (compatibly with the open immersions M N, M P ) and P = MN + nσ for some n > 0, where Σ is the divisor at infinity appearing in the definition of N (see Definition 1.22). The proof of Claim 1.24 shows that Comp(N, M)

15 MOTIVES WITH MODULUS 15 is cofinal in Comp(M). If M MCor is another object and N Comp(M ), it is easy to see that the obvious functor Comp(N, M) Comp(N, M ) Comp(N N, M M ) is cofinal. This concludes the proof of Theorem The closure of a finite correspondence. Lemma Let X be an integral Noetherian scheme, (π i : Z i X) 1 i n a finite set of proper surjective morphisms with Z i integral, and let U X be a normal open subset. Suppose that π i : π 1 i (U) U is finite for every i. Then there exists a proper birational morphism X X which is an isomorphism over U, such that the closure of π 1 i (U) in Z i X X is finite over X for every i. Proof. By induction, we reduce to n = 1; then this follows from [27, Cor ] applied with (S, X, U) (X, Z 1, U) and n = 0 (note that quasi-finite + proper finite, and that and admissible blow-up of an algebraic space is a scheme if the algebraic space happens to be a scheme). Theorem Let X, Y be two integral separated schemes of finite type over a field k. Let U be a normal dense open subscheme of X, and let α be a finite correspondence from U to Y. Suppose that the closure Z of Z in X Y is proper over X for any component Z of α. Then there is a proper birational morphism X X which is an isomorphism over U, such that α extends to a finite correspondence from X to Y. Proof. Apply Lemma 1.26, noting that Z = Z X U by [13, Lemma 2.18] The categories MSm fin and MCor fin. Definition We write MCor fin for the subcategory of MCor with the same objects and the following condition on morphisms: α MCor(M, N) belongs to MCor fin (M, N) if and only if, for any component Z of α, the projection Z M is finite. We write MSm fin := MSm MCor fin. Note that if M is normal, f MSm(M, N) belongs to MSm fin if and only if the rational map M N defined by f is a morphism (Zariski s connectedness theorem). We shall also need the following definition:

16 16 BRUNO KAHN, SHUJI SAITO, AND TAKAO YAMAZAKI Definition a) A morphism f : M N in MSm fin is minimal if f N = M. b) A morphism f : M N in MSm fin is in Σ fin if it is minimal, f : M N is a proper morphism and f o is the identity. In particular, Σ fin Σ (Proposition 1.21). Proposition a) The class Σ fin enjoys a calculus of right fractions within MSm fin and MCor fin. b) Any morphism in Σ fin is invertible in MSm (hence in MCor). c) The induced functors A : (Σ fin ) 1 MSm fin MSm and A : (Σ fin ) 1 MCor fin MCor are isomorphisms of categories. Proof. a) Same as the proof of Proposition 1.18 a), except for (2): consider a diagram in MCor fin M α M 1 M 2 with f Σ fin (in particular M 2o = M2 o ). By the properness of f, the finite correspondence α o : M1 o M 2o satisfies the hypothesis of Theorem Applying this theorem, we find a proper birational morphism f : M 1 M 1 which is the identity on M o 1 and such that α o defines a finite correspondence α : M 1 M 2. If we define M 1 = f M1, then f Σ fin and α MCor fin (M 1, M 2). If α MSm fin (M 2, M 2 ), clearly α MSm fin (M 1, M 1 ). b) is clear. To prove c), it suffices as in Corollary A.12 to show that for any M, N MCor, the obvious map f 2 lim MCor fin (M, N) MCor(M, N) M Σ fin M is an isomorphism. This map is clearly injective, and its surjectivity follows again from Theorem Corollary Let C be a category and let F : MCor fin C, G : MSm C be two functors whose restrictions to the common subcategory MSm fin are equal. Then (F, G) extends (uniquely) to a functor H : MCor C. Proof. The hypothesis implies that F inverts the morphisms in Σ fin ; the conclusion now follows from Proposition 1.30 c). Corollary Any modulus pair in MSm is isomorphic to a modulus pair M in which M is normal. Under resolution of singularities,

17 MOTIVES WITH MODULUS 17 we may even choose M smooth and the support of M to be a divisor with normal crossings. Proof. Let M 0 MSm. Consider a proper morphism π : M M 0 which is an isomorphism over M0 o. Define M := π M0. Then the induced morphism π : M M 0 of MSm fin is in Σ fin, hence invertible in MSm. The corollary readily follows. We also have the following important lemma: Lemma Let M, L, N MSm and assume M, L and N are integral. Let f : L N be a morphism of in MSm fin such that f : L N is a faithfully flat morphism. Then the diagram f MCor(N, M) MCor(L, M) Cor(N o, M o ) (f o ) Cor(L o, M o ) is cartesian. The same holds when MCor is replaced by MCor fin. Proof. As the second statement is proven in a completely parallel way, we only prove the first one. Take α Cor(N o, M o ) such that (f o ) (α) MCor(L, M). We need to show α MCor(N, M). We first reduce to the case α is integral. To do this, it suffices to show that for two distinct integral finite correspondences V, V Cor(N o, M o ), (f o ) (V ) and (f o ) (V ) have no common component. By the injectivity of Cor(N o, M o ) Cor(k(N o ), M o ), this can be reduced to the case where N o and L o are fields, and then the claim is obvious. Now assume α is integral and put β := (f o ) (α). We have a commutative diagram in which all squares are cartesian β N β ϕ β a L M L f N α N α ϕ α a f N M N. Here α (resp. β) is the closure of α (resp. β) in N M (resp. L M) and α N (resp. β N ) is the normalization of α (resp. β). By hypothesis a is proper and f is faithfully flat. This implies that a is proper [SGA1,

18 18 BRUNO KAHN, SHUJI SAITO, AND TAKAO YAMAZAKI Exp. VIII, Cor. 4.8]. We also have (f N ) (ϕ α(n M)) = ϕ β(f (N ) M)) = ϕ β(l M) ϕ β(l M ) = (f N ) (ϕ α(n M )). Now Lemma 1.4 shows ϕ α(n M) ϕ α(n M ), and we are done Quarrable morphisms. Recall [SGA3, IV.1.4.0] that a morphism f : M N in a category C is quarrable if, for any g : N N, the fibred product N N M is representable in C. We have: Proposition Let f : M N be a minimal morphism in MSm fin (Definition 1.29). If f o is smooth, f is quarrable. Proof. Let g : N N be a morphism in MSm fin ; let M be the disjoint union of the reduced irreducible components of N N M and let f : M N, g : M M be the two projections. Since f o is smooth, f 1 (N o ) is smooth, and defines M o. We define M as f N, yielding a modulus pair M and effective morphisms f : M N, g : M M with f minimal: M g f M f N g N. Indeed, g M = g f N = f g N. Since f : M N is dominant, it lifts to a morphism of normalisations f N : M N N N. If q : M N M and r : N N N denote the projections, we have q f g N = f N r g N f N r N = q f N = q M which shows that g is admissible. Let P be another modulus pair and a : P M, b : P N be effective morphisms such that fa = gb. Letc 0 : P N N M be the morphism such that f c 0 = b and g c 0 = a. Then c 0 lifts uniquely to a morphism c : P M. Indeed, it suffices to check this component by component, so we may assume P and hence P o irreducible. But c 0 (P o ) is contained in one component of M o, hence c 0 (P ) is contained in a single irreducible component of N N M. Now c M = c f N = b N so if p : P N P is the normalisation of P, we have p c M = p b N p P, and c is admissible.

19 MOTIVES WITH MODULUS 19 Corollary Let f : M N be a morphism in MSm fin. If f o is étale and f is minimal in the sense of Definition 1.29 a), then so is the diagonal f : M M N M. Proof. Thanks to Proposition 1.34, f is really a morphism in MSm fin, and it is clearly minimal. Finally, o f is a closed and open immersion as the diagonal of a separated étale morphism. 2. Presheaf theory 2.1. Modulus presheaves with transfers. Definition 2.1. By a presheaf we mean that a contravariant functor to the category of abelian groups. (1) The category of presheaves on MSm (resp. MSm, MSm fin ) is denoted by MPS (resp. MPS, MPS fin ). (2) The category of additive presheaves on MCor (resp. MCor, MCor fin ) is denoted by MPST (resp. MPST, MPST fin.) (3) We denote by Z tr (M) the image of M MSm or M MSm in MPST or MPST by the additive Yoneda functor, and Z fin tr (M) the image of M MSm fin in MPST fin by the similar functor. The categories MPST, MPST and MPST fin all have a closed monoidal structure induced by the -structures of MCor, MCor and MCor fin. We now briefly describe the main properties of the functors induced by those of the previous section MPST and PST. Proposition 2.2. The functor ω : MCor Cor of 1.6 yields a string of 3 adjoint functors (ω!, ω, ω ): MPST ω! ω ω PST where ω is fully faithful and ω!, ω are localisations; ω! is monoidal and has a pro-left adjoint, hence is exact. The pro-left adjoint ω! of ω! is monoidal. Proof. This follows from Theorems 1.16, A.19 and A.20. Let X Sm and let M MSm(X). Lemma 1.20 and Proposition A.7 show that the inclusions {M (n) n > 0} MSm(M, X) MSm(X) induce isomorphisms (see Def. 1.19)

20 20 BRUNO KAHN, SHUJI SAITO, AND TAKAO YAMAZAKI (2.1) ω! (F )(X) lim F (N) N MSm(X) lim N MSm(M,X) F (N) lim n>0 F (M (n) ). Proposition 2.3. Let M, N MCor and let X Sm. Then ω! (Hom MPST (Z tr (N), Z tr (M))(X) is the subgroup of Cor(N o X, M o ) generated by all elementary correspondences Z Cor(N o X, M o ) such that ϕ Z(M N X) ϕ Z(M N X), where ϕ Z : Z N M N X denotes the normalization of the closure of Z. Proof. (2.1) shows that ω! (Hom MPST (Z tr (N), Z tr (M))(X) agrees with ( ) MCor(N L, M) Cor(N 0 X, M o ), L MSm(X) from which the proposition follows MPST and PST. Proposition 2.4. The adjoint functors (λ, ω) of Lemma 1.15 induce a string of 4 adjoint functors (λ! = ω!, λ = ω!, λ = ω, ω ): MPST ω! ω! ω ω PST where ω!, ω are localizations while ω! and ω are fully faithful. The functors ω! and ω! are monoidal. Moreover, if X Cor is proper, we have a canonical isomorphism ω Z tr (X) Z tr (X, ). Proof. The only non obvious fact is the last claim, which follows from Lemma MPST and MPST. Proposition 2.5. The functor τ : MCor MCor of (1.2) yields a string of 3 adjoint functors (τ!, τ, τ ): MPST τ! τ MPST τ

21 MOTIVES WITH MODULUS 21 where τ!, τ are fully faithful and τ is a localization; τ! is monoidal and has a pro-left adjoint τ!, hence is exact; moreover, τ! is monoidal. There are natural isomorphisms ω! ω! τ!, etc. Proof. This follows from Theorem 1.16 and Proposition A.7. Lemma 2.6. (1) For G MPST and M MSm, we have lim G(N) = τ! G(M). N Comp(M) (2) The adjunction map id τ τ! is an isomorphism. Proof. (1) This follows from Lemma 1.23 Proposition A.7. (2) this follows from (1) since Comp(M) = {M} for M MSm. Remark 2.7. By Lemma 1.23 we have the formulas τ! Z tr (M) = lim N Comp(M) Z tr (N), τ Z tr (M) = lim the latter an inverse limit computed in MCor. Question 2.8. Is τ! exact? 2.5. MPST fin and MPST. N Comp(M) Z tr (N) Proposition 2.9. Let b : MCor fin MCor be the inclusion functor from Definition Then b is monoidal, is a localization and has a pro-left adjoint; it yields a string of 3 adjoint functors (b!, b, b ): b! MPST fin b MPST b where b!, b are localizations and b is fully faithful; b! is monoidal and has a pro-left adjoint, hence is exact. Proof. The monoidality of b is obvious; the rest follows from the usual yoga applied with Proposition The functors n! and n. We write (n!, n ) for the pair of adjoint endofunctors of MPST induced by ( ) (n) (n! is left adjoint to n and extends ( ) (n) via the Yoneda embedding). Lemma The functor n! is fully faithful and monoidal. Proof. This follows formally from the same properties of ( ) (n).

22 22 BRUNO KAHN, SHUJI SAITO, AND TAKAO YAMAZAKI Proposition For any F MPST, there is a natural isomorphism ω ω! F F where F (M) := lim F (M (n) ) (for the natural transformations (1.1)). n Proof. Let M MCor and X = ωm. Then ω ω! F (M) = and the claim follows from Lemma lim F (M ) M MSm(X) Proposition For all n 1, the natural transformation ω! ω! n stemming from (1.1) is an isomorphism. Proof. Let F MPST. For X Cor, we have ω! n F (X) = lim n F (M) = M MSm(X) lim F (M (n) ) = M MSm(X) where the last isomorphism follows from Lemma Sheaf theory lim F (M) M MSm(X) 3.1. cd-structures. We review this notion drawn out by Voevodsky in [34]. Definition 3.1 ([34, Def. 2.1]). Let C be a category with an initial object. A cd-structure on C is a collection P of commutative squares ( distinguished squares ) of the form B Y (3.1) p e A X such that if Q P and Q is isomorphic to Q, then Q P. The topology defined by P is the coarsest Grothendieck topology such that for a distinguished square (3.1), the sieve generated by (p, e) is a covering sieve and such that the empty sieve is a covering sieve of. There are important notions of complete, regular and bounded cdstructures [34, Def. 2.3, 2.10 and 2.22]. Here are sufficient conditions for complete: Lemma 3.2 ([34, Lemma 2.4]). A cd-structure P is complete provided (1) any morphism with values in is an isomorphism; (2) for any distinguished square Q of the form (3.1) and any morphism f : X X, the square Q X X is defined and distinguished.

23 and for regular: MOTIVES WITH MODULUS 23 Lemma 3.3 ([34, Lemma 2.11]). Let P be a cd-structure such that for any distinguished square Q of the form (3.1) one has (1) Q is a pull-back square (2) e is a monomorphism (3) the objects Y X Y and B A B exist and the derived square B Y B A B Y X Y where the vertical arrows are the diagonals, is distinguished. Then P is a regular cd-structure. Let P be a cd-structure on the category C. Write t P for the associated topology, Shv(C) for the category of sheaves of t P -sets, and ρ : C Shv(C) for the functor that sends X to the t P -sheaf associated with the presheaf represented by X. Proposition 3.4 ([34, Lemmas 2.8, 2.9]). If P is complete, a) For any X C, ρ(x) is a finitely presented object of Shv(C): the Yoneda functor Shv(C)(ρ(X), ) commutes with filtering colimits. b) Any presheaf on C which sends to and converts any distinguished square into a pull-back square is a sheaf. Proposition 3.5 ([34, Prop and Lemma 2.19]). If P is regular, the converse of Proposition 3.4 b) is true. In particular, applying an abelian sheaf to a distinguished square yields a long exact Mayer- Vietoris sequence of cohomology groups. The definition of bounded involves the technical notion of density structure D [34, Def. 2.20, 2.21 and 2.22] which yields a notion of dimension dim D X of an object X C. The basic example comes from the dimension of Noetherian topological spaces [34, Ex. 2.23]. The main result is: Theorem 3.6 ([34, Th. 2.27]). Let P be a complete regular cd-structure bounded by a density structure D and X an object of C. Then for any t P -sheaf of abelian groups F on C X one has H n (X, F ) = 0 for n > dim D X Grothendieck topologies on MSm fin. Definition 3.7. Let σ {ét, Nis, Zar}. We call a morphism p : U M in MSm fin a σ-covering if

24 24 BRUNO KAHN, SHUJI SAITO, AND TAKAO YAMAZAKI (i) p : U M is a σ-covering of M in the usual sense; (ii) p is minimal (that is, U = p (M )). Since the morphisms appearing in the σ-covering are quarrable by Proposition 1.34, we obtain a Grothendieck topology on MSm fin and even a complete regular cd-structure in the case σ = Nis, Zar, taking the usual distinguished squares. The category MSm fin endowed with this topology will be called the big σ-site of MSm fin σ. Definition 3.8. Let us fix M MSm fin. We define three (small) sites: (1) Let Mét be the category of morphisms f : N M in MSm fin such that f is étale, endowed with the topology induced by MSm fin ét. (2) Let M Nis be the same category as Mét, but endowed with the topology induced by MSm fin Nis. (3) Let M Zar be the category of morphisms f : N M in MSm fin such that f is open immersion, endowed with the topology induced by MSm fin Zar. The following lemma is obvious from the definitions: Lemma 3.9. Let σ {ét, Nis, Zar} and M MSm fin. Let (M) σ be the (usual) small σ-site on M. Then we have an isomorphism of sites M σ (M) σ, N N, whose inverse is given by (p : X M) (X, p (M )). (This isomorphism of sites depends on the choice of M!) Lemma Let σ {ét, Nis}. Let α : M N be a morphism in MCor fin and let p : U N be a σ-covering of MSm fin. Then there is a commutative diagram V p α U p M α N, where α : V U is a morphism in MCor fin and p : V M is a σ-covering of MSm fin. Proof. We may assume α is integral. Let α be the closure of α in M N. Since α is finite over M, we may find a σ-covering p : V M

25 MOTIVES WITH MODULUS 25 such that p in the diagram (all squares being cartesian) V M (α U) α N U U p p V M α α N V p has a splitting s. Put V := (V, p (M )) MSm. The image of s gives us a desired correspondence α. We conclude by endowing MSm fin with a density structure in the sense of [34, Def. 2.20]. Definition Let M MSm fin. We define families of morphisms D i (M) D 0 (M) MSm fin (, M) as follows: a morphism j : M M is in D i (M) if and only if j : M M is a minimal dense open immersion of coniveau i (by definition, this means that M M is of codimension i in M). (In particular, D 0 (M) = { M}.) By Lemma 3.9 and [35, 2], this defines a density structure compatible with the standard density structure on Sch considered in loc. cit., and for which the Zariski and Nisnevich cd structures are bounded. We summarize the consequences in the following theorem (cf. Propositions 3.4, 3.5 and Theorem 3.6): Theorem Let σ {Nis, Zar}. Then: a) A presheaf of sets F on MSm fin is a σ-sheaf if and only if it converts any σ-distinguished square (3.1) of MSm fin into a pull-back square. b) If F is a σ-sheaf of abelian groups, then for any σ-distinguished square (3.1) one has a Mayer-Vietoris long exact sequence H i σ(x, F ) H i σ(y, F ) H i σ(a, F ) H i σ(b, F ) M H i+1 σ (X, F )... Moreover, H i σ(x, F ) = 0 for i > dim X(= dim X o = dim X) Sheaves on MSm fin. Definition For σ {ét, Nis, Zar}, we define MPS fin σ full-subcategory of MPS fin consisting of σ-sheaves. to be the

26 26 BRUNO KAHN, SHUJI SAITO, AND TAKAO YAMAZAKI Proposition For any M MSm and σ {ét, Nis, Zar}, we have c Z fin tr (M), c b Z tr (M) MPS fin σ, where the functors (3.2) b : MPST MPST fin, c : MPST fin MPS fin. are induced by b from Proposition 2.9 and c : MSm fin MCor fin. Proof. It suffices to show the case σ = ét. Let U N be an étale covering. We have a commutative diagram 0 MCor fin (N, M) MCor fin (U, M) MCor fin (U N U, M) 0 MCor(N, M) MCor(U, M) MCor(U N U, M) 0 Cor(N o, M o ) Cor(U o, M o ) Cor(U o N o U o, M o ). The bottom row is exact by [21, Lemma 6.2]. The exactness of the top and middle row now follows from Lemma Čech complex. Theorem Let σ {ét, Nis}. If p : U M is a σ-covering in MSm fin, then the Čech complex (3.3) c Z fin tr (U M U) c Z fin tr (U) c Z fin tr (M) 0 is exact in MPS fin σ. Proof. It is adapted from [32, Prop ]. As both proofs go parallel, we only write down the one for σ = Nis. In view of Lemma 3.9, it suffices to show the exactness of (3.3) evaluated at S = (S, D) (3.4) MCor fin (S, U M U) MCor fin (S, U) MCor fin (S, M) 0 for a henselian local S and an effective Cartier divisor D on S. To a diagram S f Z g M of k-schemes with f quasi-finite, we define L(Z) to be the free abelian group on the set of irreducible components V of Z such that f V is finite and surjective over S and such that (fi V v) (D) (gi V v) (M ), where v : V N V is the normalization and i V : V Z is the inclusion. Note that L is covariantly functorial in Z. Then (3.4) is obtained as the inductive limit of (3.5) L(Z M (U M U)) L(Z M U) L(Z) 0

27 MOTIVES WITH MODULUS 27 where Z ranges over all closed subschemes of S M that is finite surjective over S. It suffices to show the exactness of (3.5). Since Z is finite over a henselian local scheme S, Z is a disjoint union of henselian local schemes. Thus the Nisnevich covering Z M U Z admits a section s 0 : Z Z M U. Define for k 1 s k := s 0 M id U : Z M U k Z M U M U k = Z M U k+1 where U k = U M M U. Then the maps L(Z M U k ) L(Z M U k+1 ) induced by s k give us a homotopy of the identity to zero. Remark We shall see in Theorem 3.21 that the same statement holds for b Z tr (M) Sheafification preserves transfers. For σ {ét, Nis, Zar}, let a fin σ : MPS fin MPS fin σ be the sheafification functor, that is, the left adjoint of the inclusion functor i fin σ : MPS fin σ MPS fin. It exists for general reasons and is exact [SGA4-I, II.3.4]. Definition Let MPST fin σ be the full subcategory of MPST fin consisting of all objects F MPST fin such that c F MPS fin σ (see (3.2) for the functor c ). Theorem Let σ {ét, Nis}. (1) Let F MPST fin. There exists an unique object F σ MPST fin such that c (F σ ) = a fin σ (c (F )) and such that the canonical morphism u : c (F ) a fin σ (c (F )) = c (F σ ) extends to a morphism in MPST fin. (2) The inclusion functor i fin tr,σ : MPST fin σ adjoint a fin tr,σ : MPST fin MPST fin MPST fin has a left σ, which is exact; in partic- is Grothendieck ( A.9). The follow- ular the category MPST fin σ ing diagram commutes MPS fin c σ MPST fin σ i fin σ i fin tr,σ id MPS fin id c MPST fin a fin σ MPS fin c σ a fin tr,σ MPST fin Proof. (2) is a consequence of (1) and the fact that MPST fin is Grothendieck abelian as a category of modules (see Theorem A.21). (1) can σ.

28 28 BRUNO KAHN, SHUJI SAITO, AND TAKAO YAMAZAKI be shown by a rather trivial modification of [32, Th ], but for the sake of completeness we include a proof. To ease the notation, put F := a fin σ c F MPS fin. First we construct a homomorphism Φ M : F (M) MSm fin σ (Z fin tr (M), F ) for any M MSm. (Recall that Z fin tr (M) MSm fin σ by Prop ) Take f F (M). There exists a σ-covering p : U M in MSm fin and g c F (U) = F (U) such that f U = u(g) in F (U) and that g U M U = 0 in F (U M U). We have a commutative diagram in which the horizontal maps are induced by a fin tr,σc 0 MPS fin σ (Z fin tr (M), F ) s MPS fin σ (Z fin tr (U), F ) s MPST fin (Z fin tr (U), F ) MPS fin σ (Z fin tr (U M U), F ) s MPST fin (Z fin tr (U M U), F ) The left vertical column is exact by Theorem Since g F (U) = MPST fin (Z fin tr (U), F ) satisfies s (g) = g U M U = 0, there exists a unique h MPS fin σ (Z fin tr (M), F ) = F (M) such that s(h) = s (g). One checks h does not depend on the choices we made. We define Φ M (f) := h. Now we define F σ. On objects we put F σ (M) = F (M) for M MSm. For α MCor fin (M, N), we define α : F (N) F (M) as the composition of F (N) Φ N MPS fin σ (Z fin tr (N), F ) α MPS fin σ (Z fin tr (M), F ) F (M). (The last map is given by f f M (id M ). One checks with this definition F σ becomes an object of MPST fin. To prove uniqueness, take F σ, F σ MPST fin which enjoy the stated properties. We have F σ (M) = F σ(m) = F (M) for any M MSm. (Recall that F := a fin σ c F MPS fin.) We also have F σ (c(q)) = F σ(c(q)) = F (q) and for any morphism q in MSm fin. Let α : M N be a morphism in MCor fin and let f F (N). Take a σ-covering p : U N of MSm fin and g cf (U) = F (U) such that f U = u(g) in F (U). Apply Lemma 3.10 to get a morphism α : V M in MCor fin

29 MOTIVES WITH MODULUS 29 and a σ-covering p : V M of MSm fin such that αp = pα. Then we have F σ (p )F σ (α)(f) = F σ (α )F σ (p)(f) = F σ (α )(u(g)) = F σ(α )(u(g)) = F σ (α ) F σ (p) (f) = F σ(p )F σ(α)(f) = F σ (p )F σ(α)(f). Since p : V M is a σ-covering and F σ is a sheaf, this implies F σ (α)(f) = F σ(α)(f). This completes the proof From MPST fin to MPST. Definition Let σ {ét, Nis}. We define MPST σ to be the full subcategory of MPST consisting of those F such that b F MPST fin σ. We denote by i tr,σ : MPST σ MPST the inclusion functor. Proposition For σ {ét, Nis}, the functor i tr,σ has an exact left adjoint a tr,σ. The category MPST σ is Grothendieck. The fully faithful functor b σ : MPST σ MPST fin σ induced by b is exact and has an exact left adjoint b σ = a tr,σ b! i fin tr,σ. Proof. We apply Lemma A.22 with D = MPST, D = MPST fin, C = MPST fin σ, f = b and i = i fin tr,σ. Let us check that its hypotheses are verified: b is fully faithful, has a right adjoint and is exact by Proposition 2.9. The functor i fin tr,σ has an exact left adjoint by Theorem Finally, b Z tr (M) MPST fin σ for any M MCor by Proposition 3.14, so that the Z tr (M) form a set of generators of MPST by strict epimorphisms which belong to MPST σ. The second claim again follows from Theorem A.21. The last one follows from Lemma A.22 c) and d). Theorem Let σ {ét, Nis}. If p : U M is a σ-covering in MSm fin, then the Čech complex (3.6) Z tr (U M U) Z tr (U) Z tr (M) 0 is exact in MPST σ. If σ = Nis, the sequence 0 Z tr (B) Z tr (Y ) Z tr (A) Z tr (X) 0 is exact in MPST Nis for any distinguished square (3.1) in MSm fin Nis. Proof. By Theorem 3.15, the complex Z fin tr (U M U) Z fin tr (U) Z fin tr (M) 0 is exact in MPST fin σ. Applying the exact functor b σ, we get (3.6) thanks to Proposition The second statement is a special case of the first (with a small computation).