Motives with modulus, II: Modulus sheaves with transfers for proper modulus pairs

We develop a theory of sheaves and cohomology on the category of proper modulus pairs. This complements [KMSY21], where a theory of sheaves and cohomology on the category of non-proper modulus pairs has been developed.


Introduction
This is a sequel to [KMSY21], where a theory of sheaves and cohomology on the category MCor of non-proper modulus pairs has been developed. This paper complements it by using work from [KM20] and [Miy20] to develop a theory of sheaves and cohomology on the category MCor of proper modulus pairs. This completes the repairs to the mistake in [KSY15]. The basic aim of both works is to lay a foundation for a theory of motives with modulus, to be completed in [KMSY20], generalizing Voevodsky's theory of motives in order to capture non A 1 -invariant phenomena.
In [KMSY21], Voevodsky's category Cor of finite correspondences on smooth separated schemes of finite type over a fixed base field k, was enlarged to the larger category of (non-proper) modulus pairs, MCor. We then define MPST (resp. MPST) as the category of additive presheaves of abelian groups on MCor (resp. MCor). We have a pair of adjunctions where τ * is induced by the inclusion τ : MCor → MCor and τ ! is its left Kan extension (see Lemma 1.2.3).
The main aim of [KMSY21] was to develop a sheaf theory on MCor generalizing Voevodsky's theory of sheaves on Cor. (2) For M ∈ MCor, let Z tr (M) = MCor(−, M) ∈ MPST be the associated representable presheaf. Then we have Z tr (M) ∈ MNST and there is a canonical isomorphism for any i ≥ 0 and F ∈ MNST: The aim of the present paper is to introduce a sheaf theory on MCor.
Definition 3. We define MNST to be the full subcategory of MPST of objects F such that τ ! F ∈ MNST.

MNST,
where τ Nis is exact and fully faithful. The functor τ Nis is also exact.
(2) The inclusion MNST → MPST has an exact left adjoint a Nis such that a Nis τ ! = τ Nis a Nis . In particular, MNST is a Grothendieck abelian category. Theorem 3 (see Proposition 6.2.1 and Theorem 6.3.2). The following assertions hold.
The functors ω ! and ω * induce a pair of adjoint functors such that ω Nis a Nis = a V Nis ω ! , ω Nis a V Nis = a Nis ω * , where a V Nis : PST → NST is Voevodsky's sheafification functor. Moreover, ω Nis and ω Nis are exact and ω Nis is fully faithful.
(2) For any F ∈ MNST and X ∈ Sm and i ≥ 0, we have a canonical isomorphism A key ingredient of the proofs is Theorem 3.2.2, which is based on the works [Miy20] and [KM20].

Acknowledgements
Part of this work was done while the first author was visiting RIKEN iTHEMS under the invitation of the second author: the first and the second authors wish to thank both for their hospitality and excellent working conditions. Part of this work was done while the third author stayed at the university of Regensburg supported by the SFB grant "Higher Invariants". The third author is grateful to the support and hospitality received there. We also thank the referee for a thorough reading.
The first author thanks Joseph Ayoub for explaining him an easy but crucial result on unbounded derived categories (Lemma A.2.7).

Notation and conventions
In the whole paper we fix a base field k. Let Sch be the category of separated schemes of finite type over k, and let Sm be its full subcategory of smooth schemes. We write Cor for Voevodsky's category of finite correspondences [Voe00].
An additive functor between additive categories is called strongly additive if it commutes with all representable direct sums. A Grothendieck topology is called subcanonical if every representable presheaf is a sheaf.
Let C and D be sites, and u : C → D a functor. We say that u is continuous (resp. cocontinuous) if the functor u * :D →Ĉ (resp. u * :Ĉ →D) between categories of presheaves carries sheaves to sheaves. (cf. [ We write MCor fin for the full subcategory of MCor whose objects are proper modulus pairs. We write MSm fin for the subcategory of MSm with the same objects and such that a morphism (2) Let f : M → N be a morphism in MSm fin . The reducedness of M, the separatedness of N and the denseness of M o in M imply that this extension f is unique. This yields a forgetful functor MSm fin → Sch, which sends M to M.
We have the following commutative diagram of inclusion functors (1.1.1) (1) The category of presheaves on MSm (resp. MSm, MSm fin ) is denoted by MPS (resp. MPS, MPS fin ).
(3) We write (4) For M ∈ MSm, we denote by Z p (M) the presheaf with values in abelian groups defined by where for any set S we denote by ZS the free abelian group on S.
(3) For G ∈ MPST and M ∈ MSm, we have (4) For G ∈ MPS and M ∈ MSm, we have where b ! , b * are localisations; b * is exact and fully faithful; b ! has a pro-left adjoint, hence is exact. For F ∈ MPST fin and M ∈ MCor, we have (see Definition 2) The same statements hold for b s from (1.1.1).
where c * is exact and faithful (but not full). The same statements hold for c fin and c from (1.1.1). We have (1) If c is faithful, so is c .
(2) Suppose that i D is strongly additive or has a strongly additive left inverse (for example, a left adjoint). If c and i C are strongly additive, so is c .
(3) Suppose that i C has a left adjoint a C . If c has a left adjoint d, then d = a C di D is a left adjoint of c . If d and a C are exact, so is d . Moreover, a C d = d a D if i D has a left adjoint a D .
(4) Suppose that i C and i D have left adjoints a C and a D , that a D is exact, and that a D c = c a C . If c is exact, then so is c .

Review of sheaf theory on non-proper modulus pairs
In this section we recall some basic definitions and properties on sheaves on categories of non-proper modulus pairs from [KMSY21, §4.1 and §4.2].
Let Sq be the product category of  (1) A Cartesian square We define MNS fin to be the full subcategory of MPS fin consisting of sheaves with respect to the Grothendieck topology associated to P MV fin . Let Let MNST fin be the full subcategory of MPST fin consisting of all objects F ∈ MPST fin such that c fin * F ∈ MNS fin , where c fin * : MPST fin → MPS fin is from (1.2.1).
We write i fin Nis : MNST fin → MPST fin for the inclusion functor and c finNis : MNST → MNS for the functor induced by c fin * . By definition, we have Proof. The first two assertions follow from the general properties of Grothendieck topologies [ We define MNST to be the full subcategory of MPST consisting of those F enjoying these equivalent conditions. We denote by i Nis

Moreover, we have
Therefore the statement follows from Proposition 2.2.9 and the known bound for Nisnevich cohomological dimension.

A cd-structure on MSm
In this section we introduce a cd-structure on MSm and describe its main properties, following the works of Miyazaki [Miy20] and Kahn-Miyazaki [KM20]. For this we need to start with the "off-diagonal" functor.

Off-diagonal
We call the functors the off-diagonal functors.

The MV cd-structure
Definition 3.2.1. Let T be an object of MSm Sq of the form Then T is called an MV-square if the following conditions hold: (1) T is a pull-back square in MSm.
(2) There exist an MV-square S (cf. Definition 2.2.1) such that S(11) ∈ MSm, and a morphism S → T in MSm Sq such that the induced morphism S o → T o is an isomorphism in Sm Sq and S(11) → T (11) is an isomorphism in MSm. In particular, T o is an elementary Nisnevich square.
We let P MV be the cd-structure on MSm consisting of MV-squares.
The following are the main results of [Miy20].

MNS,
where i s,Nis is the natural inclusion and its left adjoint a s,Nis is exact. Moreover, MNS is Grothendieck. For F ∈ MPS, the following conditions are equivalent.
(2) For any MV-square Q as (3.2.1), the associated sequence is exact.
Proof. Same as for Lemma 2.2.4: the first two assertions are general facts on Grothendieck topologies, and the equivalence (i) ⇐⇒ (ii) follows from [Voe10a, Corollary 2.17] in view of Theorem 3.2.2 (1).
The following are the main results of [KM20]. To state them, we need a definition.

Definition 3.2.4.
(1) For any square S ∈ MSm Sq , we define categories Comp(S) as the full subcategories of S ↓ MSm Sq consisting of those objects S → T such that S(ij) → T (ij) belongs to Comp(S(ij)) for any (ij) ∈ Sq.
(2) For an MV fin -square S in MSm fin , an object for the full subcategory consisting of MV-completions of S.
Theorem 3.2.5. The following assertions hold.
(1) [KM20, Theorem 1]. The functor τ s : MSm → MSm is continuous in the sense of [SGA4, exposé III] for the topologies given by P MV and P MV .
Remark 3.2.8. The essential point of the above proof is the fact that the diagram category Sq does not have a loop, and therefore the use of the graph trick terminates in finitely many steps. We remark that we can generalize the proof to a much more abstract argument, cf. [KM18, Lemma C.6]. Proof. Indeed, the sheaf condition is tested on finite diagrams, hence the presheaf given by a direct sum of sheaves is a sheaf.  Proof. First we prove (1). The first assertion follows from the continuity of τ s (Theorem 3.2.5 (1)). Similarly, the second assertion morally follows from the "continuity of τ ! s " (see [KM20, Remark 1]): we give a proof based on Theorem 3.2.5 (2).

Sheaves on MSm
Take F ∈ MNS. By Lemma 2.2.4, it suffices to show that the sequence is exact for any Q ∈ P MV as in (2.2.1). By Remark 1.1.1 (1), we may assume that M is normal. Since a filtered colimit of exact sequences of abelian groups is exact, the desired assertion follows from Lemma 1.2.3 (4), Corollary 3.2.3 and Corollary 3.2.6. Finally the adjointness follows from Lemma 1.2.3 (2). This completes the proof of (1).
All this extends to sheaves of abelian groups by [SGA4, exposé II, Proposition 6.3.1]. The one property which is missing is the right formula of (4.1.1). From (4.1.2), we deduce a base change morphism

Sheaves on MCor
The following is an analogue of Lemma-Definition 2. Proof. Let F ∈ MPST. Then (1) For any M ∈ MCor, the presheaf Z tr (M) ∈ MPST belongs to MNST.
(2) We have a natural isomorphism The second isomorphism of (4.2.3) easily implies that a Nis is left adjoint to i Nis . Then (2) follows from Lemma 1.2.5, Lemma 4.1.1 and Lemma 1.2.6 (3).
Finally, the exactness of a Nis is a consequence of the first isomorphism of (4.2.3) since c Nis is faithfully exact as we have just shown. (1) We have τ * (MNST) ⊂ MNST.
(2) Let τ Nis : MNST → MNST be the functor characterized by Then τ Nis is a right adjoint of τ Nis , and τ Nis is fully faithful, exact, and strongly additive. Moreover, τ Nis preserves injectives and is strongly additive.
Remark 4.2.6. We will see in Theorem 5.1.1 below that τ Nis is also exact.

Main result
We begin with the following. The proof will be given later in this section (see Corollary 5.5.1). We now deduce its consequences. Proof. By Theorem 5.1.1, τ Nis is exact and it preserves injectives by Lemma 4.2.5. Hence we have R q (i Nis )τ Nis G = R q (i Nis τ Nis )G = R q (τ * i Nis )G = τ * R q i Nis G for any G ∈ MNST by [KMSY21, Theorem A.9.1]. Using the projectivity of Z tr (M) in MPST and that of τ ! Z tr (M) = Z tr (M) in MPST, and using Proposition A.1.1 twice, we get isomorphisms

A generation lemma
We now start proving Theorem 5.1.1. We need some preliminaries. (1) We put Note Z tr (M) τ ∈ MNST by Lemma 4.2.5. Remark 5.3.4. We can prove that Z fin tr (M) τ lies in MNST fin (so that we may remove a fin Nis in Theorem 5.4.1(1) below). (1) TheČech complex

Exactness of a certainČech complex
Theorem 5.4.1 (2) follows from (1) by applying the exact functor b Nis from Proposition 2.2.9 and using isomorphisms b Nis a fin Nis Z fin tr (M) τ a Nis b ! Z fin tr (M) τ a Nis Z tr (M) τ = Z tr (M) τ , where the first isomorphism follows from (2.2.3), the second from Lemma 5.3.3 and the last equality follows from the fact that Z tr (M) τ ∈ MNST thanks to Lemma 4.2.5 (1).
We need some preliminaries for the proof of (1). It is inspired by that of [KMSY21,Theorem 3.4.1], with some elaboration. Take (X, D) ∈ MCor and a point x ∈ X. Let {X λ } λ∈Λ be the filtered system of connected affine étale neighborhoods of x ∈ X. Let where v : V N → V is the normalization and i V : V → Z is the inclusion. Let E τ (Z) ⊂ E(Z) be the subset of those V which belong to Z fin tr (M) τ (S, D), i.e. satisfying the following condition: there exists λ ∈ Λ such that (Z, f , g) (resp. V → Z) is the base change via S → X λ of (5.4.5) where V λ is an irreducible component of Z λ satisfying the condition: (♣) λ V λ is finite over X λ and satisfies the admissibility condition Let L τ (Z) be the free abelian group on the set E τ (Z).
Lemma 5.4.2. Let V λ be as in (♣) λ and X µ → X λ (λ, µ ∈ Λ) be a map in the system of étale neighborhoods of x ∈ X. Let Proof. The finiteness over X µ and the admissibility condition of (♣) µ are clearly satisfied. To check the last condition of (♣) µ , let X µ → X µ be the normalization in X µ of X λ from (♣) λ and letṼ µ = h µ (V µ ) with which is proper over X µ by the assumption.
HenceṼ µ is also proper over X µ , which implies the desired condition.

Lemma 5.4.3. For a commutative diagram (5.4.3), there is a natural induced map
which makes E τ a covariant functor on D.
Proof. Take V ∈ E(Z) and let V = ϕ(V Hence V λ satisfies the last condition of (♣) λ since V λ does. This implies V ∈ E τ (Z ).
Proof of Theorem 5. 4.1 (1). It suffices to show the exactness of (5.4.8) · · · → Z fin tr (U × M U ) τ (S) → Z fin tr (U ) τ (S) → Z fin tr (M) τ (S) → 0 where S = (S, D × X S) with (X, D) and S as in (5.4.1). We first note that for a closed subscheme Z ⊂ S × U × M · · · × M U finite and surjective over an irreducible component of S, the image of Z in S × M is finite over S. From this fact we see that (5.4.8) is obtained as the inductive limit of (5.4.9) where Z ranges over all closed subschemes of S × M that is finite surjective over an irreducible component of S. It suffices to show the exactness of (5.4.9). Since Z is finite over a henselian local scheme S, Z is a disjoint union of henselian local schemes. Thus the Nisnevich cover Z × M U → Z admits a section s 0 : Z → Z × M U . Define for k ≥ 0 where U k is the k-fold fiber product of U over M. Then the maps give us a homotopy from the identity to zero.
(2) The base change morphism a Nis τ * ⇒ τ Nis a Nis is an isomorphism.
(1) Since a Nis , τ ! and τ * all commute with representable colimits as left adjoints, we are reduced by Lemma 5.2.1 to G of the form Z tr (M/U ), which is equivalent to where the cokernel is taken in MPST. This follows from Theorem 5.4.1(2).
(2) Let F ∈ MNST. The base change morphism a Nis τ * F → τ Nis a Nis F is defined as the composition where η (resp. ε) is the unit (resp. counit) of the adjunction (a Nis , i Nis ) (resp. (a Nis , i Nis )). Since the second map is an isomorphism by the full faithfulness of i Nis , it remains to show that the first one is an isomorphism. By the full faithfulness of τ Nis (Lemma 4.2.5), it suffices to show it after applying this functor. But a Nis τ ! τ Nis a Nis by Theorem 4.2.4, so we are left to show that the map is an isomorphism. This follows from (1), since a Nis τ ! τ * is exact, and Ker η F and Coker η F are killed by a Nis . Finally, (3) follows from (2), Lemma 1.2.6 (4) and the exactness of τ * .
Corollary 5.5.2. The functor τ Nis has a right adjoint.
Proof. The category MNST is cocomplete and has a small set of generators, as a Grothendieck category (Theorem 2.2.7.) Moreover, τ Nis respects all representable colimits as an exact, strongly additive functor (Lemma 4.2.5 and Corollary 5. 5.1 (3)). Thus the dual hypotheses of the "special adjoint functor theorem" [Mac98, Chapter V, Section 8, Theorem 2] are verified.

MNS, MNS and NS
We consider the functors Proof. a) Since λ s is left adjoint to ω s , we have ω s,! = λ * s , hence the continuity of λ s proves the first assertion. The second one follows from Theorem 4.1.2, as ω s,! = ω s,! τ s,! . b) If ω * s F ∈ MNS, then ω * s F = τ * s ω * s F ∈ MNS by Theorem 4.1.2. If ω * s F ∈ MNS, then ω s,! ω * s F ∼ − → F ∈ NS by a) since ω * s is fully faithful. If F ∈ NS, then we have ω * s F ∈ MNS since ω s is continuous. c) The second formula of (6.1.3) follows from the cocontinuity of ω s (Proposition 6.1.1) and [SGA4, exposé III, Proposition 2.3 (2)]. We prove the rest of the assertions by using Lemma 1.2.6 as follows. In the situation of Lemma 1. Indeed, i s,Nis is strongly additive (Lemma 4.1.1), ω * s is exact as a left and right adjoint, i s,Nis has an exact left adjoint a s,Nis , and ω s,! is exact. This finishes the proof.

MNST, MNST and NST
Let PST be the abelian category of presheaves on Cor. The graph functor c V : Sm → Cor induces an exact faithful functor c V * : PST → PS. Let NST be the full subcategory of PST consisting of F ∈ PST such that c V * F ∈ NS. The functor c V * restricts to c V ,Nis : NST → NS.
One also sees from [KMSY21, (2.2.1)] (and its analogues for ω s , ω s ) that Proposition 6.2.1. Moreover, the functors ω Nis and ω Nis are both exact, ω Nis is fully faithful, strongly additive and preserves injectives. The assumptions of Lemma 1.2.6 (2) and (3) are satisfied, since i Nis is strongly additive by Theorem 4.2.4 (1), i V Nis is strongly additive by the quasi-compactness of Nisnevich cohomology, ω * is strongly additive as a left and right adjoint, i V Nis , i Nis have exact left adjoints a V Nis , a Nis by [MVW06] and 4.2.4 (2), and ω ! is exact by [KMSY21, Proposition 2.2.1]. Therefore, the assertions follow, except for the second identity of (6.2.6) and the exactness of ω Nis . (Note that the exactness of ω Nis implies that ω Nis preserves injectives.) We can prove the second identity of (6.2.6) as follows. Its first identity yields a base change morphism a Nis ω * ⇒ ω Nis a V Nis . Let F ∈ PST. We want to show that the morphism a Nis ω * F → ω Nis a V Nis F is an isomorphism. Since c Nis is conservative as MSm and MCor have the same objects, it suffices to show that the induced morphism c Nis a Nis ω * F → c Nis ω Nis a V Nis F is an isomorphism. Since = ω Nis s a V s,Nis c V * , the above morphism is rewritten as a s,Nis ω * s c V * F → ω Nis s a V s,Nis c V * F, which is an isomorphism by Proposition 6.1.2 c). Now, the formula we have proven and the exactness of ω * as a left and right adjoint show that the assumption of Lemma 1.2.6 (4) is satisfied. Hence ω Nis is exact, as desired. This finishes the proof of c).
d) The proof is completely parallel to that of c). To see this, it suffices to observe the following: i Nis is strongly additive and has an exact left adjoint by [KMSY21,Lemma 4.5 = ω Nis s a V s,Nis c V * . This finishes the proof.

Relation between cohomologies
We now prove Theorem 3 (2) from the introduction. We fix M ∈ MSm(X) and will show that the composition I(M) → ω Nis I(X) → ω Nis I(U ) is already surjective. This follows from the functoriality of ω ! , but for clarity we give an explicit argument. Take any N ∈ MSm(U ). Proof.

From D(MNS fin ) to D(MNS)
We  We are now left to show the strong additivity of the three functors. For D(b s,Nis ), this follows from the strong additivity of b s,Nis as a left adjoint, and Proposition A.2.8 a).
(2) One has isomorphisms Remark 7.5.2. One can show that the essential image of D(τ Nis ) is Since the proof involves delicate and lengthy arguments relying on the notion of left-completeness, we skip it (see [KSY15]).

D(MNST), D(MNST) and D(NST)
We leave it to the reader to produce an unbounded version of Theorem 6.3.2.

A.1. A spectral sequence
The following convenient proposition is used several times in the paper.
given by the unit map of the adjunction (λ A , ρ A ) is an isomorphism.  Here is a first application: Proposition A.2.5. Assume C = A, F right adjoint to G, and G exact. Then RF is right adjoint to RG = D(G).
Proof. This is a special case of [KS06,Theorem 14.4.5].
We come back to the general situation. Suppose that F carries injectives of A to G-acyclics. Then where σ ≥n is the stupid truncation. This isomorphism still holds in D(A), because λ A is strongly additive. By the hypothesis, this reduces us to the case where M ∈ D + (A), and therefore to Grothendieck's theorem (cf. Theorem A.2.1 c)).
Let F : A → B be a left exact functor between Grothendieck categories. In view of Lemma A.2.7, we need a practical sufficient condition to ensure that RF is strongly additive. The following ones are adapted to the context of this paper: Then RF is strongly additive.

c) Suppose that RF admits a left adjoint G which sends a set (E α ) of compact generators of D(B) to compact objects of D(A).
Then RF is strongly additive.
Proof. a) The strong additivity of F easily implies that of K(F), which in turn implies that of D(F) since λ B is strongly additive as a left adjoint. b) Let (C i ) i∈I ∈ D(A) I . We must show that the map for any D ∈ D(B); since E is compact in B, this formula shows that E[n] is compact in D(B). Therefore we must show that the homomorphisms i∈I B(E, H n (RFC i )) → B(E, H n (RF i∈I C i )).
are bijective. By (ii), the spectral sequence B(E, R p FH q (C)) ⇒ B(E, H p+q (RFC)) converges for any C ∈ D(A). Thus it suffices to show that the homomorphisms i∈I B(E, R p FH q (C i )) → B(E, R p FH q ( i∈I C i )) are bijective. By (i), this follows from the compactness of E. c) Keep the notation of b is an isomorphism for all q ∈ Z. By adjunction, it is transformed into which is an isomorphism since the G(E α ) are compact.
Finally, we need a practical sufficient condition to ensure that, in Condition (i) of Proposition A.2.8 b), the case p = 0 implies the cases p > 0. This is given by the classical Lemma A.2.9. Suppose that F is strongly additive and that, in A, infinite direct sums of injectives are F-acyclic. Then R p F is strongly additive for any p > 0.