Motives with modulus, I: Modulus sheaves with transfers for non-proper modulus pairs

We develop a theory of modulus sheaves with transfers, which generalizes Voevodsky's theory of sheaves with transfers. This paper and its sequel are foundational for the theory of motives with modulus, which is developed in [KMSY20].


Introduction
The aim of this paper is to lay a foundation for a theory of motives with modulus, which will be completed in [KMSY20], generalizing Voevodsky's theory of motives. Voevodsky's construction is based on A 1invariance. It captures many important invariants such as Bloch's higher Chow groups, but not their natural generalisations like additive Chow groups [BE03,Par09] or higher Chow groups with modulus [BS19]. Our basic motivation is to build a theory that captures such non A 1 -invariant phenomena, as an extension of [KSY16].
Let Sm be the category of smooth separated schemes of finite type over a field k. Voevodsky's construction starts from an additive category Cor, whose objects are those of Sm and morphisms are finite correspondences. We define PST as the category of additive presheaves of abelian groups on Cor (i.e. functors Cor → Ab that commute with finite sums). Let NST ⊂ PST be the full subcategory of those objects F ∈ PST whose restrictions F X to X Nis is a sheaf for any X ∈ Sm, where X Nis denotes the small Nisnevich site of X, that is, the category of all étale schemes over X equipped with the Nisnevich topology. Objects of NST are called (Nisnevich) sheaves with transfers. For F ∈ NST, we write H i Nis (X, F) = H i (X Nis , F X ). The following result of Voevodsky [Voe00, Theorem 3.1.4] plays a fundamental rôle in his theory of motives.
(1) The inclusion NST → PST has an exact left adjoint a V Nis such that for any F ∈ PST and X ∈ Sm, (a V Nis F) X is the Nisnevich sheafication of F X as a presheaf on X Nis . In particular NST is a Grothendieck abelian category.
(2) For X ∈ Sm, let Z tr (X) = Cor(−, X) ∈ PST be the associated representable additive presheaf. Then we have Z tr (X) ∈ NST and there is a canonical isomorphism for any i ≥ 0 and F ∈ NST: H i Nis (X, F) Ext i NST (Z tr (X), F). Our basic principle for generalizing Voevodsky's theory of sheaves with transfers is that the category We then define MPST (resp. MPST) as the category of additive presheaves of abelian groups on MCor (resp. MCor). We have a functor where τ * is induced by the inclusion τ : MCor → MCor and τ ! is its left Kan extension, and ω * is induced by ω and ω ! is its left Kan extension (see Propositions 2.4.1 and 2.2.1).
The main aim of this paper is to develop a sheaf theory on MCor generalizing Voevodsky's theory. is contravariantly functorial for morphisms in MCor, which does not follow immediately from the definition.
The preprint [KSY15] contained a mistake, pointed out by Joseph Ayoub: namely, Proposition 3.5.3 of loc. cit. is false. Theorem 2 (1) shows that the only false thing in that proposition is that the functor b Nis of loc. cit. is not exact, but only left exact (see Proposition 4.5.4 of the present paper.) This weakens [KSY15, Proposition 3.6.2] into Theorem 2 (2); see however Question 1 below. What we gain in the present correction is that the notion of sheaf, which was artificially developed in [KSY15] for MCor, corresponds now to a genuine Grothendieck topology.
Another proposition incorrectly proven in [KSY15] was Proposition 3.7.3. In Part II of this work [KMSY21], we correct this proof and recover the proposition in full, hence get a good sheaf theory also for proper modulus pairs. This allows us to develop the categories of motives again in [KMSY20].
In the last part of this introduction, we raise the following question. Its affirmative answer would simplify the right hand side of Theorem 2 (2) under two additional conditions (i) and (ii) below. (These conditions turn out to be essential in [Sai20].) Question 1. Assume that F ∈ MNST satisfies the following conditions: If ch(k) = 0, by resolution of marked ideals ([BM08, the case d = 1 of Theorem 1.3]), the above question is reduced to the following. an isomorphism?

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 fourth author was visiting IMJ-PRG supported by Fondation des Sciences Mathématiques de Paris. We are grateful to the support and hospitality received in all places.
We thank Ofer Gabber and Michel Raynaud for their help with Lemma 1.6.1, and Kay Rülling for pointing out an error and correcting Definition 1.8.1.
We are very grateful to Joseph Ayoub for pointing out a flaw on the computation of the sheafification functor a Nis and on the non-exactness of the functor b Nis in the earlier version. The authors believe that the whole theory has been deepened by the effort to fix it. We also thank the referees for a careful reading and many useful comments.
Finally, the influence of Voevodsky's ideas is all-pervasive, as will be evident when reading this paper.

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 [Voe00].  (2) In the first version of this paper, we imposed the condition that M is locally integral; it is now removed. The main reason for this change is that this condition is not stable under products or extension of the base field. The next remark shows that this removal is reasonable (see also Remark 1.3.8).
The following lemma will play a key rôle: Lemma 1.1.3. Let X ∈ Sch and let X be an open dense subscheme of X. Assume that X ∈ Sm and that X − X is the support of a Cartier divisor. Then for any modulus pair N we have where M ranges over all modulus pairs such that M = X and M o = X. (Note that by definition we have Proof. This is proven in [KSY16, Lemma 2.6.2]. In loc. cit. X and N o are assumed to be quasi-affine, and X and N proper and normal (see Remark 1.1.2). 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 1.2.4 below.)

Composition
To discuss composability of admissible correspondences, we need the following lemma of Krishna We also need the following "containment lemma" from [KP12, Proposition 2.4], [BS19, Lemma 2.1], [Miy19, Lemma 2.4]. We provide a proof for self-containedness.
Proof. Set Z := V N × V V and consider the following commutative diagram: To see that such a Z γ exists, it suffices to note that V N → V is finite surjective, hence so is its base change Z → V (recall that for any scheme S of finite type over k, the normalization S N → S is a finite surjective morphism). Then Z N γ is also irreducible. Since Z N γ → V is dominant, the vertical map h on the left exists by the universal property of normalization, and is finite surjective. Note that we can pullback the Cartier divisor M ∞ to any scheme except for Z in the diagram, since none of their irreducible components maps into the support |M ∞ | ⊂ M. Since the pullback of an effective Cartier divisor is effective, the assumption that M ∞ | V N is effective implies that , and¯denotes closure. The hypothesis implies thatγ →δ is proper surjective. The same holds for π N γδ appearing in the second of the two other commutative diagrams: where N means normalisation. (Note that π γα and π γβ need not extend to the normalisations, as they need not be dominant.) We have the admissibility conditions for α and β: Applying 2 Lemma 1.2.2, we get inequalities , which implies by the right half of the above diagram by Lemma 1.2.1. Finally, one trivially checks that (i) or (ii) implies that the projection α × M 2 β → M 1 × M 3 is proper, and that (iii) implies both of (i) and (ii).
and α = β = graph of the identity on A 1 . Then α and β are admissible but β • α is not admissible because ∞ ≥ 2 · ∞ does not hold. (Note that neither of α = α or β = β is proper over P 1 .) Proof. We may assume α and β are irreducible. The assumption on β means β → M 2 is proper, hence so is its base change α × M 2 β → α. The assumption on α means α → M 1 is proper, hence so is α × M 2 β → M 1 as a composition of proper morphisms. This implies the left properness of βα, since βα is the image of

Categories of modulus pairs
Definition 1.3.1. By Propositions 1.2.4 and 1.2.7, modulus pairs and left proper admissible correspondences define an additive category that we denote by MCor. We write MCor for the full subcategory of MCor whose objects are proper modulus pairs (see Definition 1.1.1 (1)).
In the context of modulus pairs, the category Sm and the graph functor Sm → Cor are replaced by the following: Definition 1.3.2. We write MSm for the category with the same objects as MCor, and a morphism of MSm(M 1 , M 2 ) is given by 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. We will need some variants of these categories. (1) 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, where Z is the closure of Z in M × N . The same argument as in the proof of Proposition 1.2.7 shows that MCor fin is indeed a subcategory of MCor. We write MCor fin for the full subcategory of MCor whose objects are proper modulus pairs.
(2) We write MSm fin for the subcategory of MSm with the same objects and such that a morphism   Proof. Let Γ be the graph of the morphism U → N , and let Γ be its closure in M × N . Then we have natural projections p 1 : Γ → M and p 2 : Γ → N . Since we have Γ U , Lemma 1.3.7 below implies that p 1 is an isomorphism over U and we have p −1 1 (U ) = Γ . Defining M 1 := (Γ , p * 1 M ∞ ), the morphism p 1 induces a morphism p 1 : M 1 → M in MSm fin such that f • p 1 : M 1 → M → N comes from MSm fin defined by p 2 . Also note that Γ → M is proper since f is, which implies that p 1 : M 1 → M is an isomorphism in MSm. This finishes the proof. Proof. Consider the commutative diagram  Proof. This follows from the definition and the fact that if A is an integral domain with quotient field K, then a ∈ K is integral over A if and only if so is a n .

Changes of categories
We now have a basic diagram of additive categories and functors . All these functors are faithful, and τ is fully faithful; they "restrict" to analogous functors τ s , ω s , ω s , λ s between MSm, MSm and Sm. Note that ω • (−) (n) = ω for any n. Moreover: Lemma 1.5.1. We have ωτ = ω. Moreover, λ is left adjoint to ω, and the restriction of λ to Cor prop (finite correspondences on smooth proper schemes over k) is "right adjoint" to ω. (i.e., Cor(ω(M), X) = MCor(M, λ(X)) for M ∈ MCor and X ∈ Cor prop .) The same statements are valid for τ s , ω s , ω s , λ s when restricted to MSm, MSm and Sm.
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 over X and M is separated; it is evidently finite (hence proper) over X. It also satisfies q * M ∞ = 0 where q is the composition 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 last claim is immediate.
The following theorem is an important refinement of Lemma 1.5.1. The proof starts from §1.7 and is completed in §1.8. Theorem 1.5.2. The functors ω, τ, ω s and τ s have pro-left adjoints ω ! , τ ! , ω ! s and τ ! s (see §A.2). General definitions and results on pro-objects and pro-adjoints are gathered in § §A.1 and A.2. We shall freely use results from there.

The closure of a finite correspondence
We shall need the following result for the proof of Theorem 1.5.2. Lemma 1.6.1. Let X be a 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 [RG71, Corollary 5.7.10] applied with (S, X, U ) ≡ (X, Z 1 , U ) and n = 0 (note that a morphism is finite if and only if it is quasi-finite and proper, and that an admissible blow-up of an algebraic space is a scheme if the algebraic space happens to be a scheme).
Theorem 1.6.2. Let X, Y ∈ Sch. 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 .
The following lemma also relies on [RG71]: it will be used several times in the sequel. Lemma 1.6.3. Let f : U → X be an étale morphism of quasi-compact and quasi-separated integral schemes. Let g : V → U be a proper birational morphism, T ⊂ U a closed subset such that g is an isomorphism over Proof. The following argument is taken from the proof of [SV00, Proposition 5.9]. Noting V is étale over X − S, we apply the platification theorem [RG71, Corollary 5.7.11] to V → X and conclude that there exists a closed subscheme Z supported in S such that the proper transform V of V under X = Bl Z (X) → X is flat over X . By the construction the induced morphism ϕ : V → U × X X is proper birational. On the other hand ϕ is flat since it becomes flat when composed with the étale morphism U × X X → X ([Har77, Chapter II, Proposition 8.11 and Chapter III, Exercise 10.3]). Hence it is an isomorphism. This proves the lemma since V → U factors V → U . (1) Identities, stability under composition: obvious.
(2) Given a diagram in MCor ∞ allows us to complete the square in MCor.
(3) Given a diagram (2) and such that sf = sg, the underlying correspondences to f and g are equal since the one underlying s is 1 M o 2 . Hence f = g. The above proof of (2) also shows that we have for any M, N ∈ MCor.
Point b) now follows from a) and Corollary A.5.5, noting that ω is essentially surjective. Indeed, any smooth k-scheme X admits a compactificationX by Nagata's theorem; blowing upX − X, we then make it a Cartier divisor. The case of ω s is exactly parallel.
Let ω ! : Cor → pro-MCor be the pro-left adjoint of ω. By Proposition A.6.2, we have for X ∈ Cor: and the same formula for the pro-left adjoint ω ! s of ω s . Let us spell out the indexing set MSm(X) of these pro-objects, and refine them: (1) For X ∈ Sm, we define a subcategory MSm(X) of MSm as follows. The objects are those M ∈ MSm (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 1.7.4. a) For any X ∈ Sm and any Cartier compactification X, MSm(X) is a cofiltered ordered set, and Proof. a) "Ordered" is obvious and "cofiltered" follows from Propositions 1.7.2 and A.5.2 a); the cofinality follows again from Lemma 1.1.3. 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 .

Proof of Theorem 1.5.2: case of τ
We need a definition: Note that any γ as above induces an isomorphism N o Proof. That it is ordered is obvious as Comp(M)(N 1 , N 2 ) has at most 1 element for any (N 1 , N 2 ). For "cofiltered", we first show that Comp(M) is nonempty. For this, choose a compactification j 0 : ; then j 0 lifts to j 1 : M → N 1 by the universality of the blowup [Har77, Chapter II, Proposition 7.14], and N 1 − M is the support of an effective Cartier divisor C 1 .
dominates N 2 , we are reduced to the case that N 1 and N 2 have the same ambient space N . Let C be the effective Cartier divisor on N such that |C| = N − M, which exists since N 1 ∈ Comp(M). Then for a sufficiently large n we have N ∞ This finishes the proof. For M ∈ MCor and L ∈ MCor we have a natural map which maps a representative α N ∈ MCor(N , L) to α N • j N . We also have a natural map for M, L ∈ MCor The following is an analogue to Lemma 1.1.3: Lemma 1.8.3. The maps Φ and Ψ are isomorphisms. In other words, the formula , defines a pro-left adjoint to τ, which is fully faithful.
Proof. We start with Φ.
Note that M ∞ 2 and C 2 are effective Cartier divisors on N 2 . By the universal property of the blowup [Har77, Chapter II, Proposition 7.14], j 1 extends to an open immersion j 2 : M → N 2 so that . Now the claim for Φ follows from the following: Claim 1.8.4. For any α ∈ MCor(M, L), there exists an integer n > 0 such that α ∈ MCor((N 2 , M ∞ 2 + nC 2 ), L). Indeed we may assume α is an integral closed subscheme of M o × L o . We have a commutative diagram , and j 1 and π are induced by j 1 : M → N 1 and π : N 2 → N 1 respectively. Now the admissibility of for a sufficiently large n > 0. Applying π * to this inclusion, we get an inequality of Cartier divisors Next we prove that Ψ is an isomorphism. Injectivity is obvious since both sides are subgroups of Cor(L o , M o ). We prove surjectivity. Take for any integer n > 0. Clearly this implies that |δ| does not intersect with L × |C| so that δ ⊂ L × M. Since δ is proper over L by assumption, this implies δ ∈ MCor(L, M) which proves the surjectivity of Ψ as desired.
We come back to the proof of Theorem 1.5.2. It remains to consider τ s . The natural maps are also bijective for any M, L ∈ MCor and L ∈ MCor. The proof is identical to Lemma 1.8.3. In particular, the inclusion functor τ s : MSm → MSm admits a pro-left adjoint given by , which commutes with the inclusions MSm → MCor and MSm → MCor. This completes the proof of Theorem 1.5.2.

More on MSm fin and MCor fin
proper morphism and f o is an isomorphism in Sm. We write Σ fin for the class of morphisms in Σ fin that belong to MSm.
In particular, we have Σ fin ⊂ Σ (see Definition 1.7.1) and The following commutative diagram of categories will become fundamental (cf. (2.7.1)): a for some s 1 , s 2 ∈ Σ fin .
All statements hold for Σ fin (without an underline) as well.
Proof. a) Same as the proof of Proposition 1.7.2 a), except for (2): consider a diagram in MCor fin and the corresponding map for b s are isomorphisms. These maps are clearly injective, and its surjectivity follows again from Theorem 1.6.2. It then follows from Proposition A.6.2 they have pro-left adjoints.
The first statement of c) is clear, and the second follows from b). The same proof works for Σ fin .
Corollary 1.9.4. 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 1.9.2 b).
The corollary readily follows.
We also have the following important lemma: 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 We first reduce to the case where α is integral. To do this, it suffices to show that for two distinct integral Now assume α is integral and put β : 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, exposé VIII, corollaire 4.8]. We also have , and we are done.

Fiber products and squarable morphisms
We need the following elementary lemma.
Lemma 1.10.1. Let X be a scheme. For two effective Cartier divisors D and E on X, the following conditions are equivalent: (1) D × X E is an effective Cartier divisor on X.
(2) There exist effective Cartier divisors D , E and F on X such that D = D +F, E = E +F and |D |∩|E | = ∅.

Moreover, the divisors D , E and F satisfying the conditions in (2) are uniquely determined by D and E.
Proof. We may suppose X = Spec A is affine and D, E are defined by non-zero-divisors d, e ∈ A, respectively. (2) immediately implies F = D × X E, whence (1). This formula also implies the uniqueness of F, hence D = D − F and E = E − F are unique as well. (1) We have | sup(D, E)| = |D| ∪ |E|.
(2) If f : Y → X is a morphism such that f (T ) |D| ∪ |E| for any irreducible component T of Y , then f * D and f * E have a universal supremum which is equal to f * sup(D, E) (hence the name "universal").
(3) If moreover Y is normal, then f * sup(D, E) agrees with the supremum of f * D and f * E computed as a Weil divisor on Y .
We write q i : W 1 → U i for the composition of the inclusion W 1 → W 0 and p i . By definition, we have effective Cartier divisors q * i (U ∞ i ) on W 1 and q 1 × q 2 restricts to an isomorphism (1.10.1) represents the fiber product of U 1 and U 2 over M in MSm fin as well as in MSm. If further U 1 , U 2 , M ∈ MSm fin , then it holds in MSm fin as well as in MSm.
(2) If u 1 is minimal and U 2 is normal, then q * 1 U ∞ 1 and q * 2 U ∞ 2 have a universal supremum, namely q * 2 U ∞ 2 , and the morphism W 1 → U 2 is a minimal morphism in MSm fin . If moreover u 1 is flat 3 , we have (3) In general, there is a proper birational morphism π : W 2 → W 1 which restricts to an isomorphism over Then the morphisms by definition and Remark 1.10.3, this follows from the admissibility of f i , that is, We have shown that W 1 represents the fiber product in MSm fin . Propositions 1.9.2 and A.5.6 show that the same holds in MSm as well. (This also follows from (3) below.) The last statement is an immediate consequence of the first.
(2) Let p W : W N 1 → W 1 and p U 1 : U N 1 → U 1 be the normalizations. By the minimality of u 1 , we have where the last inequality holds by the admissibility of u 2 and the normality of U 2 . Then q * 1 U ∞ 1 and q * 2 U ∞ 2 have a universal supremum since q * 1 U ∞ 1 ⊂ q * 2 U ∞ 2 implies Condition (1) of Lemma 1.10.1, which also implies that This shows the minimality of W 1 → U 2 .
Suppose now u 1 flat, and let T be an irreducible component of W 0 . Then p 2 : W 0 → U 2 is also flat, hence T dominates an irreducible component E of U 2 [Har77, Chapter III, Proposition 9.5] and we cannot have p 2 (T ) ⊂ |U ∞ 2 | since U ∞ 2 is everywhere of codimension 1 in U 2 . Suppose that p 1 (T ) ⊂ |U ∞ 1 |. By the minimality of u 1 , this implies u 2 p 2 (T ) = u 1 p 1 (T ) ⊂ |M ∞ |, hence u 2 (E) ⊂ |M ∞ |, contradicting the admissibility of u 2 .
(3) If π is the blow-up of W 1 with center q * 1 (U ∞ 1 ) × W 1 q * 2 (U ∞ 1 ), then r * 1 U ∞ 1 × W 2 r * 2 U ∞ 2 is precisely the exceptional divisor by definition, which is therefore an effective Cartier divisor, showing the first assertion. by (1.10.1). Now let f i : N → U i be morphisms in MSm for i = 1, 2 such that u 1 f 1 = u 2 f 2 . Then the morphisms  ( it is squarable in this category. Proof. (1) follows from Proposition 1.10.4 (1) and (2) Proof. This is the special case M = (Spec k, ∅) in Corollary 1.10.7 (2).

Modulus presheaves with transfers
Definition 2.1.1. By a presheaf we mean a contravariant functor to the category of abelian groups.
(2) The category of additive presheaves on MCor (resp. MCor, MCor fin ) is denoted by MPST (resp. MPST, MPST fin .) All these categories are abelian Grothendieck, with projective sets of generators: this is classical for those of (1) and follows from Theorem A.10.2 for those of (2). (See also proof of Proposition 2.6.1 below.)  MCor(N , M), etc.) We shall use the common notation Z tr but they will be distinguished by the context. We now briefly describe the main properties of the functors induced by those of the previous section.

MPST and PST
We say (f 1 , f 2 , . . . , f n ) is a string of adjoint functors if f i is a left adjoint of f i+1 for each i = 1, . . . , n − 1.
Let X ∈ Sm and let M ∈ MSm(X). Lemma 1.7.4 and Proposition A.4.1 show that the inclusions
Proof. The only non obvious statement is the last claim, which follows from Lemma 1.5.1.
(3) For F ∈ PST and M ∈ MCor, we compute We are done.
Remark 2.4.3. By Lemma 1.8.3 we have the formulas where the latter inverse limit is computed in MPST.  Proof. This follows from the usual yoga applied with Proposition 1.9.2 and Lemma A.3.1.

With and without transfers
Proposition 2.6.1. Let c : MSm → MCor be the functor from (1.3.1). Then c yields a string of three adjoint functors (c ! , c * , c * ): where c * is exact and faithful (but not full). We have The same statements hold for c : MSm → MCor and c fin ; MSm fin → MCor fin from (1.3.1). Precisely, they yield strings of three adjoint functors (c ! , c * , c * ) and (c fin ! , c fin * , c fin * ); c * and c fin * are exact and faithful. (The analogue of (2.6.1) also holds for c and c fin , but we will not need it.) Proof. To define c ! , c * and c * , we use the free additive category Z MSm on MSm [Mac98, Chapter VIII, Section 3, Exercises 5 & 6]: it comes with a canonical functor γ : MSm → Z MSm and is 2-universal for contravariant functors to additive categories. In particular: • The functor c induces an additive functorc : Z MSm → MCor. As usual,c induces a string of three adjoint functors (c ! ,c * ,c * ) (see §A.4). We then define c ! asc ! • (γ * ) −1 , etc. Everything follows from this except the faithfulness of c * , which is a consequence of the essential surjectivity of c. The cases of c fin and c are dealt with similarly.
(1) We have Proof. The first two equalities of (1) follows from the equality b c fin = c b s (see (1.9.2)). Similarly, the first equality of (2) follows from τc = cτ s . By (2.5.1), we have for any F ∈ MPST fin and M ∈ MSm. (Note that all morphisms of Σ fin ↓ M are in MSm fin , and that both of b ! and b s! can be computed by using the same Σ fin ↓ M.) This proves the last formula of (1). Lemma 2.4.2 (1) shows that for any F ∈ MPST and M ∈ MSm. The last one of (2) is similar.

A patching lemma
By the previous lemma, we obtain a commutative diagram of categories (cf. (1.9.2)): (2.7.1) All vertical arrows are faithful and horizontal ones fully faithful.
Lemma 2.7.1. Both squares of (2.7.1) are "2-Cartesian". More precisely, the following assertions hold. ( It is straightforward to see that (F s , F t , ϕ) → F gives a quasi-inverse.
2.8. The functors n ! and n * As in §A.4, the functor (−) (n) of Definition 1.4.1 induces a string of adjoint endofunctors (n ! , n * , n * ) of MPST, where n * is given by n * (F)(M) = F(M (n) ). We shall not use n * in the sequel. Proof. This follows formally from the same properties of (−) (n) . Proof. Let F ∈ MPST. For X ∈ Cor, we have

Proposition 2.8.2. For any F ∈ MPST, there is a natural isomorphism
where the last isomorphism follows from Lemma 1.7.4.

Nisnevich topology on MSm fin
Definition 3.1.1. We call a morphism p : U → M in MSm fin a Nisnevich cover if (i) p : U → M is a Nisnevich cover of M in the usual sense; (ii) p is minimal (that is, U ∞ = p * (M ∞ )).
Since the morphisms appearing in the Nisnevich covers are squarable by Corollary 1.10.7 (1), we obtain a Grothendieck topology on MSm fin . The category MSm fin endowed with this topology will be called the big Nisnevich site of MSm fin and denoted by MSm fin Nis . The following lemma is obvious from the definitions: where α : V → U is a morphism in MCor fin and p : V → M is a Nisnevich cover in 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 Nisnevich cover p : V → M such thatp in the diagram (all squares being cartesian) has a splitting s. Put V := (V , p * (M ∞ )) ∈ MSm. The image of s gives us a desired correspondence α .
Remark 3.1.5. One can also define the Zariski and étale topologies on MSm fin . Most results of this section (notably Theorems 3.4.1, 3.5.3, and Corollary 3.5.6) remain true for the étale topology, but not for the Zariski topology (e.g. Lemma 3.1.4 already fails for it). However, from the next section onward we will make essential use of cd-structures. As the étale topology cannot be defined by a cd-structure, we decide to stick to the Nisnevich topology from the beginning.

A cd-structure on MSm fin
Let Sq be the product category of For any category C, denote by C Sq for the category of functors from Sq to C. A functor f : C → C induces a functor f Sq : C Sq → C Sq . We refer to §A.7 for the notion of cd-structure, and its properties. (1) A Cartesian square Proof. We show the stronger statement that Z tr (M) restricts to an étale sheaf on Né t for any N ∈ MCor fin . Let p : U → N be an étale cover and let U := (U , p * N ∞ ). We have a commutative diagram The bottom row is exact by [MVW06, Lemma 6.2]. The exactness of the top and middle row now follows from Lemma 1.9.6. Before starting the proof of Theorem 3.4.1, it is convenient to generalize the notion of relative cycles to the modulus setting.

3.4.Čech complex
where the bottom horizontal map is composition by the graph of u. This yields subcomplexes This proves that (3.4.2) is obtained as the filtered inductive limit of the complexes (3.4.3) when Z ranges over C f (M). It suffices to show the exactness of (3.4.3) for such a Z.
Since Z is finite over the 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 ≥ 1 where U k := U × M · · · × M U . Then the maps induced by s k via Lemma 3.4.5 give us a homotopy from the identity to zero.

Sheafification preserves finite transfers
Let a fin sNis : MPS fin → MNS fin be the sheafification functor, that is, the left adjoint of the inclusion functor i fin sNis : MNS fin → MPS fin . It exists for general reasons and is exact [SGA4, exposé II, théorème 3.4].
Definition 3.5.1. Let MNST fin be the full subcategory of MPST fin consisting of all objects F ∈ MPST fin such that c fin * F ∈ MNS fin (see Proposition 2.6.1 for c fin * ).

Lemma 3.5.2. The category MNST fin is closed under infinite direct sums in MPST fin , and the inclusion functor
i fin Nis : MNST fin → MPST fin is strongly additive (Definition 3.3.3 ). The objects Z fin tr (M) and b * Z tr (M) belong to MNST fin for any M ∈ MCor.
Proof. This follows from Lemma 3.3.4, because c fin * is strongly additive as a left adjoint. The last claim follows from Proposition 3.3.5.
We write c finNis : MNST → MNS for the functor induced by c fin * . By definition, we have (3.5.1) c fin * i fin Nis = i fin sNis c finNis .
Theorem 3.5.3. The following assertions hold.
(1) Let F ∈ MPST fin . There exists a unique object F Nis ∈ MPST fin such that c fin * (F Nis ) = a fin sNis (c fin * (F)) and such that the canonical morphism u : c fin * (F) → a fin sNis (c fin * (F)) = c fin * (F Nis ) extends to a morphism in MPST fin .
(2) The functor i fin Nis has an exact left adjoint a fin Nis : MPST fin → MNST fin satisfying (3.5.2) c finNis a fin Nis = a fin sNis c fin * . In particular the category MNST fin is Grothendieck ( §A.10).
(3) The functor c finNis has a left adjoint c fin Nis = a fin Nis c fin ! i fin sNis . Moreover, c finNis is exact, strongly additive (Definition 3.3.3 ), and faithful.
Proof. This can be shown by a rather trivial modification of [Voe00, Theorem 3.1.4], but for the sake of completeness we include a proof. To ease the notation, put F := a fin sNis c fin * F ∈ MPS fin . First we construct a homomorphism Since p : V → M is a Nisnevich cover and G is separated, this implies G(α)(f ) = G (α)(f ). This completes the proof or (1).
(2) is a consequence of (1) and the fact that MPST fin is Grothendieck as a category of modules (see Theorem A.10.1 d)). Then (3) follows from Lemma A.8.1.

Cohomology in MNST fin
Notation 3.6.1. Let M ∈ MSm fin and let F ∈ MNS fin (resp. F ∈ MNST fin ). We write F M for the sheaf on (M) Nis induced from F (resp. c finσ F) via the isomorphism of sites from Lemma 3. (1) Let S be a scheme. We say a sheaf (2) We say F ∈ MNS fin is flasque if F M is flasque for any M ∈ MSm fin (see Notation 3.6.1). Again, flasque sheaves are flabby by (3.6.1).   Proof. This follows from Propositions 3.2.3 and A.7.6.    By definition, a strict Nisnevich cover is evidently a cover in MSm Nis . Up to isomorphism, any cover of MSm Nis can be refined to such a cover. More precisely, we have the following lemma. Proof. By Definition 4.2.1 and Proposition 1.9.2, there is a refinement of U → M of the form

Sheaves on MSm
where for each i we have either (i) f i ∈ Σ fin , (ii) f i = g −1 for some g ∈ Σ fin , or (iii) f i is a strict Nisnevich cover. We proceed by induction on n, the case n = 0 being trivial. Suppose n > 0. By induction, we have a refinement of U n → U 1 of the form V → N → U 1 where V → N is a strict Nisnevich cover and N → U 1 is in Σ fin .
If f 1 ∈ Σ fin , then we can take V = V and N = N , as the composition N → U 1 → U 0 belongs to Σ fin . Next, suppose f 1 = g −1 with g ∈ Σ fin . Then we can take V = V × U 1 U 0 and N = N × U 1 U 0 , where U 0 is regarded as a U 1 -scheme by g. Finally, suppose f 1 is a strict Nisnevich cover. By Lemma 1.6.3, we may find a morphism N → U 0 in Σ fin such that N := N × U 0 U 1 → U 1 factors through N . Then we can take V = V × N N . This completes the proof.
is also an MV fin -square. By Proposition 3.2.3 (2) and by [Voe10a, Corollary 2.17], the sequence is exact. By Lemma 1.6.3, the pullback of Σ fin ↓ M via U → M is cofinal in Σ fin ↓ U , and similarly for V → M and W → M. Hence, by taking its colimit over N ∈ Σ fin ↓ M, the above exact sequences and (2.5.1) imply the desired exact sequence In view of Lemma 4.2.3, this finishes the proof of (1).
(2) The adjunction (b sNis , b Nis s ) follows from the adjunction (b s! , b * s ) (see Proposition 2.5.1), by the full faithfullness of i sNis and i fin sNis , and by the formulas (4.3.1) and (4.3.2). The full faithfulness of b Nis s follows from that of b * s (see Proposition 2.5.1), i sNis and i fin sNis . Then the counit map b sNis b Nis s → Id is an isomorphism by Lemma A.3.1.
We prove the exactness of b sNis as follows. Since it is right exact as a left adjoint, it suffices to show its left exactness.
Assume given an exact sequence in MNS fin :   (2) For M ∈ MSm, let Z p (M) ∈ MPS be the associated representable additive presheaf (see (2.6.1)) and let  Proof. This follows from (2.6.2) and Definitions 3.5.1 and 4.2.1.

Sheaves on MCor
Definition 4.5.2. We define MNST to be the full subcategory of MPST consisting of those F enjoying the conditions of Lemma 4.5.1. We denote by i Nis : MNST → MPST the inclusion functor. (1) We have b ! (MNST fin ) ⊂ MNST.
(2) Let b Nis : MNST fin → MNST be the restriction of b ! so that we have (1). In (2), the adjointness and the full faithfulness are seen by using Proposition 2.5.1, (4.5.1) and (4.5.2). This proves that b Nis is right exact, and it is also exact by (4.5.2) and Proposition 2.5.1 (see also the proof of the exactness of b sNis in Proposition 4.3.3 (2)). (3) is a consequence of (2).   Moreover, c Nis is faithful, exact, strongly additive (Definition 3.3.3 ) and has a left adjoint c Nis = a Nis c ! i sNis such that c Nis a sNis = a Nis c ! .
Proof. The first equality follows from the first formula of (4.5.1) by adjunction. For the second, we use Theorems 4.2.2 and 4.5.5, together with (2.6.2), (3.5.2) and (4.5.1). The last statement follows from Lemma A.8.1 (3).

A. Categorical toolbox, I
This appendix gathers known and less-known results that we use constantly.

A.1. Pro-objects ([SGA4, exposé I, §8], [AM69, Appendix 2])
Recall that a pro-object of a category C is a functor F : A → C, where A is a small cofiltered category (dual of [Mac98, Chapter IX, §1]). They are denoted by {X α } α∈A or by " lim ← − − " α∈A X α (Deligne's notation), with X α = F(α). Pro-objects of C form a category pro-C, with morphisms given by the formula There is a canonical full embedding c : C → pro-C, sending an object to the corresponding constant pro-object (A = { * }).
For the next lemma, we recall a special case of comma categories from Mac Lane [Mac98, Chapter II, §6]. If ψ : A → B is a functor and b ∈ B, we write b ↓ ψ for the category whose objects are pairs (a, f ) ∈ A × B(b, ψ(a)); a morphism (a 1 , f 1 ) → (a 2 , f 2 ) is a morphism g ∈ A(a 1 , a 2 ) such that f 2 = ψ(g)f 1 .
The category ψ ↓ b is defined dually (objects: systems ψ(a) f − → b, etc.) According to [Mac98,Chapter IX,§3], ψ is final if, for any b ∈ B, the category ψ ↓ b is nonempty and connected; here we shall use the dual property cofinal (same conditions for b ↓ ψ). As usual, we abbreviate Id A ↓ a and a ↓ Id A by A ↓ a and a ↓ A.
Let F = (F : A → C) = {X α } α∈A ∈ pro-C. For each α ∈ A, we have a "projection" morphism π α : F → c(X α ) in pro-C. This yields an isomorphism in pro-C (explaining Deligne's notation) and a functor where we take A = C and B = pro-C in the above setting.
(Warning: the use of co in (co)final and (co)filtered is opposite in [Mac98] and in [KS06]. We use the convention of [Mac98].) [SGA4,exposé I,§8.11.5] Let u : C → D be a functor: it induces a functor pro-u : pro-C → pro-D.
The same holds if G is right adjoint to D (replacing the counit by the unit). Proof. Defineū X = u X : F(X) → G(X) for X ∈ ObC[Σ −1 ] = ObC. We must show thatū commutes with the morphisms of C[Σ −1 ]. This is obvious, since u commutes with the morphisms of C and the morphisms of C[Σ −1 ] are expressed as fractions in the morphisms of C.

A.4. Presheaves and pro-adjoints
Let C be a category. We writeĈ for the category of presheaves of sets on C (i.e. functors C op → Set); it comes with the Yoneda embedding y : C →Ĉ which sends an object to the corresponding representable presheaf. If u : C → D is a functor, we have the standard sequence of three adjoint functors  ., (5.1.1)). If u has a left adjoint v, the sequence (u ! , u * , u * ) extends to (v ! , v * = u ! , v * = u * , u * ) (ibid., Remark 5.5.2). Let A be an essentially small additive category. Instead of presheaves of sets on A, one usually uses the category Mod -A of additive presheaves of abelian groups; the above results transfer to this context, mutatis mutandis.
a) The functor u ! (resp. u * , u * ) commutes with all representable colimits (resp. limits, limits and colimits). If u has a left adjoint, then u ! also commutes with all limits. If u has a pro-left adjoint v (Definition A.2.3 ), so does u ! which is therefore exact. Moreover, u ! is then given by the formula c) If u is a localisation or is full and essentially surjective, then u ! is a localisation. d) In the case of c), for C ∈ C the following conditions are equivalent: Proof. a) follows from general properties of adjoint functors, except for the case of a pro-left adjoint. Let u admit a pro-left adjoint v, and let Y ∈ D: so there is an isomorphism of categories Y ↓ u v(Y ) ↓ c. Hence, we get by Lemma A.1.1 a cofinal functor where A is the indexing set of v(Y ). Thus, for F ∈Ĉ, u ! F(Y ) may be computed as The first equality is the formula in the proposition. The second one shows that the pro-left adjoint v ! of u ! is defined at y D (Y ) by y C (v(Y )); since any object ofD is a colimit of representable objects, this shows that v ! is defined everywhere.
For b), see [SGA4, exposé I, proposition 5.6]. In c), it is equivalent to show that u * is fully faithful by Lemma A.3.1. Let F, G ∈D, and let ϕ : u * F → u * G be a morphism of functors. In both cases, u is essentially surjective: given X ∈ D and an isomorphism α : X ∼ − → u(Y ), we get a morphism The fact that ψ X is independent of (Y , α) and is natural in X is an easy consequence of each hypothesis (see Lemma A.3.3 in the first case).
In d), the equivalence (ii) ⇐⇒ (iii) is tautological and (iii) ⇒ (i) is obvious. The implication (i) ⇒ (iii) was proven in [GZ67, Chapter I, §4.1.2] assuming that u is a localisation enjoying a calculus of left fractions; let us prove (i) ⇒ (ii) in general. Under (i), we have y C (C) u * F for some F ∈D; the unit map becomes On the other hand, the counit map ε F : u ! u * F → F is invertible by the full faithfulness of u * . By the adjunction identities, we have u * (ε F ) • η u * F = 1 u * F . Hence the conclusion.
We shall usually write u ! for the pro-left adjoint of u ! , when it exists. Corollary A.5.5. Let (C, Σ) be a localiser such that Σ enjoys a calculus of right fractions. Let F : C → D be a functor. Suppose that F inverts the morphisms of Σ and that, for any c, d ∈ C, the obvious map is an isomorphism. Then the functor Σ −1 F : Σ −1 C → D induced by F is fully faithful. A.6. Pro-Σ-objects Definition A.6.1. Let (C, Σ) be a localiser. We write pro Σ -C for the full subcategory of the category pro-C of pro-objects of C consisting of filtered inverse systems whose transition morphisms belong to Σ. An object of pro Σ -C is called a pro-Σ-object.
Proposition A.6.2. Suppose that Σ has a calculus of right fractions and, for any c ∈ C, the category Σ ↓ c contains a small cofinal subcategory. Then Q : C → Σ −1 C has a pro-left adjoint Q ! , which takes an object Y ∈ Σ −1 C to " lim ← − − " X∈Σ↓Y X (see §A.1 for the notation " lim ← − − "). In particular, Proof. In view of Corollary A.5.4 and Proposition A.5.6, this follows from Proposition A.5.2 b).
Remark A.6.3. Consider the localisation functor Q : C → Σ −1 C: it has a left Kan extension Q : pro sat(Σ) -C → Σ −1 C [Mac98, Chapter X] along the constant functor C → pro sat(Σ) -C, given by the formulâ (The right hand side makes sense as an inverse limit of isomorphisms.) Then one checks easily that Q ! is left adjoint toQ. (1) Q ! has a pro-left adjoint, and is therefore exact.
If (A, Σ) is a localiser with A additive and Σ enjoys a calculus of right fractions, then Σ −1 A is additive and so is the functor Q : A → Σ −1 A [GZ67, Chapter I, Corollary 3.3]. For future reference, we give the additive analogue of Theorem A.6.4 (see the paragraph before Proposition A.4.1 for Mod -A): Theorem A.6.5. Let (A, Σ) be a localiser; assume that A is an additive category and that Σ has a calculus of right fractions. Let Q : A → Σ −1 A denote the localisation functor, as well as the string of adjoint functors (Q ! , Q * , Q * ) between Mod -A and Mod -Σ −1 A. Then: (1) Q ! has a pro-left adjoint, and is therefore exact.
(2) For F ∈ Mod -A and Y ∈ Σ −1 A, we have

A.7. cd-structures
Let C be a category with an initial object. According to [Voe10a], a cd-structure on C is given by a collection of commutative squares stable under isomorphisms, called distinguished squares. Any cd-structure defines a topology on C: the smallest Grothendieck topology such that for a distinguished square of the form the sieve generated by the morphisms {p : V → X, u : U → X} is a cover sieve and such that the empty sieve is a cover sieve of the initial object ∅.
Recall from [Voe10a] some important properties of cd-structures.
Definition A.7.1. Let C be a category with an initial object ∅.
(1) Let P be a cd-structure on C. The class S P of simple covers is the smallest class of families of morphisms of the form {U i → X} i∈I satisfying the following two conditions: • for any isomorphism f , {f } is in S P • for a distinguished square Q of the form (A.7.1) and families {p i : V i → V } i∈I and {q j : U j → U } j∈J in S P the family {p • p i , u • q j } i∈I,j∈J is in S P .
(2) A cd-structure on C is called complete if any cover sieve of an object X ∈ C which is not isomorphic to ∅ contains a sieve generated by a simple cover.
(3) A cd-structure P is called regular if for S ∈ P of the form (A.7.1) one has • S is a pullback square (i.e., is cartesian) • u is a monomorphism • the morphisms of sheaves is surjective, where for C ∈ C we denote by ρ(C) the sheaf associated with the presheaf represented by C, and ∆ is induced by the diagonal map.
Remark A.7.5. The square (A.7.2) is cartesian. This is a formal consequence of Lemma A.7.3, since any distinguished square with respect to a regular cd-structure is cartesian by definition. However, there is a more direct proof: let Z a − → V and Z b − → W × U W be two morphisms making the corresponding square commute. Then b amounts to two morphisms b 1 , b 2 : Z → W such that (with the notation of (A.7.1)) qb 1 = qb 2 and a = vb 1 = vb 2 . Since S is cartesian by (1), we have b 1 = b 2 : Z → W , which is a solution to the universal problem.
Proposition A.7.6. Let (C, Σ) be a localiser such that Σ admits a calculus of right fractions.
(2) Assume (1) and let Q : C → Σ −1 C be the localisation functor. Suppose given a cd-structure P on C, and let P be the cd-structure on Σ −1 C given by all squares isomorphic to a square of the form Q(S), where S ∈ P . If P is strongly complete (resp. strongly regular), so is P . Proof.
(1) Let ∅ be an initial object of C. Since Q is (essentially) surjective, Q(∅) admits a morphism to any object; Condition (1) of Lemma A.7.2 for ∅ implies that this morphism is unique, and this in turn implies the same condition for Q(∅).
(2) By Proposition A.5.6 a), Q commutes with finite limits. This implies Condition (2) of Lemma A.7.2. Conditions (1), (3) of Lemma A.7.3 for P follow from the same conditions for P (note that the diagonals are preserved by Q, since they are finite limits). It remains to show that Q carries a monomorphism u : U → X to a monomorphism. Let f , g : V → U be two morphisms in Σ −1 C such that Q(u)f = Q(u)g. By calculus of fractions, we may write f = Q(f )Q(s) −1 and g = Q(g)Q(s) −1 for somef ,g ∈ Ar(C) and s ∈ Σ. Then Q(uf ) = Q(ug). By Proposition A.5.2 c), we may find t ∈ Σ such that uf t = ugt, which impliesf t =gt since u is a monomorphism. This shows f = g, as desired.

A.8. A pull-back lemma
We shall use the following elementary lemma several times. (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 . Proof.
(2) Let {F i } i∈I be a family of objects of C . We must show that the natural map f : is an isomorphism by the strong additivity of c and i C . If i D is strongly additive, g is also an isomorphism and we are done. If now i D has a strongly additive left inverse a D , we apply it to the diagram and get a composition (4) Let us take an exact sequence 0 → F → G → H → 0 in C . Put K := Coker(i C G → i C H) ∈ C. Since a D cK = c a C K = 0, we get an exact sequence 0 → a D ci C F → a D ci C G → a D ci C H → 0 by the exactness of c and a D . Using a D c = c a C and a C i C = Id (Lemma A.3.1), we conclude 0 → c F → c G → c H → 0 is exact.
The proof of Lemma A.8.1 (2) implicitly used the following (trivial) lemma, which we state for the sake of clarity.
Lemma A.8.2. Let D ⊆ C be a full embedding of categories. Suppose that a direct (resp. inverse) system (d α ) of objects of D has a colimit (resp. a limit) in C, which is isomorphic to an object d of D. Then d represents the (co)limit of (d α ) in D. Examples A.9.2. If F has an exact left adjoint, it carries injectives to injectives. If G is exact, the hypothesis on F is automatically verified.

A.9. Homological algebra
The following is a slight generalization of [Mil80, Chapter III, Proposition 2.12], (where the underlying category of S is supposed to be a category of schemes). Lemma A.9.3. Let F be a sheaf of abelian groups on a site S. The following conditions are equivalent.
(1) We have H q (X, F) = 0 for any X ∈ S and q > 0.
(3) We haveȞ q (U /X, F) = 0 for any cover U → X in S and q > 0.
(4) The sheaf F is i S -acyclic, where i S is the inclusion functor of the category of sheaves to that of presheaves.
Proof. For X ∈ S, we write Γ X (resp. Γ pr X ) for the functor F → F(X) from the category of sheaves (resp. presheaves) to Ab. We have Γ X = Γ pr X i S . Since Γ pr X is exact, Theorem A.9.1 implies R q Γ X = Γ pr X R q i S , and hence H q (X, F) = R q i S F(X). This proves the equivalence of (1) and (4). The rest is shown in the same way as [Mil80, Chapter III, Proposition 2.12].
Definition A.9.4. We say F is flabby if the conditions of Lemma A.9.3 are satisfied. Lemma A.9.5. Let S be the category of abelian sheaves on a site C, T an abelian category, and c * : T → S an additive functor which has a left adjoint c ! : S → T . Suppose that any cover in C admits a refinement U → X such that c ! (Č(U /X)) is exact in T , wherě C(U /X) = (· · · → y(U × X U ) → y(U ) → y(X) → 0) is theČech complex associated to U → X (y denotes the Yoneda functor). Then c * I is flabby for any injective object I ∈ T .
Proof. (Compare [Voe00, Proposition 3.1.7].) It suffices to showȞ q (U /X, c * I) = 0 for any q > 0 and for any U → X as in the assumption. If we denote by U n X the n-fold fiber product of U over X, thenȞ q (U /X, c * I) is computed as the cohomology of the complex which is acyclic by the assumption and the injectivity of I.

A.10. Grothendieck categories
Recall that a Grothendieck abelian category (for short, a Grothendieck category) is an abelian category verifying Axiom AB5 of [Gro57]: small colimits are representable and exact, and having a set of generators (equivalently, a generator). These generators are generators by strict epimorphisms. We have the following basic facts: