Duality for complexes of tori over a global field of positive characteristic

If K is a number field, arithmetic duality theorems for tori and complexes of tori over K are crucial to understand local-global principles for linear algebraic groups over K. When K is a global field of positive characteristic, we prove similar arithmetic duality theorems, including a Poitou-Tate exact sequence for Galois hypercohomology of complexes of tori. One of the main ingredients is Artin-Mazur-Milne duality theorem for fppf cohomology of finite flat commutative group schemes.


Introduction
Let K be a global field of characteristic p 0 and let A K denote the ring of adèles of K. Let G be a reductive group over K, and X be a torsor under G. We are interested in rational points on X, and more precisely, on various local-global principles associated to X: does X satisfy the Hasse principle, i.e., does X(A K ) = ∅ imply X(K) = ∅? If not, can we explain the failure using the so-called Brauer-Manin obstruction to the Hasse principle? Assuming that X(K) = ∅, can we estimate the size of X(K) by studying the so-called weak and strong approximation on X (with a Brauer-Manin obstruction if necessary), i.e., the closure of the set X(K) in the topological space X(A S K ), where S is a (not necessarily finite) set of places of K and A S K is the ring of S-adèles (with no components in S)?
The answer to those questions is known in the case where K is a number field, see for instance [San81,Cor. 8.7 & 8.13] for the Hasse principle and weak approximation, and [Dem11a,Th. 3.14] for strong approximation. Note that in the number field case, similar results are known for certain non principal homogeneous spaces of G (see [Bor96] or [BD13]).
In the case of a global field of positive characteristic, the answer is known for semisimple simply connected groups (thanks to works by Harder, Kneser, Chernousov, Platonov, Prasad), but the general case is essentially open (see [Ros18,Th. 1.9] for some related results). One strategy to attack the remaining local-global questions is similar to one that worked for number fields: arithmetic duality theorems for tori, and abelianization of Galois cohomology (see for instance [Dem11b] and [Dem11a] for the case of strong approximation over number fields). Indeed, given a reductive group G over a field L (e.g. L is a global or a local field), one can construct a complex of tori of length two C := [T 1 → T 2 ], (1) together with "abelianization maps" H i (L, G) → H i (L, C) (cohomology sets here are Galois cohomology or hypercohomology sets), such that the cohomology sets of G can be computed via the abelian cohomology groups of C and the Galois cohomology of a semisimple simply connected group associated to G. The latter is well-understood when L is a local or global field.
Motivated by the discussion above, this paper deals with arithmetic duality theorems for complexes of tori over global fields K of positive characteristic; in characteristic 0, we also get refinements of previously known results.
The aforementioned applications to the arithmetic of reductive groups and homogeneous spaces will be given in a future paper.
The main object is a two-term complex C := [T 1 → T 2 ] of K-tori T 1 and T 2 , and we are particularly interested in its Galois hypercohomology groups H i (K, C). The main result of the paper can be summarized as follows: we get Poitou-Tate exact sequences relating global Galois cohomology groups H i (K, C) and local ones H i (K v , C) -for any place v of K -via the cohomology of the dual object C of C. To be more precise, let us introduce some notation: K is the function field of a smooth, projective, and geometrically integral curve X over a finite field k. Let X (1) denote the set of closed points in X. If A is a discrete abelian group, then A * is the Pontryagin dual of homomorphisms from A to Q/Z, and A ∧ denotes the completion A ∧ := lim ← −n∈N * A/n.
(1) Throughout the paper, the piece of notation := means that the equality is a definition.
J.É.P. -M., 2020, tome 7 We can now state one of the main results in the paper (see Theorem 5.10): Theorem. -Let C := [T 1 → T 2 ] be a two-term complex of K-tori, and let C := [ T 2 → T 1 ] be the dual complex, where T is the module of characters of a torus T (T 1 and T 2 are in degree −1). Then there is an exact sequence In the case where K is a number field, we also recover a generalization of [Dem11b, Th. 6.1 & 6.3]. In the function field case, some partial results related to this exact sequence for one single torus can be deduced from [GA09,§6].
The main ingredient to prove the theorem above is the so-called Artin-Mazur-Milne duality theorem for the fppf cohomology of finite flat commutative group schemes over open subsets of X (see [Mil06,Th. III.8.2] and [DH19, Th. 1.1]). It is worth noting that although the complexes that appear in the previous theorem consist of smooth group schemes (hence the result can be stated using only Galois cohomology), it is essential for the proof to involve finite group schemes (which are not smooth in general) over Zariski open subsets of X. That is why [DH19, Th. 1.1] is required instead of the Artin-Verdier duality theorem ([Mil06, Cor. II. 3.3]) in étale cohomology. Likewise for Theorems 4.11 and 4.9.
Also, since it is necessary at some point to work with the fppf topology, the approach of duality theorems via Ext groups (like [Mil06, Th. II. 3.1]) does not seem to work, the difficulty being the lack of good notion of constructible sheaf for the fppf topology (see [Mil06, Introduction to Chap. III]).
The structure of the paper is the following: Section 2 extends the construction and properties given in [DH19] of fppf cohomology with compact support to the case of bounded complexes of finite flat group schemes. General properties of étale cohomology of complexes of tori and of their dual complexes are given in Section 3. Section 4 deals with applications of Artin-Mazur-Milne duality theorem to various duality statements for the étale cohomology of complexes of tori over open subsets U of X. In Section 5, one deduces several Poitou-Tate exact sequences for Galois cohomology from the results of Section 4.

Compact support hypercohomology
Let K be the function field of a smooth, projective, and geometrically integral curve X over a finite field k. Let U be a non empty Zariski open subset of X. Denote by U (1) the set of closed points of U .
Let C = (C p ) p∈Z be a bounded complex of fppf sheaves over U . In this text, we define the dual of C to be the Hom-complex C defined by following the sign conventions in [Stacks,Tag 0A8H] or in [Bou07, X.5.1]. Note that there is a functorial morphism of complexes With those conventions, if C is concentrated in degree 0, i.e., C = F with F an fppf sheaf, then C is the same as the Cartier dual F D := Hom(F , G m ) attached in degree −1, i.e., C = F D [1] and the above pairing coincides with the obvious pairing Note also that for any bounded complex C , we have a natural isomorphism of (2) If N is a commutative group scheme (over U or over K), its Cartier dual is denoted N D . The Pontryagin dual of a topological abelian group A (consisting of continuous homomorphism from A to Q/Z) is denoted A * . Unless explicitly specified, the topology used for sheaves (resp. complex of sheaves) and cohomology (resp. hypercohomology) is the fppf topology.
For each closed point v of X, the completion of K at v is denoted by K v : it is a local field of characteristic p with finite residue field F v (observe the slight difference of notation with [DH19], where K v stands for the henselization and K v for the completion). Denote by O v the ring of integers of K v . For every fppf sheaf F over U with generic fiber F , recall ([DH19, Prop. 2.1]) the long exact sequence (where the piece of notation v ∈ U means that we consider all closed points of X U ).
There is also a long exact sequence J.É.P. -M., 2020, tome 7 associated to every short exact sequence Let us now extend the construction of the groups H i c (U, . . . ) and [DH19, Prop. 2.1] to the case of bounded complexes. Let C := [· · · → F i → F i+1 → · · · ] be a bounded complex of fppf sheaves over U . Let C → I • (C ) be an injective resolution of the complex C , in the sense of [Stacks,Tag 013K]. Following [DH19, §2], let Z := X U and Z : We now define Γ c (U, I • (C )) to be the following object in the category of complexes of abelian groups: and H r c (U, C ) := H r (Γ c (U, I • (C ))). We will also denote by RΓ c (U, C ) the complex Γ c (U, I • (C )). Similarly, one can define, for any closed point v ∈ X, complexes As in [DH19], similar definitions could be made when K is a number field (taking into account the real places), but in this article we will focus on the function field case. However, we will make remarks regularly throughout the text explaining similarities and differences appearing in the number field case.
We will need the analogue of [DH19, Prop. 2.1 & 2.12] for bounded complexes C : by construction, the first two points of loc. cit., Prop. 2.1 (i.e., exact sequence (3) and (4)) still hold for bounded complexes. (1) There is a canonical commutative diagram of abelian groups: where the long row and the columns are exact.
(2) Let V ⊂ U be a non empty open subset. Then there is an exact sequence Proof. -There is a commutative diagram such that the second line and the left column are exact (by (3) and Prop. 2.1 2.): For v ∈ U V , the localization exact sequence (cf. proof of Proposition 2.1, 2.) yields that the second column is a complex. Since j is injective by definition, the required exact sequence follows by diagram chasing.
For a complex C of finite flat group schemes, let us now endow the groups H * c (U, C ) with a natural topology, compatible with the one defined in [DH19] in the case where C is a finite flat group scheme.
Let F be a local field (that is: a field complete for a discrete valuation with finite residue field) and let C := [C r fr −→ C r+1 → · · · → C s ] be a bounded complex of finite commutative group schemes over Spec F , with C i in degree i. We assume that F is of positive characteristic p (if F is p-adic, then all groups H r (F, C ) are finite by [Mil06, Cor. I.2.3]).
is an open map (where the topology on f (G 1 ) is induced by G 2 ). This is equivalent to saying that f induces an isomorphism of the topological quotient G 1 / ker f with the topological subspace f (G 1 ) ⊂ G 2 .
Let A := ker(f r ), and let C := Cone(A[−r] j −→ C ). Then there is an exact triangle In addition, we have a natural quasi-isomorphism ϕ : C → C , where C := [C r+1 / Im(f r ) → C r+2 → · · · → C s ] has a smaller length than C . There is an alternative dévissage for the complex C , given by the exact triangle: Recall that for a finite and commutative F -group scheme N , the fppf groups H i (F, N ) are finite if i = 1 ([Mil06, §III.6]) and they are equipped with a locally compact topology for i = 1 by [Ces15]. By induction on the length of C , one deduces that if C i = 0, then H i+1 (F, C ) is finite. In particular, with the previous notation, we get that H i (F, C ) is finite if i r or i s + 2.
We now define a natural topology on H i (F, C ) by induction on the length of C , such that any morphism of such complexes induces a strict map between hypercohomology groups. Using the dévissages given by (5) and (6), one gets the following exact sequences where all the groups except H i (F, C ) are endowed with a natural topology via the induction hypothesis (observe that C r = 0).
-Assume that i = r + 1. Equip Im f ∼ = H i−r (F, A)/ Im H i−1 (F, C ) with the quotient topology. Then the two rightmost groups in exact sequence (7) are finite, and we can endow H r+1 (F, C ) with the topology such that Im f is an open subgroup (see [DH19, beginning of §3]). In other words it is the finest topology such that f is continuous. Then f and g are strict.
-Assume that i = r + 1. Then H i−r (F, C r ) is finite (and discrete), and using exact sequence (8) one can similarly endow H i (F, C ) with the finest topology making f continuous. Then both maps f and g are strict.
By construction, this topology is functorial in C , i.e., if C 1 → C 2 is a morphism of complexes, then the induced map H i (F, C 1 ) → H i (F, C 2 ) is strict. In addition, given a quasi-isomorphism C 1 → C 2 , the induced morphism on cohomology groups is a homeomorphism.
Let us now deal with the topology on the groups H * c (U, C ), where C is a complex of finite flat commutative group schemes defined over U . Recall that we have an exact sequence analogous to (3): We endow the groups H i (U,C ) with the discrete topology, and the groups H i−1 (K v ,C ) with the topology defined above. Proof.
-We prove the result by induction on the length of C , using the dévissages induced by the exact triangles (5) and (6).
-Assume that i = r + 1. Using the exact triangle (6), we get the following commutative diagram of long exact sequences of topological groups (where all maps are strict): The groups on the left hand side are finite, and by [Ces17, Lem. 2.7], the image of the right hand side map is discrete in is Hausdorff, an easy topological argument implies that the image of the central vertical map is discrete.
-Assume that i = r + 1. Using the exact triangle (5), we get the following commutative diagram of long exact sequences of topological groups: The groups on the left hand side are finite, and by induction on the length of the complex, the image of the right hand side map is discrete in v / ∈U H 1 (K v , C ). A similar topological argument as before implies that the central vertical map is discrete.
As a consequence of this Lemma, one can endow H i c (U, C ) with the following topology: we put the quotient topology on the group v / ∈U H i (K v , C )/ Im H i (U, C ) (this topology is Hausdorff), and since H i (U, C ) is discrete, there is a unique topology on H i c (U, C ) such that the maps in the exact sequence (9) are strict.
Proof. -We prove this Lemma by induction on the length of the complex C . By [DH19, Prop. 3.5], the lemma is proved when C is a complex of length one, i.e., concentrated in one given degree. Given a complex C , consider the previous dévissages: -Assume that i = r + 1. Then the exact sequence (7) implies that the group H r+1 c (U, C ) is an extension (the maps being strict) of a (discrete) finite group by a profinite group (which is a quotient of H 1 c (U, A) by a closed subgroup), hence H r+1 is an extension of a finite (discrete) group by a profinite group (which is a quotient of H i c (U, C ) by a closed subgroup), hence it is profinite.

Cohomology of tori and short complexes of tori
Let U be a non empty Zariski open subset of X. Recall that for every U -torus T (in the sense of [SGA3, IX, Déf. 1.3]), there is a finite étale covering (that can be taken to be connected and Galois) V of U such that T V := T × U V is split, that is: isomorphic to some power G r m (r ∈ N) of the multiplicative group ([SGA3, X, Th. 5.16]). The group of characters T of T is a U -group scheme locally isomorphic to Z r for the étale topology, namely it is a torsion-free and finite type Gal where by convention T 1 is in degree −1 and T 2 in degree 0, we can apply the construction of Section 2. Namely we have dual complexes C = [ T 2 → T 1 ] and C = [ T 2 ρ −→ T 1 ] (concentrated in degrees −1 and 0), which are respectively defined over U and over K. Fix a separable closure K of K. Denote by S the finite set X U and by G S = π ét 1 (U ) the étale fundamental group of U , which is the Galois group of the maximal field extension K S ⊂ K of K unramified outside S; then each T i (i = 1, 2) can be viewed as a discrete G S -module.
Recall that fppf and étale cohomology coincide for sheaves represented by smooth group schemes ([Mil80, §III.3]) like a torus T , its group of characters T , or finite flat group schemes of order prime to p. In particular (by [Mil80, Lem. III.1.16]) we have for every integer i: (where the limit is over all non empty Zariski open subsets U of X), and likewise for the complex C .
For such 2-term complexes, the pairings of Section 2 can be made explicit (see [Dem11b,§2]; note that the sign conventions are slightly different here), and give maps in the bounded derived category D b (U ) (resp. D b (Spec K)) of fppf sheaves over U (resp. over Spec K). In the case T 1 = 0 or T 2 = 0, we recover (up to shift) the classical pairings T ⊗ T → G m and T ⊗ T → G m associated to one single torus T . We also have for each positive integer n the n-adic realizations and likewise for C and C. The fppf sheaf T Z/n (C ) is representable by a finite group scheme of multiplicative type over U (in the sense of [SGA3, IX, Déf. 1.1]) with Cartier dual T Z/n ( C ), and similarly for T Z/n (C) and T Z/n ( C) over K. Besides we have exact triangles ([Dem11b, Lem. 2.3]), where for every abelian group (or group scheme) A, the piece of notation n A stands for the n-torsion subgroup of A: and (11) in D b (U ), and similar triangles for C, C in D b (Spec K). Note also that the objects C ⊗ L Z/n and C ⊗ L Z/n in the derived category have canonical representatives as complexes of fppf sheaves given by We also have an exact triangle in D b (U ): where coker ρ is a torus and M := ker ρ is a group of multiplicative type, and the dual exact triangle For every integer i, there are exact sequences and we also have similar exact sequences for the compact support fppf cohomology groups H i c (U, . . . ). Proof (a) Let V be an étale and finite connected Galois covering of U such that Then the Hochschild-Serre spectral sequence provides an exact sequence Thus H 1 (U, T ) is of finite type (resp. finite if U = X) as well.
(b) We can assume (by ([Mil06, Lem. III.8.9] and [DH19, Cor. 4.9]) that U = X. Since every finitely generated Galois module is a quotient of a torsion-free and finitely generated Galois module, the assumption that N is of multiplicative type implies (by [SGA3, X, Prop. 1.1]) that there is an exact sequence of U -group schemes where T 1 and T 2 are U -tori. Therefore, there is an exact sequence of abelian groups By (a), we know that H 1 (U, T 1 ) is finite. Let n be the order of N ; then the map For an fppf sheaf (or a bounded complex of fppf sheaves) F on U with generic fiber F , we set (cf. exact sequence (3)) By the Kummer sequence in the fppf topology Using the exact sequence in the fppf topology . It is therefore sufficient to show that H 2 c (U, T ) is of finite exponent, and by a restrictioncorestriction argument, we reduce to the case T = Z. As Therefore, H 2 c (U, Z) is a subgroup of H 1 (Gal(L/K), Q/Z), where L ⊂ K S is the maximal abelian extension of K that is unramified outside S and totally decomposed at every v ∈ S = X U . The group Gal(L/K) is isomorphic to Pic U by class field theory, which implies that it is finite because U = X. Hence H 2 c (U, Z) is finite, which proves (c).
(the equality holds because the X-group scheme n T is finite and X is connected). But K is a global field, hence T (K) tors is finite: indeed, if L is a finite extension of K such that T splits over L, then T (K) ⊂ T (L) with T (L) (L * ) r for some r, and L * contains only finitely many roots of unity. Therefore, T (K) has trivial Tate module, which yields the result.
(e) Using exact sequence (16), we get a surjection H 2 (X, T /n) → n H 3 (X, T ), so it is sufficient (by Remark 3.2) to show that H 3 (X, T ) is of finite exponent. By restriction-corestriction, we can therefore assume that T = Z. By the same method as in (d), we get that the dual of H 3 (X, Z) is lim ← −n H 1 (X, µ n ). As H 0 (X, G m ) = k * because X is a proper and geometrically integral curve, we get an exact sequence of finite groups Since Pic X is of finite type, we have lim ← −n ( n Pic X) = 0, hence lim ← −n H 1 (X, µ n ) is the inverse limit of the k * /k * n , which is k * itself because k is finite. Thus H 3 (X, Z) is the dual of k * , which is finite (but not zero).
The commutative diagram with exact lines and the five lemma now give that the restriction map H i (U, C ) → H i (K, C) is an isomorphism for i ∈ {−1, 0}, and is injective for i = 1. The fact that H 1 c (U, C ) is of finite type if U = X is immediate by dévissage thanks to Lemma 3.4 (c).
(b) The first two assertions follow from Lemma 3.1, using exact sequence (14). For U = X, every X-torus T satisfies that H 1 (X, T ) is of finite type (Lemma 3.1) and H 2 (X, T ) is finite (Lemma 3.4 (b), whence the result.
(c) By functoriality, the image of D 1 (U, C ) by the map u : H 1 (U, C ) → H 2 (U, T 1 ) is a subgroup of D 2 (U, T 1 ). The latter is finite by Lemma 3.4 (b), because it is a quotient of H 2 c (U, T 1 ). As the kernel of u is a quotient of H 1 (U, T 2 ) (which is finite by Lemma 3.4 (a), this means that D 1 (U, C ) is finite.
The group H 2 c (U, T 2 ) is finite by Lemma 3.4 (c). Hence D 2 (U, T 2 ) is finite. On the other hand, the kernel of the map H 1 (U, C ) → H 2 (U, T 2 ) is a quotient of the group H 1 (U, T 1 ), which is finite by Lemma 3.4 (a). Thus D 1 (U, C ) is finite.
Remark 3.7. -The same argument as in Proposition 3.6 (a) shows that for v ∈ U , the restriction map is an isomorphism for i ∈ {−1, 0}, and is injective for i = 1.
Recall ([Dem11b, §3]) that for v ∈ X (1) , given a complex C of K v -tori, the groups H −1 (K v , C) and H 0 (K v , C) are equipped with a natural Hausdorff topology (and the groups H i (K v , C) are endowed with the discrete topology for i 1, as are all groups H r (K v , C) for −1 r 2).
Proof. -We can assume U = X. Let us start with the case when C = G m .
which is finite. Consider now a U -torus T . Let W be a connected Galois finite covering of U (with function field L ⊃ K) that splits T . Let G := Gal(L/K). By the case T = G m , In the general case, exact triangle (12) yields a commutative diagram with exact lines where M is a U -group of multiplicative type and T is a U -torus. Since U = X, the right vertical map is injective. As the lemma holds for is a subgroup of ker u. As H 1 (U, M ) is finite (Remark 3.2), we also have that ker u is finite and so is j −1 (H 1 ). Therefore, I ∩ H 1 = j(j −1 (H 1 )) is finite, which implies that I is discrete.
The same result for the image of H −1 (U, C ) into v ∈U H −1 (K v , C) follows immediately because H −1 (U, C ) is a subgroup of H 0 (U, T 1 ) (which has just been shown to be a discrete subgroup of v ∈U H 0 (K v , T 1 )), Remark 3.9. -The analogue of Lemma 3.8 does not hold over a number field as soon as at least one finite place of K is not in U and K has at least two non archimedean places: indeed, for the exact sequence to hold (cf. [DH19, beginning of §2]), the groups H 0 (K v , G m ) at the archimedean places must be understood as the modified Tate group

the archimedean places) is countable and infinite
(it is isomorphic to O * K ), hence I is not compact by Baire's Theorem. Therefore, the Equip the finitely generated group H 0 (U, C ) with the discrete topology. We give H 0 c (U, C ) the unique topology such that all maps in the exact sequence (17) are strict (by Lemma 3.8, the left map is strict and the quotient of v ∈U H −1 (K v , C) by the image of H −1 (U, C ) is a locally compact Hausdorff group). We also give the finite group (cf. Proposition 3.6 (c) D 1 (U, C ) the discrete topology, and topologize H 1 c (U, C ) such that all maps in the exact sequence Definition 3.10. -Define E as the class of those abelian topological groups A that are an extension (the maps being continuous) of a finitely generated group F (equipped with the discrete topology) by a profinite group P (this implies that all maps in this exact sequence are strict by [DH19, Lem.

3.4]).
It is easy to check that for every group A in E , a closed subgroup of A and the quotient of A by any closed subgroup of A are still in E . Also a topological extension of a (discrete) finitely generated group by A stays in E . Finally, every group A in E is isomorphic to the direct product of a finitely generated group (equipped with discrete topology) by a profinite group: indeed, up to replacing F by F/F tors and P by π −1 (F tors ) in the extension (19), we can assume that F = Z r for some r 0. Since F is free, the morphism π has a section s : F → A, which is automatically continuous because F is discrete. Setting B = s(F ), we get a topological isomorphism A ∼ = P ×B. We also set The piece of notation A{ } stands for the -primary torsion subgroup of A.
For A finitely generated, we have A ( ) = A ⊗ Z Z ; since Z is a torsion-free (hence flat) Z-module, the functors A → A ( ) and A → A ∧ are exact in the category of finitely generated abelian groups.
Proof (a) Since the functor . ⊗ Z Z/ m is right exact, the sequence is exact. Therefore, the projective system (ker[B/ m → E/ m ]) m 1 has surjective transition maps, which implies that the map lim (b) Assume that E is finite. Then (by induction on m) we also have that m E is finite for every positive integer m thanks to the exact sequence Let I ⊂ B be the image of A by the map A → B. By (a), the map A ( ) → I ( ) is surjective, so it is sufficient to prove that the sequence is exact. By the snake lemma, we have an exact sequence Taking projective limit over m yields the required exact sequence because the kernel of the map I/ m → B/ m is finite (it is a quotient of m E). Similarly, if A/ is finite, then A/ m (hence also I/ m ) is finite for every positive m and the same argument works.
If we assume further that A is a topological abelian group, its profinite completion is A ∧ := lim

← −H (A/H), where H runs over all open subgroups of finite index in
Proof. -Let T be a U -torus with generic fiber T . Let v be a closed point of X and let L be a finite Galois extension of K v such that T splits over L. As L * Z × O * L is in E , so is T (L). Then H 0 (K v , T ) is in E as a closed subgroup of T (L) (the subgroup of Gal(L/K v )-invariants). The exact sequence J.É.P. -M., 2020, tome 7 and the definition of the topology on H 0 (K v , C) now imply that H 0 (K v , C) is in E . The exact sequence (18) and Lemma 3.
is in E thanks to the exact sequence (17), the group H 0 (U, C ) being of finite type by Proposition 3.6.
Chap. XIII, Prop. 2]). This proves the lemma. Zariski open subset of X. Also the piece of notation v ∈ U means that we consider the closed points of X U and the real places of K; for v ∈ Ω R and i 0, the groups H i (K v , . . . ) must be understood as the modified Tate groups (cf. Remark 3.9). More precisely: there are at least two archimedean places. Also, there is no more counterexample as in Remark 3.5.
-Proposition 3.6 (a) and (c) hold (without the condition U = X), (b) is also true (and when U = X, the group H 1 (X, C ) is even finite).

Duality theorems in fppf cohomology
In order to state and prove duality results for the cohomology of complexes of fppf sheaves, we need to extend some constructions from [DH19] to the context of bounded complexes. Let A and B be two bounded complexes of fppf sheaves over U . Following Following Section 2, given a bounded complex C of finite flat commutative group schemes over U , there is a natural topology on the abelian groups H i c (U, C ). This topology is profinite via Lemma 2.5, and considering H j (U, C ) as discrete torsion groups, the pairings   We now prove the proposition by induction on the length of the complex C .
-if C is concentrated in a given degree n, then the proposition is a direct consequence of [DH19, Th. 1.1].
-assume that C : We apply the functor • to this exact triangle. Then, using (2), we get that the natural triangle is exact.
Since the pairing between a complex and its dual is functorial, we get from the previous triangles a commutative diagram of topological groups with exact rows, where the vertical maps comes from the pairings (21): The second line remains exact as the dual of an exact sequence of discrete groups. Note that we have a quasi-isomorphism ϕ : C → C , where C := [C r+1 / Im(f r ) → C r+2 → · · · → C s ] has a smaller length than C , hence by induction, we can assume that the natural maps H r c (U, C ) → H 2−r (U, C ) * are isomorphisms. Since all the C i 's are finite flat group schemes, we see that the dual morphism ϕ : C → C is a quasi-isomorphism, hence by functoriality of the pairings, we deduce that the maps H r c (U, C ) → H 2−r (U, C ) * are isomorphisms too. Hence in diagram (22), all vertical morphisms, except perhaps the central one, are isomorphisms. Then the five lemma implies that the central morphism is an isomorphism.
By induction on the length of C , the proposition is proved.
From now on, we denote by C = [T 1 → T 2 ] a complex of U -tori with generic fiber C = [T 1 → T 2 ] and dual C (cf. Section 3). As a consequence of the previous proposition, we can get the global duality results for the cohomology of the complex C ⊗ L Z/n: (1) There is a perfect pairing of finite groups Moreover, all the groups involved are zero if |i| > 2.
Proof. -Recall that there is a quasi-isomorphism of complexes ψ = C → C ⊗ L Z/n, where C := [ n T 1 ρ −→ n T 2 ], with n T 1 in degree −2, and that the dual morphism Since C is a bounded complex of finite flat commutative group schemes, then Proposition 4.1 implies that the pairings in the statement of the proposition are perfect pairings of topological groups.
Let us now check the finiteness and vanishing results. Using the exact triangle (10), we get an exact sequence: As the finite U -group schemes T Z/n (C ) and n ker ρ are of multiplicative type, Lemma 3.1 and Remark 3.2 imply that the groups H r (U, T Z/n (C )) and H r (U, n ker ρ) are finite for every integer r. Thus H i (U, C ⊗ L Z/n) is finite.
Similarly, using the exact triangle (10), the group H i c (U, C ⊗ L Z/n) is finite for i ∈ {0, 1} (or if p and n are coprime) by Remark 3.2.
Recall that for a finite and flat group scheme N over U , we have H r (U, N ) = 0 for r < 0 (obvious), for r 4, and also for r = 3 if U = X: indeed, by loc. cit., N D ); the latter is clearly zero if r 4 and if U = X, we also have H 0 c (U, N D ) = 0 thanks to the exact sequence the last map being injective by the assumption U = X. The previous dévissages now yield the vanishing assertions of the proposition.
we get an exact sequence of abelian groups since H i (U, C ⊗ L Z/n) is finite (Proposition 4.2 (1)), the finiteness of the groups n H i+1 (U, C ) and H i (U, C )/n follows for all i.
To prove that H i (U, C ) is torsion if i 2 (resp. if U = X and i = 1), we can restrict by dévissage (using exact sequence (14)) to the case when C = T is one single torus. If U = X and i = 1, this follows from Lemma 3.1 (a), so assume i 2. We can also assume by a restriction-corestriction argument that T = G m because the torus T is split by some finite étale covering of U . Now H 2 (U, G m ) = Br U is torsion because it injects into Br K; also H 3 (U, G m ) is torsion (it is even 0 if U = X) and H i (U, G m ) = 0 for i 4 by [Mil06, Prop. II.2.1], the group scheme G m being smooth (hence étale and fppf cohomology coincide).
For every U -torus T , we have H i (U, T ) = 0 for negative i (obvious), hence by dévissage H i (U, C ) = 0 for i < −1. Let i 3; as seen before H i (U, C ) is torsion and n H i (U, C ) is a quotient of H i−1 (U, C ⊗ L Z/n) by exact sequence (23). The latter is zero if i 4, and also if i = 3 when U = X by the vanishing assertions in Proposition 4.2 (1). Thus H i (U, C ) is zero if i 4, and also if i = 3 if we assume further U = X.
(b) Similarly, the finiteness statements follow from the exact sequence combined with Proposition 4.2 (1). Let i 2. To prove that H i c (U, C ) is torsion we can assume that C is the dual of a torus (via exact sequence (15)), then that C = Z (by a restriction-corestriction argument). Using the exact sequence v ∈U it is sufficient to prove that H i (U, Z) is torsion because the Galois cohomology groups Let T be a U -torus. For each integer i, there is an exact sequence Therefore, H i c (U, T ) = 0 for i < 0, hence H i c (U, C ) = 0 (by dévissage) for i < −1.    (b) There is an exact sequence By Proposition 4.2 (2), the group n H i c (U, C ) is finite for i ∈ {1, 2} and the group H i c (U, C )/n is finite for i ∈ {0, 1}. To prove that the groups H i c (U, C ) are torsion for i 2, we can assume as usual that C = T is a torus. Then we apply Proposition 4.3 (a) and the exact sequence v ∈U Besides We now prove a key lemma.
Lemma 4.7. -Let be a prime number (possibly equal to p). Let i be an integer.
(a) The maps only for i ∈ {−1, 0} (for i ∈ {−1, 0} and U = X, the diagram would consist of profinite but possibly infinite groups if = p, so direct limits would not necessarily behave well; in particular, -adic completions involved would not necessarily be profinite).
The same argument as in (a) shows that ϕ is surjective with divisible kernel, and this kernel is trivial for i = −2 because the finitely generated abelian group H 0 (U, C ) (cf. Proposition 3.6 (a)) has trivial -adic Tate module.
direct sum over all prime , yields the duality between the torsion group of cofinite type (cf. Proposition 4.3) H 1 (U, C ) and the finite type Z-module (cf. Proposition 3.6) H 1 c (U, C ) ∧ . Lemma 4.7 (a) for i = 1 yields that for U = X, the map ψ an isomorphism, which immediately gives the duality between the torsion group of finite type (cf. Proposition 4.3) H 2 (U, C ) and the finite type Z-module (cf. Proposition 3.6) Remark 4.10. -In the case U = X, the first assertion of Theorem 4.9 (b) should be replaced by a duality between H 1 (X, C ) and H 1 (X, C ) ∧ (see Theorem 4.11 (b) below in the case U = X). The second assertion (duality between H 2 (X, C ) and H 0 (X, C ) ∧ ) actually still holds, cf. Theorem 4.11 (a).
The following duality theorem has the same flavor as [Mil06, Th II.4.6 (b)] (but one should be careful that in the number field case, the case r = 3 of the latter does not hold in general, see also Remark 4.18).
The maps g 1 and g 2 are isomorphisms by the first point applied to C = T 1 , C = T 2 . The maps f 1 and f 2 are isomorphisms by Lemma 4.7 (b) applied to the same complexes (map ϕ in the case i = −1): indeed, for a U -torus T , the groups H 1 (U, T ) and H 2 c (U, T ) are finite (Lemma 3.4 (a) and (b), hence they coincide with their -adic completions and do not contain a non trivial divisible subgroup. Therefore, h is an isomorphism by the five-lemma, whence the second point.
(b) Consider diagram (32) for i = 0. By (a), the left vertical map is an isomorphism and the middle vertical map is an isomorphism by Proposition 4.2 (1), hence ϕ m is an isomorphism from m H 1 (U, C ) to (H 1 c (U, C )/ m ) * . Taking direct limit over m, then direct sum over all prime , yields the duality between H 1 (U, C ) (which is torsion by Proposition 4.4, but not necessarily of cofinite type, cf. Remark 4.5) and H 1 c (U, C ) ∧ . Now consider diagram (32) for i = 1. By the previous duality, the left vertical map induces an isomorphism between H 1 (U, C )/ m and ( m (H 1 c (U, C ) ∧ )) * . Since H 1 c (U, C ) is in the class E (that is: it is the product of a finite type group by a profinite group) by Proposition 3.13, the m -torsion of H 1 c (U, C ) and of H 1 c (U, C ) ∧ coincide, hence the left vertical map is actually an isomorphism and the right vertical map ψ m is an isomorphism as well (the middle vertical map is an isomorphism by Proposition 4.2 (2)). Taking direct limit and direct sum over all prime , we get the duality between the torsion group (cf. Proposition 4.4) H 2 (U, C ) and the profinite group H 0 c (U, C ) ∧ .
Proof. -Using the exact triangle (12) and the fact that coker ρ := T is a torus, we know that D 2 (U, T ) is finite and is sufficient to show that H 3 (U, ker ρ) is finite to get the finiteness of D 2 (U, C ). But ker ρ is a group of multiplicative type, so there is an exact sequence where F is a finite group of multiplicative type and T 1 is a torus. Since H 3 (U, T 1 ) = 0 by Proposition 4.3 (a) and H 3 (U, F ) = 0 (cf. Remark 3.2; it is dual to H 0 c (U, F ), which is zero because U = X), the group H 3 (U, ker ρ) is actually zero.
The group D 0 (U, ker ρ) is trivial thanks to the assumption U = X. Thus the exact triangle (13) shows that D 0 (U, C ) is finite because so is H 1 (U, T ) (Lemma 3.4 (a)).  . Since all D i (U, C ) are finite by Proposition 3.6 (c) and Lemma 4.14, the decreasing sequence of positive integers #D i (U, C ) (when U becomes smaller and smaller) must stabilize for some U = U 1 . We get an isomorphism from D i (U 1 , C ) to D i (V, C ) for all V ⊂ U 1 . Since H i (K, C) is the direct limit over V of the H i (V, C ), we get an injective map u : D i (U 1 , C ) → H i (K, C). As D i (U 1 , C ) is the same as D i (V, C ) for every V ⊂ U 1 , the image of u is contained in X i (C) (because its restriction to H i (K v , C) is zero for all v ∈ V and V can be taken arbitrarily small). Conversely, every element of X i (C) can be lifted to an a ∈ H i (V, C ) for some V , and by definition a ∈ D i (V, C ) = D i (U 1 , C ), so the image of u contains X i (C).
(b) Let V ⊂ U 0 be an arbitrary non empty Zariski open subset. Let r ∈ {0, 1}. The injectivity of H r (V, C ) → H r (K, C) has been proved in Proposition 3.6 (a). Identifying now D r (V, C ) with a subgroup of H r (K, C), we get (again using the maps H r (V, C ) → H r (U, C ) for V ⊂ U ⊂ U 0 ) a decreasing sequence of finite subgroups (when V becomes smaller and smaller), which stabilizes for some U 1 . Since D r (U 1 , C ) is also D r (V, C ) for every V ⊂ U 1 , we have D r (U 1 , C ) ⊂ X r ( C). On the other hand, every element of X r ( C) comes from H r (V, C ) for some V ⊂ U 1 , and it is then automatically in D r (V, C ) = D r (U 1 , C ) because it is everywhere locally trivial. stabilize for some V = U 1 ⊂ U 0 . Then the maps i U1,V for V ⊂ U 1 are isomorphisms, which implies (passing to the limit) that the restriction map N U1 → H 1 (K, T ) is injective. By definition of N U1 , this means that N U1 = 0, hence N V = 0 for every V ⊂ U 1 . This gives the first point.
For W ⊂ V ⊂ U 1 , the restriction map H 0 (V, C ) → H 0 (W, C ) is injective because so is its composition with H 0 (W, C ) → H 0 (K, C ). As H 0 (V, C ) and H 0 (W, C ) are finitely generated by Proposition 3.6 (b), the induced map H 0 (V, C ) ∧ → H 0 (W, C ) ∧ is still injective. By Theorem 4.9, the dual map H 2 . The same notation stands for C. The groups v∈X (1) H i (K v , C) and v∈X (1) H i (K v , C) are equipped with their restricted product topology (associated to the topology previously defined on the H i (K v , C) and H i (K v , C)). All groups H i (K, C) (resp. H i (K, C)) are equipped with the discrete topology.
Lemma 5.5. -Let i be an integer. Then the image of H i (K, C) in v∈X (1) H i (K v , C) is discrete for the subspace topology. The same holds if C is replaced by C.
Proof. -As the local fields K v are of strict cohomological dimension 2, the statement is obvious except for −1 i 2. Fix a Zariski open subset U ⊂ U 0 with U = X. All groups H i (K v , C) are discrete, so the subgroup is open in v∈X (1) H i (K v , C). Let I be the image of H i (K, C) in v∈X (1) H i (K v , C). Every element of H 1 (K, C) comes from H 1 (V, C ) for some V ⊂ U , hence by Lemma 2.2, there is a surjection D i (U, C ) → I ∩ E. Since all groups D i (U, C ) are finite by Proposition 3.6 (c), Lemma 4.14 and Lemma 4.15, this implies that I ∩ E is finite, hence I is discrete. The same argument shows that the image J of H i (K, C) in v∈X (1) H i (K v , C) is discrete for i 1. For i ∈ {−1, 0}, this is an immediate consequence of Lemma 3.8 (again combined with Lemma 2.2). (a) There are exact sequences

Proof
(a) Let V ⊂ U be a non empty Zariski open subset. Let i ∈ {−1, 0}. By Lemma 2.2, we have an exact sequence By Proposition 3.13, the map H i+1 c (V, C ) → H i+1 c (V, C ) ∧ is injective, thus by Theorem 4.11 we get an exact sequence C ) is a discrete torsion group. Besides, the kernel of the first map is a subgroup of D i (U, C ), hence it is finite for i = 0 by Lemma 4.15. This kernel is also obviously zero for i = −1 as soon as V = U . This implies that the inverse limit of this exact sequence (when V runs over all non empty Zariski open subsets of U ) remains exact, which yields the result.
-The groups D i (U, C ) are finite (Lemma 4.15 and Proposition 3.6 (c)).
Now the same method as in (a) gives the exactness of The right column is also exact by definition of D i+1 (U, C ). By diagram chasing, this yields an exact sequence J.É.P. -M., 2020, tome 7 -Lemma 5.3 still holds. Therefore, Theorem 5.4 is also true: indeed, since the pairing (38) has divisible right kernel and trivial left kernel, taking the direct limit over U (and using the facts that the sequence of finite groups D 0 (U, C ) stabilizes for U sufficiently small) yields a pairing X 0 (C) × X 2 ( C) with divisible right kernel and trivial left kernel. But it is known that X 2 ( C) is finite (see [Dem11b, Proof of Th. 5.14]), whence the result (which extends [Dem11b, Th. 5.23]). Actually the image of H 2 c (U, C ) into H 2 (K, C) is finite by dévissage thanks to exact triangle (13): indeed, H 3 (K, T ) ∼ = v∈Ω R H 3 (K v , T ) is finite for a torus T , and H 2 c (U, M ) is also finite for a group of multiplicative type M because we already saw (cf. Remark 3.15 and Lemma 3.4 (c) that this holds when M is a torus or a finite group.
-In Lemma 5.5, one has to restrict to i 1 for the assertion about C. The result about C holds for an arbitrary i (although the group D 2 (U, C ) might be infinite, we just saw that its image in H 2 (K, C) is finite, which is sufficient).
-Lemma 5.6 (a) is not valid any more (one has to complete the first two terms in the exact sequences); the second exact sequence of (b) still holds (same proof), as does the first one except for the surjectivity of the last map, which must be replaced by the exact sequence v ∈U H 2 (K v , C) × v∈U H 2 nr (K v , C) −→ H −1 (K, C) * −→ D 3 (U, C ) −→ 0 because we lack the vanishing of D 3 (U, C ). Also, since D 2 (U, C ) is in general infinite, the proof of the exactness of H 2 (U, C ) −→ v ∈U H 2 (K v , C) × v∈U H 2 nr (K v , C) −→ H −1 (K, C) * is a little bit more complicated (using exact triangle (13) one reduces to the case when C is quasi-isomorphic to M [1], where M is a group of multiplicative type; then one proceeds as in Lemma 5.6 (b), the group D 2 (U, M ) being finite because H 2 c (U, M ) is finite).
-By the previous observations, the end of sequence (41) starting with H 1 (K, C) * → H 1 (K, C) → · · · remains exact. Theorem 5.8 is valid with one single slight complication in the proof: we do not know in general that D 3 (U, C ) = 0, but the direct limit over U of the D 3 (U, C ) is X 3 ( C), which is zero. Theorem 5.10 is therefore also unchanged, which extends [Dem11b, Th. 6.1 & 6.3].
Remark 5.14. -In the case of one single torus T with module of characters T , some of our results of Section 4 and 5 can be deduced from similar theorems on 1-motives proved by González-Avilés ([GA09, Th. 6.6]) and González-Avilés/Tan ([GAT09, Th. 3.11]).