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**A new approach to Tate’s local duality.
(Une approche nouvelle de la dualité locale de Tate.)**
*(French)*
Zbl 1160.11364

From the introduction: Let \(K\) be a complete field for a discrete valuation, of characteristic 0, with perfect residue field \(k\) of characteristic \(p>0\). Choose a separable closure \(K^-\) of \(K\), and denote its residue field by \(k^-\) and denote \(G_K = \text{Gal}(K^-/K)\). Let \(l\) be a prime number. For all integers \(n \geq 1\) let \(\mu_{l^n}\) denote the group of \(l^n\)th roots of unity in \(K^-\) and \(\mu_{l^\infty}\) the inductive limit of \(\mu_{l^n}\). We call the \(\mathbb Z_l\)-adic representation of \(GK\) the one given by a \(\mathbb Z_l\) \(V\) of finite type with a linear continuous action of \(G_K\). These representations form an abelian category denoted \(\text{Rep}_{\mathbb Z_l} (G_K)\). If \(V\) is annihilated by a power of \(l\), it is called \(l\)-torsion representation. If \(V\) is an \(F_l\) vector space, it is called representation mod \(l\) of \(G_K\). Let \(\text{Rep}_{l-\text{tor}}(G_K)\) (resp. \(\text{Rep}_{F_l} (G_K)\)) denote the complete subcategories of \(\text{Rep}_{\mathbb Z_l} (G_K)\) formed by \(l\)-torsion (resp. mod \(l\)) representations.

Let \(V\) be an \(l\)-torsion representation of \(GK\). Then one can define its cohomology groups \(H^i(G_K, V)\). The following are classical results of Tate:

Theorem A (Vanishing of the cohomology, cf. [J.-P. Serre, Cohomologie Galoisienne, 5th ed. Berlin: Springer (1994; Zbl 0812.12002), chapter II, proposition 12]). The groups \(H^i(G_K, V)\) are zero for \(i \geq 3\).

Theorem B (Finiteness of the cohomology, cf. [Serre, loc. cit.], chapter 5, proposition 14 and theorem 5). If \(k\) is finite, the groups \(H^i(G_K, V)\) are also finite and the Euler-Poincaré characteristic is defined by \[ \chi(V)=\prod_{0\leq i\leq 2} \text{card}(H^i(G_K, V))(-1)^i \] takes the values: \(\chi(V) = p^{-[K:\mathbb Q_p]\;\text{length}_{\mathbb Z_p} (V)}\) if \(l = p\) and \(1\) otherwise.

Theorem C (Local duality, cf. [Serre, loc. cit.], chapter 5, theorem 2). Let \(V^*=\operatorname{Hom}_{\mathbb{Z}_l}(V ,\mu_{l^\infty})\) the dual of \(V\) twisted à la Tate. If \(k\) is finite, there exists a canonical isomorphism of \(H^2(G_K,\mu_{l^\infty})\) on \(\mathbb Q_l/\mathbb Z_l\). For \(i \in\{0, 1, 2\}\), the cup-product then induces a perfect duality between the groups \(H^i(G_K,V)\) et \(H^{2-i}(G_K,V^*)\).

In this paper we are only interested in the case \(l = p\) which is the most delicate case.

The first two results can be proved via an equivalence of categories explicitly shown by Fontaine in the Grothendieck Festschrift, by an explicit calculation of the groups \(H^i(G_K, V)\) (cf. the author [Bull. Soc. Math. Fr. 126, 563–600 (1998; Zbl 0967.11050)]). The proofs are simpler than the usual approaches of these difficult theorems. In particular, one does not use local class fields, essentially because one reduces everything to the study of the cyclotomic \(\mathbb Z_p\)-extensions by the theory of norm fields of Fontaine-Wintenberger.

The aim of this article is to give a proof of Theorem C basing on similar techniques. Because one does not use the reciprocity map, this allows also to give a new proof of the \(p\)-part of the local class fields (the most difficult one); now the Tate duality gives for any integer \(n\geq 1\) a functorial isomorphism between the finite groups \(\operatorname{Hom}(G_K,\mathbb Z/p^n\mathbb Z) = H^1(G_K, \mathbb Z/p^n\mathbb Z)\) and the dual of \(H^1(G_K,\mu_{p^n})\), this last one can be identified by Kummer theory with \(K^*/(K^*)^{p^n}\). Passing to the projective limit, one finds the \(p\)-part of the reciprocity map (see for ex. [J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of number fields. Berlin: Springer (2000; Zbl 0948.11001)], propositions 7.2.12 and 7.2.13).

The plan of this article is as follows:

– in §1 properties of the continuous differentials on certain Cohen rings are exposed; one introduces residues whose importance in duality questions is well-known;

– in §2, the definition of different Cohen rings introduced by Fontaine in his “Grothendieck Festschrift”, as well as their associated structures are recalled;

– in §3, we recall Fontaine’s construction that allows to associate with each \(\mathbb Z_p\)-adic representation \(V\) of \(G_K\) an étale \(\Phi-\Gamma_K\)-module \(M\) over \(\mathcal O_{\mathcal E(K)}\) and find using differentials the étale \(\Phi-\Gamma_K\)-module corresponding to the Tate module of the multiplicative group of \(K\);

– in §4, we recall the description of the groups \(H^i(G_K, V)\) treated in [the author, loc. cit.] and we give an explicit calculus of cup-products, all this via the category of “étale \(\Phi-\Gamma_K\) torsion modules over \(\mathcal O_{\mathcal E(K)}\)”;

– finally in §5, we define a canonical isomorphism between \(H^2(G_K,\mu_{p^\infty})\) and \(\mathbb Q_p/\mathbb Z_p\); then we prove Theorem C using a Pontryagin duality between complexes associated with \(M\).

Let \(V\) be an \(l\)-torsion representation of \(GK\). Then one can define its cohomology groups \(H^i(G_K, V)\). The following are classical results of Tate:

Theorem A (Vanishing of the cohomology, cf. [J.-P. Serre, Cohomologie Galoisienne, 5th ed. Berlin: Springer (1994; Zbl 0812.12002), chapter II, proposition 12]). The groups \(H^i(G_K, V)\) are zero for \(i \geq 3\).

Theorem B (Finiteness of the cohomology, cf. [Serre, loc. cit.], chapter 5, proposition 14 and theorem 5). If \(k\) is finite, the groups \(H^i(G_K, V)\) are also finite and the Euler-Poincaré characteristic is defined by \[ \chi(V)=\prod_{0\leq i\leq 2} \text{card}(H^i(G_K, V))(-1)^i \] takes the values: \(\chi(V) = p^{-[K:\mathbb Q_p]\;\text{length}_{\mathbb Z_p} (V)}\) if \(l = p\) and \(1\) otherwise.

Theorem C (Local duality, cf. [Serre, loc. cit.], chapter 5, theorem 2). Let \(V^*=\operatorname{Hom}_{\mathbb{Z}_l}(V ,\mu_{l^\infty})\) the dual of \(V\) twisted à la Tate. If \(k\) is finite, there exists a canonical isomorphism of \(H^2(G_K,\mu_{l^\infty})\) on \(\mathbb Q_l/\mathbb Z_l\). For \(i \in\{0, 1, 2\}\), the cup-product then induces a perfect duality between the groups \(H^i(G_K,V)\) et \(H^{2-i}(G_K,V^*)\).

In this paper we are only interested in the case \(l = p\) which is the most delicate case.

The first two results can be proved via an equivalence of categories explicitly shown by Fontaine in the Grothendieck Festschrift, by an explicit calculation of the groups \(H^i(G_K, V)\) (cf. the author [Bull. Soc. Math. Fr. 126, 563–600 (1998; Zbl 0967.11050)]). The proofs are simpler than the usual approaches of these difficult theorems. In particular, one does not use local class fields, essentially because one reduces everything to the study of the cyclotomic \(\mathbb Z_p\)-extensions by the theory of norm fields of Fontaine-Wintenberger.

The aim of this article is to give a proof of Theorem C basing on similar techniques. Because one does not use the reciprocity map, this allows also to give a new proof of the \(p\)-part of the local class fields (the most difficult one); now the Tate duality gives for any integer \(n\geq 1\) a functorial isomorphism between the finite groups \(\operatorname{Hom}(G_K,\mathbb Z/p^n\mathbb Z) = H^1(G_K, \mathbb Z/p^n\mathbb Z)\) and the dual of \(H^1(G_K,\mu_{p^n})\), this last one can be identified by Kummer theory with \(K^*/(K^*)^{p^n}\). Passing to the projective limit, one finds the \(p\)-part of the reciprocity map (see for ex. [J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of number fields. Berlin: Springer (2000; Zbl 0948.11001)], propositions 7.2.12 and 7.2.13).

The plan of this article is as follows:

– in §1 properties of the continuous differentials on certain Cohen rings are exposed; one introduces residues whose importance in duality questions is well-known;

– in §2, the definition of different Cohen rings introduced by Fontaine in his “Grothendieck Festschrift”, as well as their associated structures are recalled;

– in §3, we recall Fontaine’s construction that allows to associate with each \(\mathbb Z_p\)-adic representation \(V\) of \(G_K\) an étale \(\Phi-\Gamma_K\)-module \(M\) over \(\mathcal O_{\mathcal E(K)}\) and find using differentials the étale \(\Phi-\Gamma_K\)-module corresponding to the Tate module of the multiplicative group of \(K\);

– in §4, we recall the description of the groups \(H^i(G_K, V)\) treated in [the author, loc. cit.] and we give an explicit calculus of cup-products, all this via the category of “étale \(\Phi-\Gamma_K\) torsion modules over \(\mathcal O_{\mathcal E(K)}\)”;

– finally in §5, we define a canonical isomorphism between \(H^2(G_K,\mu_{p^\infty})\) and \(\mathbb Q_p/\mathbb Z_p\); then we prove Theorem C using a Pontryagin duality between complexes associated with \(M\).