On the structure of certain Galois cohomology groups.

*(English)*Zbl 1138.11048Let \(K\) be a number field and let \(\Sigma\) be a finite set of primes of \(K\) containing all Archimedean primes and all primes lying above some fixed rational prime \(p\). Let \(K_{\Sigma}\) be the maximal extension of \(K\) which is unramified outside \(\Sigma\) and write \(G_{\Sigma}=\text{Gal}(K_{\Sigma}/K)\). Further, let \(K_{\infty}\subset K_{\Sigma}\) be the cyclotomic \({\mathbb Z}_{p}\)-extension of \(K\) and write \(\Gamma=\text{Gal}(K_{\infty}/K)\). Now let \(V\) be an \(n\)-dimensional \({\mathbb Q}_{p}\)-vector space equipped with a continuous \({\mathbb Q}_{p}\)-linear action of \(G_{\Sigma}\) and let \(D=V/T\), where \(T\) is a Galois-invariant \({\mathbb Z}_{p}\)-lattice in \(V\). Thus \(D\) is a discrete \(G_{\Sigma}\)-module which is isomorphic to \(({\mathbb Q}_{p}/{\mathbb Z}_{p})^{n}\) as a \({\mathbb Z}_{p}\)-module and the Galois action defines a representation \(\rho_{o}: G_{\Sigma}\to \text{Aut}_{{\mathbb Z}_{p}}(T)\simeq GL_{n}( {\mathbb Z}_{p})\). There is a natural action of \(\Gamma\) on the \({\mathbb Z}_{p}\)-module \(H^{i}(K_{\Sigma}/K_{\infty},D)\) (for any \(i\geq 0\)) and the latter may therefore be regarded as a discrete \(\Lambda\)-module, where \(\Lambda={\mathbb Z}_{p}[[\Gamma]]\). As is well-known, \(\Lambda\simeq{\mathbb Z}_{p}[[\,t\,]]\), where \(t\) is a variable, and this is a complete Noetherian local domain of Krull dimension 2. The module \(H^{i}(K_{\Sigma}/K_{\infty},D)\) is cofinitely generated over \(\Lambda\), i.e., its Pontryagin dual is finitely generated over \(\Lambda\).

In [R. Greenberg, “Iwasawa theory for \(p\)-adic representations”, Adv. Stud. Pure Math. 17, 97–137 (1989; Zbl 0739.11045)], it was shown that, if \(p\neq 2\), then \(H^{2}(K_{\Sigma}/K_{\infty},D)\) is a cofree \(\Lambda\)-module, and if \(H^{2}(K_{\Sigma}/K_{\infty},D)=0\), then the Pontryagin dual of \(H^{1}(K_{\Sigma}/K_{\infty},D)\) contains no nonzero finite \(\Lambda\)-submodules. The objective of the paper under review is to generalize these results.

Let \(R\) be a commutative, Noetherian, complete local domain with finite residue field of characteristic \(p\). Assume furthermore that \(R\) is reflexive, i.e., \(R=\bigcap_{\,\text{height}(\mathcal P)=1} R_{\mathcal P}\), where \(R_{\mathcal P}\) denotes the localization of \(R\) at \(\mathcal P\) and the intersection takes place inside the field of fractions \(\mathcal K\) of \(R\). This holds, for instance, if \(R\) is integrally closed. If the Krull dimension of \(R\) is \(m+1\), where \(m\geq 0\), then \(R\) contains a subring \(\Lambda\) isomorphic to \(B[[\,t_{1},\dots,t_{m}\,]]\), where \(B={\mathbb Z}_{p}\) or \({\mathbb F}_{p}[[\,t\,]]\) according as \(\text{char}\,R=0\) or \(\text{char}\,R=p\), such that \(R\) is finitely generated as a \(\Lambda\)-module.

Let \(\rho: G_{\Sigma}\to \text{GL}_{n}(R)\) be a continuous representation and let \(\mathcal T\) be the underlying free \(R\)-module on which \(G_{\Sigma}\) acts via \(\rho\). Set \(\mathcal D=\mathcal T\otimes_{R}\widehat{R}\), where \(\widehat{R}=\text{Hom}(R,{\mathbb Q}_{p}/{\mathbb Z}_{p})\) is the Pontryagin dual of \(R\) endowed with the trivial action of \(G_{\Sigma}\). Thus \(\mathcal D\) is a discrete abelian group which is isomorphic to \(\widehat{R}^{n}\) as an \(R\)-module and has a continuous \(R\)-linear action of \(G_{\Sigma}\). Further, set \({\mathcal T}^{*}=\text{Hom}(\mathcal D, \mu_{p^{\infty}})\). The groups \(H^{i}(K_{\Sigma}/K_{\infty},\mathcal D)\), for \(i\geq 0\), inherit an \(R\)-module structure and cofinitely generated over \(R\). Now set

\[ \text{Ш}^{\,i}(K,\Sigma,\mathcal D)=\text{Ker}\left[H^{i}(K_{\Sigma}/K_{\infty},\mathcal D)\to \prod_{v\in\Sigma}H^{i}(K_{v},\mathcal D)\right], \]

where, for each \(v\in\Sigma\), \(K_{v}\) denotes the completion of \(K\) at \(v\). A finitely generated, torsion-free \(R\)-module \(X\) is said to be reflexive if \(X=\bigcap_{\text{height}(\mathcal P)=1} X_{\mathcal P}\), where \(X_{\mathcal P}=X\otimes_{R}R_{\mathcal P}\) and the intersection takes place inside \(X\otimes_{R}\mathcal K\). A discrete \(R\)-module \(A\) is said to be coreflexive if its Pontryagin dual \(X\) is reflexive. Further, a finitely generated, torsion \(R\)-module \(Z\) is said to be “pseudo-null” if \(Z_{\mathcal P}=0\) for every prime ideal \(\mathcal P\) of \(R\) of height 1.

The following is the main result of the paper. Assume that: (a) \(R\) is a reflexive domain, (b) \({\mathcal T}^{*}/({\mathcal T}^{*})^{G_{v}}\) is reflexive for every \(v\in\Sigma\), where \(G_{v}\) denotes the absolute decomposition group of \(v\), and (c) \(({\mathcal T}^{*})^{G_{v_{0}}}=0\) for at least one non-Archimedean prime \(v_{0}\in\Sigma\). Then \(\text{Ш}^{\,2}(K,\Sigma,\mathcal D)\) is a coreflexive \(R\)-module. Further, if \(\text{Ш}^{\,2}(K,\Sigma,\mathcal D)=0\), then the Pontryagin dual of \(H^{1}(K_{\Sigma}/K,\mathcal D)\) has non nonzero pseudo-null \(R\)-submodules.

This is indeed a (broad) generalization of the results from [Greenberg (loc. cit.)] mentioned at the beginning, for if we take \(\mathcal T=T\otimes_{{\mathbb Z}_{p}}\Lambda\) and \(\mathcal D=T\otimes_{{\mathbb Z}_{p}}\widehat{\Lambda}\), then \(H^{i}(K_{\Sigma}/K,\mathcal D)=H^{i}(K_{\Sigma}/K_{\infty},D)\) for all \(i\geq 0\) and similarly for the Tate-Shafarevich groups. In fact, the above general statement implies the following significant strengthening of the results from the author’s paper mentioned above: let \(p\) be any prime (e.g., \(p=2\)) and let \(K_{\infty}/K\) be any Galois extension such that \(\Gamma\simeq {\mathbb Z}_{p}^{m}\) for some \(m\geq 1\). Then \(\text{Ш}^{\,2}(K_{\infty},\Sigma,D)\) is a coreflexive \(\Lambda\)-module. Further, if \(\text{Ш}^{\,2}(K_{\infty},\Sigma,D)=0\), then the Pontryagin dual of \(H^{1}(K_{\Sigma}/K_{\infty}, D)\) has no nonzero pseudo-null \(\Lambda\)-submodules.

The paper discusses conditions under which both of the assumptions (b) and (c) above hold, and gives examples where they fail. Further, the vanishing condition \(\text{Ш}^{\,2}(K_{\infty},\Sigma,D)=0\) is related to a certain lower bound for the \(R\)-corank of \(H^{1}(K_{\Sigma}/K_{\infty},\mathcal D)\) becoming an equality. In addition, results are obtained on the \(\Lambda\)-divisibility or “almost divisibility” of the cohomology groups appearing above.

In [R. Greenberg, “Iwasawa theory for \(p\)-adic representations”, Adv. Stud. Pure Math. 17, 97–137 (1989; Zbl 0739.11045)], it was shown that, if \(p\neq 2\), then \(H^{2}(K_{\Sigma}/K_{\infty},D)\) is a cofree \(\Lambda\)-module, and if \(H^{2}(K_{\Sigma}/K_{\infty},D)=0\), then the Pontryagin dual of \(H^{1}(K_{\Sigma}/K_{\infty},D)\) contains no nonzero finite \(\Lambda\)-submodules. The objective of the paper under review is to generalize these results.

Let \(R\) be a commutative, Noetherian, complete local domain with finite residue field of characteristic \(p\). Assume furthermore that \(R\) is reflexive, i.e., \(R=\bigcap_{\,\text{height}(\mathcal P)=1} R_{\mathcal P}\), where \(R_{\mathcal P}\) denotes the localization of \(R\) at \(\mathcal P\) and the intersection takes place inside the field of fractions \(\mathcal K\) of \(R\). This holds, for instance, if \(R\) is integrally closed. If the Krull dimension of \(R\) is \(m+1\), where \(m\geq 0\), then \(R\) contains a subring \(\Lambda\) isomorphic to \(B[[\,t_{1},\dots,t_{m}\,]]\), where \(B={\mathbb Z}_{p}\) or \({\mathbb F}_{p}[[\,t\,]]\) according as \(\text{char}\,R=0\) or \(\text{char}\,R=p\), such that \(R\) is finitely generated as a \(\Lambda\)-module.

Let \(\rho: G_{\Sigma}\to \text{GL}_{n}(R)\) be a continuous representation and let \(\mathcal T\) be the underlying free \(R\)-module on which \(G_{\Sigma}\) acts via \(\rho\). Set \(\mathcal D=\mathcal T\otimes_{R}\widehat{R}\), where \(\widehat{R}=\text{Hom}(R,{\mathbb Q}_{p}/{\mathbb Z}_{p})\) is the Pontryagin dual of \(R\) endowed with the trivial action of \(G_{\Sigma}\). Thus \(\mathcal D\) is a discrete abelian group which is isomorphic to \(\widehat{R}^{n}\) as an \(R\)-module and has a continuous \(R\)-linear action of \(G_{\Sigma}\). Further, set \({\mathcal T}^{*}=\text{Hom}(\mathcal D, \mu_{p^{\infty}})\). The groups \(H^{i}(K_{\Sigma}/K_{\infty},\mathcal D)\), for \(i\geq 0\), inherit an \(R\)-module structure and cofinitely generated over \(R\). Now set

\[ \text{Ш}^{\,i}(K,\Sigma,\mathcal D)=\text{Ker}\left[H^{i}(K_{\Sigma}/K_{\infty},\mathcal D)\to \prod_{v\in\Sigma}H^{i}(K_{v},\mathcal D)\right], \]

where, for each \(v\in\Sigma\), \(K_{v}\) denotes the completion of \(K\) at \(v\). A finitely generated, torsion-free \(R\)-module \(X\) is said to be reflexive if \(X=\bigcap_{\text{height}(\mathcal P)=1} X_{\mathcal P}\), where \(X_{\mathcal P}=X\otimes_{R}R_{\mathcal P}\) and the intersection takes place inside \(X\otimes_{R}\mathcal K\). A discrete \(R\)-module \(A\) is said to be coreflexive if its Pontryagin dual \(X\) is reflexive. Further, a finitely generated, torsion \(R\)-module \(Z\) is said to be “pseudo-null” if \(Z_{\mathcal P}=0\) for every prime ideal \(\mathcal P\) of \(R\) of height 1.

The following is the main result of the paper. Assume that: (a) \(R\) is a reflexive domain, (b) \({\mathcal T}^{*}/({\mathcal T}^{*})^{G_{v}}\) is reflexive for every \(v\in\Sigma\), where \(G_{v}\) denotes the absolute decomposition group of \(v\), and (c) \(({\mathcal T}^{*})^{G_{v_{0}}}=0\) for at least one non-Archimedean prime \(v_{0}\in\Sigma\). Then \(\text{Ш}^{\,2}(K,\Sigma,\mathcal D)\) is a coreflexive \(R\)-module. Further, if \(\text{Ш}^{\,2}(K,\Sigma,\mathcal D)=0\), then the Pontryagin dual of \(H^{1}(K_{\Sigma}/K,\mathcal D)\) has non nonzero pseudo-null \(R\)-submodules.

This is indeed a (broad) generalization of the results from [Greenberg (loc. cit.)] mentioned at the beginning, for if we take \(\mathcal T=T\otimes_{{\mathbb Z}_{p}}\Lambda\) and \(\mathcal D=T\otimes_{{\mathbb Z}_{p}}\widehat{\Lambda}\), then \(H^{i}(K_{\Sigma}/K,\mathcal D)=H^{i}(K_{\Sigma}/K_{\infty},D)\) for all \(i\geq 0\) and similarly for the Tate-Shafarevich groups. In fact, the above general statement implies the following significant strengthening of the results from the author’s paper mentioned above: let \(p\) be any prime (e.g., \(p=2\)) and let \(K_{\infty}/K\) be any Galois extension such that \(\Gamma\simeq {\mathbb Z}_{p}^{m}\) for some \(m\geq 1\). Then \(\text{Ш}^{\,2}(K_{\infty},\Sigma,D)\) is a coreflexive \(\Lambda\)-module. Further, if \(\text{Ш}^{\,2}(K_{\infty},\Sigma,D)=0\), then the Pontryagin dual of \(H^{1}(K_{\Sigma}/K_{\infty}, D)\) has no nonzero pseudo-null \(\Lambda\)-submodules.

The paper discusses conditions under which both of the assumptions (b) and (c) above hold, and gives examples where they fail. Further, the vanishing condition \(\text{Ш}^{\,2}(K_{\infty},\Sigma,D)=0\) is related to a certain lower bound for the \(R\)-corank of \(H^{1}(K_{\Sigma}/K_{\infty},\mathcal D)\) becoming an equality. In addition, results are obtained on the \(\Lambda\)-divisibility or “almost divisibility” of the cohomology groups appearing above.

Reviewer: Cristian D. Gonzales-Aviles (La Serena)