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On the structure of certain Galois cohomology groups. (English) Zbl 1138.11048
Let $$K$$ 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.

##### MSC:
 11R23 Iwasawa theory 11R34 Galois cohomology
Iwasawa theory
Magma; SageMath
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