# Questions tagged [galois-cohomology]

For questions on Galois cohomology, the study of the group cohomology of Galois modules.

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### Galois fixed part of modules. Why does $|(1-\sigma)A||A^G|=|A|$ hold?

Let $L/K$ be a quadratic extension of number field $K$. Let $\sigma$ be a generator of $G=Gal(L/K)$. Let $A$ be a finite $Gal(L/K)$ module. Then, Why does $|(1-\sigma)A||A^G|=|A|$ hold ? Here, $|A|$ ...
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### Almost purity for perfectoid fields applied on a computation of Galois cohomology.

Let $K=\mathbf{Q}_p(\zeta_{p^{\infty}})$ and $C:=\widehat{\overline{\mathbf{Q}_P}}$. When I wrote an answer for a question a question answer, I have in my mind that "$\widehat{K}$ is a ...
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### $H^1(\text{Gal}(L/K), O_L)=0$ for local fields

Let $K$ be a localfield and $O_K$be its ring of integers, and $L/K$ be a quadratic extension. It is known that $H^1(\text{Gal}(L/K), L) = 0$ according to Hilbert's Theorem 90. However, what is known ...
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### Relation between Galois Cohomology $H^1(\text{Gal(L/K)}, L) = 0$ and $H^1(\text{Gal}(L/K), O_L)=0$

Let $K$ be a number field and $O_K$be its ring of integers, and $L/K$ be a quadratic extension. It is known that $H^1(\text{Gal}(L/K), L) = 0$ according to Hilbert's Theorem 90. However, what is known ...
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### Why is coboundary $\sigma \to \sigma m-m$ automatically continuous?

Let $K$ be a field. Let $G_K := Gal(\overline{K}/K)$ be absolute Galois group of $K$. Let $M$ be a $G_K$-module, that is, $G_K$ (with the Krull topology) acts continuously on $M$ with discrete ...
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### Proof of Proposition 2.27 in Diamond, Darmon, Taylor's "Fermat's Last Theorem"

See page 67 of the free online text for Proposition 2.27 below. All cohomology is Galois cohomology. Page 62 defines goodness, ordinaryness, and semistability. Page 65-66 defines $H^1_f$. The ...
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### The first Galois cohomology commutes with projective limits

I am reading Serre's paper "Sur les groupes de congruence des variétés abéliennes" (here is the link to this paper: https://www.mathnet.ru/links/016949238724700ec2209f00e507a40f/im3061.pdf). ...
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### Tate Twists of Z/nZ

Let $k$ be a $p$-adic local field with absolute Galois group $G_k$. In Cohomology of Number Fields, the authors define the $n$-th Tate twist of a finite $G_k$-module $A$ as the $G_k$-module $A(n)$ ...
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### Is the subgroup of $WC(E/K)$ of curves with an $L$-rational point finite for $K$ local, $L/K$ finite Galois?

User BrauerManinobstruction's question "Weil–Châtelet group of a real elliptic curve is isomorphic to Z/2Z when Δ>0" relates to exercise 10.7 of Silverman's AEC, which asks the reader to ...
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### Restriction map in semilocal Galois cohomology

Let $F/K$ be a finite extension of number fields, let $v$ be a prime of $K$ that is totally split in $F$. Call $w_1,\dots, w_n$ the primes of $F$ above $v$. Let $T$ be a $Gal(\bar{K}/K)$-module. Then ...
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### Why discrete modules?

The profinite group cohomology of discrete modules can be defined by right derived functors. Its application includes Galois cohomology, Brauer groups etc. These facts demonstrate that defining ...
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### In definition of Galois cohomology, is continuity necessary?

Let $K$ be a number field. Galois cohomology $H^1(Gal(\overline{K}/K), E)$ is composed of representative of continuous maps $Gal(\overline{K}/K) \to E$ which satisfies cocycle relation. If we discard ...
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### Invariants of Iwasawa Modules

Let $\Lambda$ denote Iwasawa algebra $\mathbb{Z}_p[[\Gamma]]$, where $\Gamma$ is a group isomorphic to $\mathbb{Z}_p$(ring of $p$-adic integers) . The structure theorem of the Iwasawa module tells; If ...
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### Local invariant map in the case of closed points of a curve

In class field theory, we have the well-known local invariant map $\mathrm{inv}_v: \mathrm{Br}(k_v) \rightarrow \mathbb{Q}/\mathbb{Z}$, where $k$ is a field and $v$ is a place of $k$. Similarly, we ...
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### Is $\mathbb{Q}$ an injective $\mathbb{Z}[G]$-module

Here $G$ is a finite group. I can show that $\mathbb{Q}$ (as a trivial $G$-module) is cohomologically trivial, since the inclusion $\mathbb{Q} \to \mathbb{Q}[G]$, $a\mapsto a(\sum_{g\in G} g)$ splits (...
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### Restriction vs. multiplication by $n$ in Tate cohomology

$\DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\Cor}{Cor}$ Let $H$ be a subgroup of index $n$ of a finite group $G$, and let $M$ be a $G$-module, that is, an abelian group on which $G$ acts. ...
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### Real forms of $\mathbb{G}_m$ as a variety

Notation: Denote $k=\mathbb{R}$, $K=\mathbb{C}$ and let $G:=\text{Gal}(K/k)=\{\text{id},\sigma\}$, where $\sigma$ is complex conjugation. Background: In the Notices article A Gentle Introduction to ...
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### Splitting of exact sequence of G-modules

Suppose we have an exact sequence of $G$(finite)-modules; $$0\to A=B^G \to B \to C \to 0$$ where $A,B$ and $C$ are $p$-primary groups. If p does not divide the order of $G$, can I say that the above ...
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