The galois-cohomology tag has no usage guidance.

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### Generic triviality of $G$-bundles

Let $k$ be an algebraically closed field and $X$ a curve over $k$. Then any $G$-bundle on a curve (where $G$ is reductive and connected) is generically trivial. This is the one of the main results of ...

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### Relations between the cohomology of discrete groups and of profinite groups

Let $G$ be a discrete group and $K$ be the profinite completion of $G$. Let $C_K$ denote the category of contionuous $K$-modules and ${C_K}'$ denotes category of finite continuous $K$-modules. Now for ...

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### Motivic Pfister type varieties and norm varieties

Due to results of Rost it is known that the Grothendieck-Chow motiv of a Pfister quadric $X$ belonging to a pure $\alpha \in H^n(k,\mu_2)$ is decomposable in the following way
$M(X) = ...

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### Torsors trivializing over a fixed finite etale cover

Let $S$ be an integral regular scheme and let $T\to S$ be a finite etale morphism. Let $G$ be a smooth affine finite type group scheme over $S$.
Is the set of $S$-isomorphism classes of $G$-torsors ...

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### Lifting torsors in characteristic $p$ to characteristic zero

Let $R$ be a local integral domain with residue field $k$ such that $R$ is of characteristic zero and $k$ is of characteristic $p>0$. Let $G$ be a smooth finite type affine group scheme with ...

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### Computability of Brauer groups

A friend of mine and I were talking about computable algebra, and this question came up. The answer may already be known, but I couldn't find it with Google:
Suppose I have a countable field, $k$. ...

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### Lifting projective Galois representation with condition

Let $\bar{\rho}: G_K\to PGL_n(\mathbb{C})$ be projective representation of the absolute Galois group of a number field $K$ and $\varphi\in Aut(G_K)$.
A theorem of Tate tells us that we can always ...

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### Classification of 3-forms in dimension 7

I'm looking for a classification of $3$-forms over a real vector space of dimension $7$ as for the $3$-forms in dimension $6$. References on the latter case are R. Bryant On the geometry of almost ...

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### explicit zero 2-cocycle

Let $G$ be a group which acts linearly on a vector space of dimension $n$ over a field $k$. Denote by $\rho$ this representation and consider the associated adjoint representation $Ad\rho$ which is ...

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### A question on the cohomology of elliptic curves over local fields

Let $K$ be a number field,$\nu$ a nonarchimedian prime of $K$, $K_{\nu} $ the completion of $K $ at $\nu $ with maximal unramified extension $K_{\nu}^{unr} $. Let $E $ be an elliptic curve defined ...

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### Cohomological dimension of transcendental p-adic extensions

Let $q$ denote a quadratic form over a field $k$.
The u-invariant of a field $u(k)$ is defined by $u(k):=\{ max (\mathrm{rank}(q)) $ | $ q $ is anisotropic over $k\}$.
Let $k = \mathbb{Q}_p$ for any ...

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### Twists of projective automorphisms

Let $X$ be a projective variety over a perfect field $k$. Recall that a twist of $X$ is a variety $Y$ over $k$ such that $$X_{\bar k} \cong Y_{\bar k}.$$
The twists of $X$ are classified by the Galois ...

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### Motives of a variety of type D4

Over the last decade Nikita Semenov, Skip Garibaldi and others have made some progress in the theory of cohomological invariants, (Rost)-motives and motivic decompositions of algebraic groups. For ...

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### Local duality for abelian varieties

Let $A$ be an abelian variety over a p-adic field $K$. Let $I$ be the inertia group of $K$. There is a Yoneda pairing $$H^n(\hat{\mathbb{Z}},A^I) \times Ext^{2-n}_{\hat{\mathbb{Z}}}(A^I,\mathbb{Z}) ...

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### Exactness on rational points of algebraic groups

Let $k$ be a finite extension of the p-adic number field $Q_p$ and G be a connected algebraic (not affine) group over $k$. It is well-known (see e.g. [1] Proposition 3.1) that G decomposes as
...

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### extensions of crystalline representations

Denote by $G_p$ a choice of an absolute Galois group of $Q_p$, the field of $p$-adic numbers. Consider a continuous representations of $G_p$ on a $3$-dimensional $Q_p$ vector space that is a ...

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### When does the continuous Galois(=etale) cohomology of fields coincide with the naive one? Often true by the Bloch-Kato conjecture?

For a field $F$ I am interested in its $l$-adic (Galois=\'etale) cohomology; here $l$ is a prime distinct from the characteristic of $F$ (for simplicity one may assume that the latter is $0$).
For ...

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### Splitting varieties of two Galois cohomology symbols

One characteristic property of the so called norm varieties defined by Rost, is that they split pure symbols of Galois cohomology groups $H^n(k,\mu_p)$, meaning:
For some $\alpha \in H^n(k,\mu_p)$ ...

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### On Serre's problem regarding the injectivity of Albert-Algebra cohomological invariants

In these Lecture Notes http://molle.fernuni-hagen.de/~loos/jordan/archive/cohinv/cohinv.pdf from 2006 by Garibaldi on page 21. 7.5 there is the following open problem mentioned:
Is the map
$g_3 ...

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### “Forms” of quadrics

The theory of Severi-Brauer varieties is well-known. Let $k$ be a (perfect) field. There may exist varieties not isomorphic to $\mathbf{P}^n$ over $k$, which are isomorphic to $\mathbf{P}^n$ over ...

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### What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?

We denote by $F^{\mathbb{R}}_{p,q}$ the quadratic form over the field ${\mathbb{R}}$
$$
F^{\mathbb{R}}_{p,q}(x)=x_1^2+\dots+x_p^2-(x_{p+1}^2+\dots+x_{p+q}^2)
$$
on the vector space ...

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### The cardinality of first non-abelian Galois cohomology

Let $G$ be a linear algebraic group over a non-archimedean local field $F$. Let $H^1(F,G)$ be the first non-abelian Galois cohomology. It is known that when $F$ is of characteristic 0, i.e. finite ...

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### Rank four quadratic Form with non trivial discriminant in I(k)

Im sure this is a beginners question.
Let $k$ be a field and $I(k)$ the fundamental ideal in the Witt-ring W(k).
The Arason-Pfister-Hauptsatz states:
"If $\varphi$ is any anisotropic class in ...

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### torsors on quasi-split groups

Let $\mathbf{G}$ be a split connected reductive group scheme over a scheme $X$.
Let $X'\rightarrow X$ an étale Galois cover of group $\Gamma$.
We consider $G$ a quasi-split group scheme over $X$ ...

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### $H^1$ and fractional ideals group

Let $L/K$ be a Galois extension with Galois group and $\mathfrak p$ be a prime of the ring of integers $\mathcal O_K$.
I would like to prove that $H^1(G, I_{\mathfrak p})=1$ where $I_{\mathfrak p}$ is ...

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### Variant of Hilbert 90 for Galois extensions

Let $K/\mathbb F_q(x)$ be a finite Galois extension with Galois group $G$. Let $Aut(K)$ be the group of $\mathbb F_q$-automorphisms of $K$.
Obviously, $G\subseteq Aut(K)$. It is well known that
...

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### Set of isomorphisms of Pfister forms corresponding to first cohomology of algebraic group

Assume $k_0$ is a field with char($k_0$) not $2$. Let us define functors from $\rm Field_{/k_0}\to \rm Sets$ as $\rm Pfister_n(k):=\{\text{isomorphism classes of n-fold Pfister forms over k}\}$;
...

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### Continuity of l-adic cohomology: is the cohomology of the generic point isomorphic to the completion of the limit of cohomology of open subvarieties?

Let $X$ be a variety over an algebraically closed field $k$. Denote by $\eta$ its generic point; it is the inverse limit of the open subvarieties $X_i$ of $X$. It is well known that the etale ...

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### rationality question while dealing with an isogeny

I don't think that the following is known, but before going to other things, I would like to know what can be said about it. Thanks in advance for any relevant comment !
So here is the situation. Let ...

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### Forms of algebraic varieties

Let $X$ be an algebraic variety (say, projective, irreducible and smooth), defined over a field $K$, and let $L$ be a Galois extension. I am interested in algebraic varieties $Y$, defined over $K$, ...

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### Open subgroups of the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$

Let $G$ be the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$. Then $G$ is isomorphic to a semidirect product of $\widehat {\mathbb Z}(1)$ by $ Gal_\mathbb Q$.
Is it true that ...

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### Unramified Galois cohomology of number fields

I'm trying to understand unramified Galois cohomology of number fields a bit better.
Set-up: Let $k$ be a number field and $S$ a finite set of places of $k$ which contains all the archimedean ...

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### What is the interpretation of this galois cohomology set?

Let $K$ be a field of characteristic zero. Let $G_K:=Gal(\bar{K}/K)$
The nontrivial elements of the set $H^1(G_K,PGL_2)$ correspond to $\bar{K}/K$-forms of $\mathbb{P}^1$; i.e. curves that are ...

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### References for Gauss Composition using Galois Cohomology

Note: I have already posted this on stackexchange, but have not yet gotten a response.
What are some good references for the Gauss composition law on binary quadratic forms in terms of Galois ...

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### Cohomology after completion

I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if ...

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### Projectives in the category of discrete G-modules

If $G$ is a profinite group, then the category $Mod(G)$ of discrete $G$-modules has sufficiently many injectives (Neukirch, Schmidt, Wingberg: Cohomology of Number Fields, 2.6.5).
Since the cited ...

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### p-adic Hodge theory for varieties defined over \C _p ?

I have a question on p-adic Hodge theory:
When e.g. $X$ is a smooth proper scheme over a finite extension $K$ of $\mathbf{Q}_{p}$ then e.g. one variant of $p$-adic Hodge theory says that there is a ...

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### $L/k$ forms for affine schemes of finite type

Notations and terminology: Let $k$ be a field and $X$ be a $k$-scheme. Denote by $X_L$ the scheme $X\times_k\rm Spec(L)$. For a field extension $L/k$, a $L/k$ form is a $k$-scheme $Y$ such that there ...

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### on the Galois cohomology of reductive groups

Let $G$ a simply connected group over an algebraically closed field.
$F=k((t))$ and $\mathcal{O}=k[[t]]$.
Let $\gamma\in G(\mathcal{O})\cap G(F)^{rs}$.
Let $E=k((t^{1/n}))$ with $n$ prime to the ...

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### Peu or très ramifiée extension

Let $p$ be a prime number. Let $\mathbb{F}$ be a finite extension of $\mathbb{F}_p$. Let $\omega$ be the mod $p$ cyclotomic character and let $V$ be a representation of $G_{p} = Gal(\bar{\mathbb{Q}}_p ...

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### How is etale cohomology of integer rings related to Galois cohomology?

In the paper of Bloch and Kato in the Grothendieck Festschrift, and some other papers relating to the Bloch-Kato conjecture and the ETNC, the cohomology groups
$H^i_{\mathrm{et}}(\operatorname{Spec} ...

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### Descent for group actions

Suppose I have a finite Galois extension of fields $K/k$, as well as a finite group $G$ with a surjection $f: G \rightarrow \mathrm{Gal}(K/k)$.
Finally, suppose I have an action $\sigma$ of $G$ on a ...

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### Isomorphism between the set of classes of Principal Homogeneous spaces and non-Abelian H^1(G,A) cohomology

Let A be a G-group, i.e. a set on which G acts on, has a group structure and satisfies $^s(xy)=^s x ^s y$ for all $x,y \in A \ , s \in G$. A homogeneous principal space P is a non-empty G-set on which ...

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### Biprincipal spaces and their composition, following Serre's “Galois Cohomology”

When reviewing my notes and Serre's book "Galois Cohomology" Chapter 5 dealing with non-abelian group cohomology, I realized that I don't fully understand the concept of biprincipal spaces such as ...

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### Generalization of Kummer isomorphism?

This is a question I asked on math.stackexchange without success.
Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action ...

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### the sixth morphism in the long exact sequence associated to the Hochschild-Serre spectral sequence

The long exact sequence associated to the Hochschild-Serre spectral sequence for extension of groups $1 \to H \to G \to G/H \to 1$ is
$$
\begin{array}[t]{lll}
1 \to & H^1(G/H, A^{G/H}) ...

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### Galois cohomology of the field of Laurent series

Let $k$ a separably closed field. Do we have that $k((t))$ is of cohomological dimension one?

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### do commutative groups torsors have a point in an Abelian extension of the base field?

Let $A$ be a principal homogeneous space for a commutative algebraic group defined over a field $k$ that contains all roots of unity. Is it true that $A$ has a $K$-point for an extension $K \supset k$ ...

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### gluing gerbes over a spectrum of a field

A theorem of Giraud says that gerbes over a scheme $X$ bounded by a sheaf of Abelian groups $A$ are classified by elements of the etale cohomology group $H^2(X,A)$. Similar statements hold in other ...

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### geometric interpretation of the transgression map

Let $X$ be an algebraic variety over an algebraically closed field $k$ and let a finite group $G$ act on it so that it acts freely on the generic fibre of the projection $X \to X/G$, so ...