# Questions tagged [galois-descent]

The galois-descent tag has no usage guidance.

29
questions

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### Galois action on morphism between $\overline{k}$ schemes

I have a question on a certain property of morphisms between schemes endowed with Galois action. The motivation arises from a comment by Phil Tosteson on this question.
Phil wrote: "If the map ...

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123 views

### Application of Galois descent

I not understand an assumption done at the beginning of the proof of Rigidity lemma in Moonens and van der Geers book about Abelian variaties (page 12). Here is it:
Question: Why the assumption $k= \...

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241 views

### Non-linear Galois descent

This question is about Galois theory. So let $K / k$ be a Galois extension of fields. Let us assume that $K / k$ is finite dimensional, though everything can be made to work in the profinite case by ...

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113 views

### The reductive $p$-adic group $^{2}\!A_3''$ via Galois decent

I am running into some confusion when trying to explicitly describe the group $^{2}\!A_3''$ (using the naming convention that Tits gives in his Corvallis notes). If anyone can give me any advice, I ...

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300 views

### Number Rings and (Galois) Descent

In algebraic number theory, one chooses for each finite étale $\mathbb{Q}$-algebra $K$ a finite $\mathbb{Z}$-algebra $\mathcal{O}_K$. Usually one simply speaks of the finite $\mathbb{Q}$-algebras ...

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327 views

### Infinite Galois descent for finitely generated commutative algebras over a field

Let $k_0$ be a field of characteristic 0, and let $k$ is a fixed algebraic closure of $k_0$.
Write $G={\rm Gal}(k/k_0)$.
Let $A_0$ be a finitely generated commutative $k_0$-algebra with a unit.
Then ...

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323 views

### Galois descent in motivic cohomology

Let $X_N$ denote the Fermat curve defined over $\mathbb{Q}$ by the equation $x^N+y^N-z^N=0$ and let $X_{N,\mathbb{Q}(\mu_N)}$ be the base change. Let $G$ be the Galois group of $\mathbb{Q}(\mu_N)/\...

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148 views

### Weil Pairing and Galois descent

One way the Weil pairing for an Abelian Variety $A/k$ is phrased is the following (for simplicity, let me only deal with the multiplication by $m$ map ($[m]: A\to A$) instead of arbitrary isogenies):
...

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142 views

### Base change, descent theory and coherent sheaves

Let $k$ be a field of characteristic zero and $X$ a smooth, projective $k$-variety. Let $E_{\overline{k}}$ be a coherent sheaf on $X_{\overline{k}}$ ($\overline{k}$ denotes the algebraic closure of $k$...

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742 views

### Galois descent for schemes over fields

Let $K\subset L$ be a finite galois extension of fields (the case I have in mind is $K=\mathbb{R},L=\mathbb{C}$). Given a scheme $X$ over $K$ by pulling back to $L$ we get a scheme $Y=X\times _K L$ ...

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551 views

### Galois descent for absolute Galois group

Let $K$ be a field of characteristic zero, $\bar{K}$ its algebraic closure and $X$ a smooth, projective $K$-scheme. We know the Galois descent theory for quasi-coherent sheaves defined on $X_L$ for a ...

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393 views

### Galois descent for etale motivic cohomology

I am interested in the map from etale motivic cohomology of a smooth and projective variety over a field $K$ to the Galois invariants of etale
motivic cohomology over the algebraic closure $\bar K$:
$...

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356 views

### Neat applications of Galois descent?

I'm enjoying reading about Janelidze's categorical Galois theory, which gives as a special case the usual theorems of Galois descent (along torsors). The approach I took was just with covering space ...

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457 views

### Galois descent for dimension of vector spaces

Let $L/K$ be a Galois extension (I am interested in $\overline{\mathbb{Q}}/\mathbb{Q}$ so I do not assume it to be finite).
Let $V\subset L^n$ be a $L$-subvector space, of dimension $d$, such that $g(...

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557 views

### Geometric intuition for the condition of Galois descent

Continuing in my attempts to understand bits and pieces of Borceux and Janelidze's Galois Theories, I've just realized that I don't have any geometric intuition for the most convenient ...

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434 views

### Descent of sheaves under galois covering

Let $\pi: Y\rightarrow X$ be a finite Galois covering between normal projective varieties with Galois group $G$. Let $E$ be a coherent sheaf on $Y$ with a $G$-linearisation, i.e., there are ...

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### Why are monadicity and descent related?

This question is probably too vague for experts, but I really don't know how to avoid it.
I've read in several places that under mild conditions, a morphism is an effective descent morphism iff the ...

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541 views

### Reinterpreting Galois descent over finite fields

This question is indirectly related to my previous question Is an elliptic curve that is isomorphic to its Frobenius conjugate defined over $\mathbb{F}_p$?
Let $\mathbb{F}_{q^n}/\mathbb{F}_q$ be an ...

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313 views

### Is an elliptic curve that is isomorphic to its Frobenius conjugate defined over $\mathbb{F}_p$?

Let $p$ be prime and $q = p^n$. Let $E$ be an elliptic curve over $\mathbb{F}_q$, and let $E^{(p)}$ be the pullback of $E$ by the $p$-power Frobenius of $\mathbb{F}_q$. If $E$ is isomorphic (over $\...

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225 views

### Obstruction to get a galois invariant cycle

Let $X$ be a smooth projective variety over a finite field $k$, $G=Gal(\bar{k}/k)$ and $\Gamma\in CH^i(\bar{X})$ such that:
$cl(\Gamma) \in H_{et}^{2i}(\bar{X},\mathbb{Z}_l(i))^G$ and
$\exists$ $...

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409 views

### Descend of etale morphism

I am not sure whether the title is appropriate for this question or not. I am sorry if there is anyone who is confused with the title and the contents.
What I want to ask is the following: let $k$ be ...

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376 views

### Which morphisms of ring spectra are of effective descent for modules?

There is a well understood bifibration of $\infty$-categories over the $\infty$-category of commutative ring spectra whose fiber over a ring $R$ is the category of $R$-module spectra. This is in ...

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671 views

### Stack of Tannakian categories? Galois descent?

I'm having trouble finding a reference for something that I'm guessing the experts worked out long ago. Let's take a local or global field $F$ for this post, and fix a separable algebraic closure $\...

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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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393 views

### Galois descent for semilinear endomorphisms

Let $K \subset L$ be a finite Galois extension, $\sigma$ an automorphism of $L$ (not necessarily fixing $K$) and let $E$ be a finite-dimensional vector space over $L$ together with an $\sigma$-linear ...

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1k views

### Applications of Descent?

The technique of faithfully flat descent, and, in the case of vector spaces, Galois descent has been used quite a bit in Algebraic Geometry. However, the question of whether, say, a given $k$-vector ...

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### Reference wanted - etale sheaves on $X$ versus on $\overline{X}$

Hello,
Let $X$ be a scheme of finite type over a field $k$. Let $l$ be an Galois extension of $k$ with Galois group $\Gamma$, and $\overline{X}$ be the base change of $X$ from $k$ to $l$. Then If I ...

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1k views

### What kind of structures allow Galois descent?

EDIT: Question solved.
Let me explain what I mean.
The classical formulation of Galois descent, e. g. in Crawley-Boevey's "Cohomology and central simple algebras", uses the following notion:
...

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714 views

### Galois descent, explicit inverse map

Let $L/K$ be a finite Galois extension with Galois group $G$ and $V$ a $L$-vector space, on which $G$ acts by $K$-automorphisms satisfying $g(\lambda v)=g(\lambda) g(v)$. It is known that the ...