Questions tagged [galois-descent]
The galois-descent tag has no usage guidance.
39
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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 ...
- 10.3k
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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 $\...
- 13k
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1
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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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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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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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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 $\...
- 2,623
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3
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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 ...
- 13.4k
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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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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:
...
- 32.7k
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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 ...
- 1,083
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463
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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 $\bar K$:
$...
- 1,413
6
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3
answers
1k
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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 ...
- 630
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Are the Galois actions on automorphisms of twists isomorphic?
This might be a trivial question and I might be overlooking something:
Suppose $k$ is a field with algebraic closure $\overline k$ and absolute Galois group $\Gamma$. Let $X,Y$ be two distinct ...
- 7,498
6
votes
2
answers
505
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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)/\...
- 63
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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(...
- 7,582
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1
answer
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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 ...
- 10.3k
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1
answer
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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 ...
- 4,647
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4
answers
780
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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 ...
4
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1
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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 ...
- 5,442
4
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1
answer
141
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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 ...
4
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0
answers
142
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Uniqueness of Galois descent
Let $Y,Z$ be $\mathbb{F}_p$-schemes, they are both models of scheme $X$ over $\overline{\mathbb{F}_p}$ . Let $F$ be the absolute Frobenius of $\overline{\mathbb{F}_p}$, If $1_Y\times F$ and $1_Z\times ...
- 477
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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):
...
- 7,498
3
votes
2
answers
246
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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
$\exists$ $...
- 101
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0
answers
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How the Galois group acts on a Néron–Severi group of a variety?
Let $K/k$ be a Galois extension with Galois group $\Gamma$ and let $X$ be a variety over $k$. Assume that either $X(k)\neq\varnothing$ or $\mathrm{Br}(k)=0$, the Brauer group of $k$. By the Hochschild-...
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Galois descent of a Hopf algebra
In a question here on why the classification of commutative Hopf algebras requires the field to be algebraically closed, a brilliant answer is given discussing the notion of Galois descent.
As I ...
- 201
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1
answer
168
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A conceptual explanation for a simple fact about twists of objects with abelian automorphism group
Suppose $X,X'$ are two objects (say, genus 1 curves) over a field $k$ such that over the algebraic closure $X_{\overline k} \cong X'_{\overline k}$ and moreover, $Aut_{\overline k}(X)$ is abelian.
...
- 7,498
2
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1
answer
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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 ...
- 5,716
2
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1
answer
743
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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 ...
- 2,185
2
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0
answers
127
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Homotopy fixed points vs coalgebras
Referring to the last part of this answer https://mathoverflow.net/a/225403/170683, I would like to understand how in the case of a Galois cover $f\colon X\to Y=X/G$ with Galois group $G$ (I guess ...
- 321
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Characterization of effective descent morphism
A faithfully flat morphism of commutative rings $A \rightarrow B$ is an effective descent morphism. So is a regular monomorphism (right?). What is a characterization of effective descent morphisms?
...
- 4,647
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0
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172
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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 $k$...
- 2,106
2
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0
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467
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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 ...
- 10.3k
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1
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Fpqc-locally constant if and only if étale-locally constant?
Also in SE.
Let $\mathcal{F}$ be sheave over $S_\mathrm{fpqc}$. We say $\mathcal{F}$ is a fpqc-locally constant sheaf (of finitely generated abelian groups) if there exists a fpqc covering $(S_i\to S)...
- 311
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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 ...
- 271
1
vote
1
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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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Can we check smoothness of a morphism after base change to the algebraic closure?
I know that smoothness is fppf local on the base, but this is not enough because taking algebraic closures is not finitely presented. The reason I'm asking this is because I want an easy/quick ...
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Question regarding Galois descent of sections of a vector bundle
Let $\pi: Y\rightarrow X$ be a finite 'etale Galois morphism between two smooth projective varieties with Galois group $G$. Let $\mathcal{E}$ be a vector bundle on $X$. Then $\pi^*\mathcal{E}$ is a $G$...
- 667
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0
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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= \...
- 5,716
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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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