All Questions
Tagged with descent galois-descent
15 questions
3
votes
0
answers
122
views
Weil restriction of a bunch of points or more general disjoint unions
$\DeclareMathOperator\Spec{Spec}$For a finite extension of fields $k'/k$, let $R_{k'/k}$ denote the Weil restriction functor from quasiprojective $k'$-schemes to quasiprojective $k$-schemes, defined ...
- 472
1
vote
0
answers
182
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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 $G$...
- 737
2
votes
1
answer
616
views
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)...
- 452
3
votes
0
answers
340
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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?
...
user30211
5
votes
1
answer
439
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 ...
user30211
7
votes
3
answers
674
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 ...
- 14.1k
8
votes
0
answers
1k
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 $Y=X\times _K L$ ...
- 953
2
votes
0
answers
527
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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 ...
- 10.5k
6
votes
1
answer
1k
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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 ...
- 10.5k
10
votes
1
answer
810
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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 ...
- 247
30
votes
2
answers
3k
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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 ...
- 10.5k
10
votes
1
answer
1k
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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 ...
- 2,663
1
vote
1
answer
606
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 ...
- 601
7
votes
1
answer
526
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 ...
- 10.4k
1
vote
1
answer
441
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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 ...
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