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Results tagged with ag.algebraic-geometry
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user 47722
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
10
votes
Accepted
What is a "split $SO(n)$"?
In fact, to properly understand what is $SO(n)(\mathcal{O})$, you need to know what are split groups over a ring. A summary of this story goes as follows:
Given a ring $R$, and given a finite Dynkin …
- 1,017
1
vote
Accepted
Reference for Hensel's Lemma in Algebraic Geometry
The question has been answered in the comments
5
votes
Is the action of $\textrm{Gal}(\overline{k}/k)$ on $G \times_k \overline{k}$ a group homomor...
Here is the story for a general (group) scheme: let $k$ be a field (no need for perfectness) and let $k_{s}$ be a field extension (in practice, the separable closure of $k$). Let $X$ be a scheme defin …
- 1,017
1
vote
0
answers
45
views
Hypersurface whose "square" level sets intersect all linear subspace of "high" dimension
Let $k$ be an infinite perfect field (e.g. I'm happy to assume that $k$ has characteristic $0$. On the other hand, the algebraically closed case is not interesting for this question). The question is …
- 1,017
7
votes
1
answer
1k
views
Reference for Hensel's Lemma in Algebraic Geometry
The following form of Hensel's Lemma in Algebraic Geometry is well-documented in the literature:
$\textbf{Theorem 1}$: Let $R$ be an Henselian local ring with maximal ideal $\mathfrak{m}$, and let …
- 1,017
12
votes
Accepted
Conjugacy of Borel subgroups over arbitrary fields
Yes. This follows directly from Theorem 20.9 (i) in "Armand Borel, Linear Algebraic Groups, Second enlarged edition, 1991" which goes like this:
Theorem: Let $ G $ be a connected reductive group over …
- 1,017
4
votes
$(X_0,R_0)$ is a root system
EDIT : After the answer of Friedrich Knop, I see that his interpretation is surely the right one (the notation $R_0$ for "roots restricting to $0$" is an indication). What I was describing was in fact …
- 1,017
5
votes
1
answer
363
views
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 …
- 1,017
2
votes
1
answer
215
views
Notation for the automorphisms of a $S$-scheme over automorphisms of $S$
Here is a slightly anecdotical notational question.
Let $S$ be a scheme and let $X$ be a scheme over $S$, with structural morphism $s\colon X\to S$. Is there a good suggestive notation for the group …
- 1,017
2
votes
0
answers
115
views
Converging sequence of base change
Here is a natural question that I hope will be of interest to some.
Let $\mathbf{F}_p(\!(T)\!)$ be the field of formal Laurent series over $\mathbf{F}_p$. An automorphism of $\mathbf{F}_p(\!(T)\!)$ i …
- 1,017
5
votes
$G(k)/H(k)$ as a submanifold of $G/H(k)$
From the number of votes, it seems that the useful comment of Laurent Moret-Bailly has been under appreciated, so I thought it would be useful to explicitly record what the main theorem in the linked …