All Questions
Tagged with ag.algebraic-geometry algebraic-groups
91
questions
29
votes
2
answers
10k
views
When is fiber dimension upper semi-continuous?
Suppose $f\colon X \to Y $ is a morphism of schemes. We can define a function on the topological space $Y$ by sending $y\in Y$ to the dimension of the fiber of $f$ over $y$.
When is this function ...
- 23.9k
23
votes
1
answer
3k
views
What is the status of the Friedlander-Milnor conjecture today?
For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense:
Conjecture ...
- 231
9
votes
1
answer
1k
views
Nonabelian $H^2$ and Galois descent
I would like to know whether the following metatheorem on nonabelian $H^2$ has been ever stated and/or proved.
Let $k$ be a perfect field and $k^s$ its fixed separable closure.
Let $X^s$ be a variety ...
- 13.8k
31
votes
7
answers
10k
views
Quotients of Schemes by Free Group Actions
I've often seen people in seminars justify the existence of a quotient of a scheme by an algebraic group by remarking that the group action is free. However, I'm pretty sure they are also invoking ...
- 5,488
27
votes
3
answers
3k
views
Why is this not an algebraic space?
This question is related to the question Is an algebraic space group always a scheme? which I've just seen which was posted by Anton. His question is whether an algebraic space which is a group object ...
- 27.3k
20
votes
7
answers
8k
views
Elementary reference for algebraic groups
I'm looking for a reference on algebraic groups which requires only knowledge of basic material on the theory of varieties which you could find in, for example, Basic Algebraic Geometry 1 by ...
- 15.5k
14
votes
1
answer
1k
views
Uniform proof of dimension formula for minimal special nilpotent orbit?
Given a simple Lie algebra over an algebraically closed field of good characteristic such
as $\mathbb{C}$, its subvariety $\mathcal{N}$ of nilpotent elements has dimension $2N$ (where $N$ is the ...
- 52.5k
13
votes
2
answers
2k
views
Is the fixed locus of a group action always a scheme?
Suppose $G$ is an algebraic group with an action $G\times X\to X$ on a scheme. Does the fixed locus (the set of points x∈X fixed by all of $G$) have a scheme structure? You can obviously define the ...
- 23.9k
12
votes
1
answer
868
views
Pointless groups III
This question is a sequel to Pointless groups, to which @DanielLitt produced an elegant and easy-to-understand counter-example, and Pointless groups II, where @R.vanDobbendeBruyn pointed out that my ...
- 12k
12
votes
3
answers
4k
views
Books on reductive groups using scheme theory
Prof. Conrad mentioned in a recent answer that most of the (introductory?) books on reductive groups do not make use of scheme theory. Do any books using scheme theory actually exist? Further, are ...
- 19.5k
11
votes
1
answer
1k
views
Pointless groups
This question now has two sequels, Pointless groups II (to which @R.vanDobbendeBruyn gave a counterexample for an infinite, imperfect field) and Pointless groups III, both using revised wording ...
- 12k
9
votes
1
answer
1k
views
Optimal definition of "paving by affine spaces"?
Cell decompositions have been used in topology for a long time as a tool in computing cohomology, but the notion in algebraic geometry and arithmetic geometry of paving by affine spaces (or "affine ...
- 52.5k
8
votes
1
answer
303
views
Algebraic points of uniformly bounded degree on an algebraic variety
Let $k$ be a perfect field, and let $\bar k$ be a fixed algebraic closure of $k$.
Let $\overline{X}$ be a nonempty smooth algebraic variety over $\bar k$.
Does there exist a natural number $d=d(\...
- 13.8k
7
votes
1
answer
607
views
Pointless groups II
This question is a sequel to Pointless groups, where I asked for a certain kind of counterexample. @DanielLitt produced an elegant and easy-to-understand counterexample, but also suggested a sense in ...
- 12k
7
votes
3
answers
986
views
homomorphism into reductive groups
Let $k$ be an algebraically closed field with char($k$)$= p > 0$.
Let $P$ be a finite $p$-group. For any homomorphism
$\rho : P \rightarrow GL(n,k)$ we know that the image $im(\rho)$ can be
put ...
- 135