All Questions
Tagged with ag.algebraic-geometry algebraic-groups
914
questions
11
votes
4
answers
2k
views
The category of finite locally-free commutative group schemes
I'm trying to understand the properties of the category $\mathcal{FL}/S$ of finite locally-free commutative group schemes over an arbitrary base-scheme $S$. I know it is not in general an abelian ...
- 111
17
votes
2
answers
1k
views
Can Hom_gp(G,H) fail to be representable for affine algebraic groups?
Let $G$ and $H$ be affine algebraic groups over a scheme $S$ of characteristic 0 and let $\textbf{Hom}_{S,gp}(G,H)$ be the functor $T \mapsto \text{Hom}\_{T,gp}(G,H)$
Theorem (SGA 3, expose XXIV, 7....
- 10.4k
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
14
votes
2
answers
956
views
Do orbits and stable loci of group actions have natural scheme structures?
Suppose G is an algebraic group with an action G×X→X on a scheme. Then many of the usual constructions you make when you talk about group actions on sets can be made scheme-theoretically. ...
- 23.9k
8
votes
2
answers
472
views
Division Algebras as Algebraic Groups
If I'm given a division algebra D with Z(D)=F, then how can I view Dx as an algebraic group defined over F? I'd like to see first how Dx can be given the structure of a variety defined over F, and ...
- 2,789
16
votes
3
answers
5k
views
What is an Oper?
Given a curve C, and a reductive group G, there is a moduli stack Loc_G(C), the stack of G-local systems. I keep reading that there's a substack of "opers" but am having trouble locating a definition....
- 15.9k
18
votes
7
answers
6k
views
Langlands Dual Groups
Can someone explain, explicitly, how to, given a reductive complex algebraic group construct the Langlands dual group? I know it is a group with the cocharacters of G as its characters, but how does ...
- 15.9k
14
votes
6
answers
2k
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Does every morphism BG-->BH come from a homomorphism G-->H?
Given a homomorphism f:G→H between smooth algebraic groups, we get an induced homomorphism of algebraic stacks Bf:BG→BH, given by sending a G-torsor P over a scheme X to the H-torsor PxGH, ...
- 23.9k
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
6
votes
4
answers
1k
views
When is a map given by a word surjective?
Let $w(x,y)$ be a group word in $x$ and $y$.
Let $x$ and $y$ now vary in $\operatorname{SL}_n(K)$, where $K$ is a field. (Assume, if you wish, that $K$ is an algebraically complete field of ...
- 19.3k
25
votes
4
answers
2k
views
algebraic group G vs. algebraic stack BG
I've gathered that it's "common knowledge" (at least among people who think about such things) that studying a (smooth) algebraic group G, as an algebraic group, is in some sense the same as studying ...
- 23.9k
29
votes
5
answers
4k
views
Deformation theory of representations of an algebraic group
For an algebraic group G and a representation V, I think it's a standard result (but I don't have a reference) that
the obstruction to deforming V as a representation of G is an element of H2(G,V&...
- 23.9k
13
votes
3
answers
1k
views
How to topologize X(R) when R is a topological ring?
Given a topological ring $R$, under what conditions and in what way, can one induce a topology on the $R$-points of a scheme $X$? For example, if $X$ is $P^n$ or $A^n$, one has natural topology on ...
- 671
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