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Results tagged with algebraic-groups
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user 4008
Algebraic varieties with group operations given by morphisms, or group objects in the category of algebraic varieties, the category of algebraic schemes, or closely related categories.
6
votes
Quotient of a reductive group by a non-smooth central finite subgroup
The quotient by any finite flat subgroup scheme always exists (see for example SGA 3:Exp V, Thm 7.1). In
your case the subgroup scheme is of multiplicative type, the dual of a finite abelian group $A …
- 22.6k
3
votes
Accepted
Is a double centralizer type theorem ( encountered in semisimple algebras) true for algebra...
When $G=\mathrm{GL}_n$, then the centraliser $C$ of a subgroup scheme $H$ form the invertible elements of the algebra $M$ of matrices commuting with $H$. The group of such elements is Zariski dense in …
- 22.6k
3
votes
Proof of Steinberg's tensor product theorem
[This was intended to be comment to Ben's reply but I exceeded the allowable limit for comments.]
Actually it doesn't work over any ring. Just take any ring $R$ for which
$GL(V)(R) \to PGL(V)(R)$ is …
- 22.6k
16
votes
Accepted
Is every homogeneous G-variety of the form G/H?
It depends on what you mean by "closed subgroup". If you mean a Zariski closed subset which forms a subgroup then the answer is no. If you mean a closed subgroup scheme, then the answer is yes. An exa …
- 22.6k
10
votes
Accepted
Comparing algebraic group orbits over big and small algebraically closed fields
I think this will work. There are a finite number of orbits of the action of $G$
on $X$ precisely when there is an open orbit and a finite number of orbits on
the complement of the orbit. Hence, it is …
- 22.6k
18
votes
Accepted
Suzuki and Ree groups, from the algebraic group standpoint
It is not really a question of inner forms. What happens is that the
algebraic group $G_2$ has an extra endomorphism $\varphi$ whose square
is the Frobenius map (over the appropriate finite field). Ju …
- 22.6k
2
votes
Simplicity of (complex) orthogonal groups
A reference with a small amount of patching to do is Bourbakie: Groupes et algèbres de Lie, Chap IV, 2.7, it uses the theory of BN-pairs (which there are called Tits systems). It is shows that the onl …
- 22.6k
4
votes
Accepted
Principal bundles over groups
There is the Grothendieck theory of bitorsors (see e.g. SGA 7, Exp VII) which gives an abstract answer to this question. The key point is that when you have an actual group extension you do not just h …
- 22.6k
14
votes
Accepted
Central extensions of group schemes
If we have a central extension of group schemes $1\rightarrow B \rightarrow C\rightarrow
A\rightarrow1$ with $A$ abelian, then we get a commutator mapping
$\Lambda^2A\rightarrow B$ (of sheaves as $\La …
- 22.6k
6
votes
Accepted
Connected extensions of finite by connected algebraic groups
In Groupes algébriques et corps de classes Serre classifies the $2$-dimensional commutative unipotent connected algebraic groups $G$ (VII:11). With the exception of the product of the additive group w …
- 22.6k
22
votes
What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a no...
After some thought my pessimism (as expressed in my concurrence with the answer
of Milne) has abated somewhat. If I were bold enough I would conjecture the
following (assuming that the characteristic …
- 22.6k