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Results tagged with algebraic-groups
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user 52824
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.
7
votes
Accepted
Compactness of adelic quotients for unipotent groups over global fields
You don't mention if $U$ is assumed to be smooth or connected, but it doesn't matter. In general, if $H$ is any affine group scheme of finite type over a global field $K$ and if $H$ does not contain $ …
- 1,285
5
votes
Accepted
The cardinality of first non-abelian Galois cohomology
Since the fppf cohomology group ${\rm{H}}^1(F, \alpha_p) = F/F^p$ is visibly uncountable (where $p = {\rm{char}}(F) > 0$), perhaps you meant to assume $G$ is smooth (and then fppf cohomology coincides …
- 1,285
5
votes
Openness of finite index subgroups of $\mathrm{GL}_n(\prod O_v)$
In view of Laurent's answer, one may ask more generally for a criterion on a connected reductive $K$-group $G$ to ensure that for all compact open subgroups $U$ of $G(\mathbf{A}_K^S)$ (for a finite se …
3
votes
Accepted
on lifting extensions
There is always the uninteresting example $\widetilde{G} = G \times Z$ relative to a choice of splitting of $\widetilde{T}$ as a central extension of $T$ by $Z$ (as may be chosen since you assumed the …
- 1,285
1
vote
Compact elements in $G(K)$ for a reductive group $G$ over a nonarchimedean local field $K$
It should be that $G(K)^0$ is the group of $g \in G(K)$ such that $|\chi(g)| = 1$ for all $K$-rational characters $\chi:G \rightarrow {\rm{GL}}_1$.
Once this is shown, it follows that $G(K)/G(K)^0$ …