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Results tagged with ag.algebraic-geometry
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user 43107
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
5
votes
Accepted
Why is the semisimple quotient of a reductive group with semisimple rank 1 equal to PGL2?
Your question comes down to knowing that if $G$ (your $G/Z(G)$) is a smooth connected affine extension of ${\rm{PGL}}_2$ by a finite group then the quotient map $q:G \rightarrow {\rm{PGL}}_2$ is eithe …
- 1,623
6
votes
Accepted
homotopy exact sequence for the étale fundamental group
Your second question has a negative answer because you've got some variances backwards. When $f:G \rightarrow G'$ is a map of groups, by composition with $f$ one gets a functor from the category of $G …
- 1,623
7
votes
Accepted
rationality question while dealing with an isogeny
The answer is negative (even under all of the given hypotheses); this expresses a standard difficulty in the arithmetic aspects of connected semisimple groups over local function fields in contrast wi …
- 1,623
11
votes
Accepted
Variant of Hilbert 90 for Galois extensions
This is an application of Artin's Lemma to reduce to Hilbert 90. Namely, let $K$ be any field at all, and $G$ any finite subgroup of ${\rm{Aut}}(K)$ (as noted in this comments, such finiteness holds …
- 1,623
9
votes
Accepted
Does regular field extension preserve regularity?
Yes, and it is only necessary to assume $K$ is separable over $k$ (i.e., not necessary to assume in addition that $k$ is algebraically closed in $K$). The idea is to use Serre's regularity criterion …
- 1,623
23
votes
Accepted
Local factors of Hasse-Weil zeta function - what do they have in common?
This is an elaboration on ACL's answer, way too long for a comment, which highlights a technical ingredient (well-known to all experts) that underlies the precise sense in which the $\ell$-adic etale …
- 1,623
13
votes
Accepted
Orbits of group scheme action
Presumably you meant to assume the schemes are finite type over $k$. To work naturally with orbit questions for such schemes one just has to bring in appropriate use of flatness to adapt intuition an …
- 1,623