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Jul
25
comment Relationship between étale and topological $K(\pi,1)$s
Question 2 is reminiscent of the Malcev-Grothendieck theorem : if the étale fundamental group of a smooth complex projective variety is trivial, then there is no non-trivial bundle with a flat connection. See the introduction of arxiv.org/abs/1112.4603 .
Jul
13
answered Is there a “free abelian group of rank 1” in the category of affine group schemes?
Jun
6
comment Kähler differentials, intuition behind $\text{div}(\omega)$, canonical divisor on algebraic curves?
Well reading Hartshorne's chapter on curves should be more than enough. You will learn more doing this job by yourself than taking the easy option of asking the question here, where it is not really appropriate.
Jun
6
comment Intuition for thinking about $R$-module of Kähler differentials, universal receptacles, derivations?
not a research level question in my opinion.
Jun
6
comment Kähler differentials, intuition behind $\text{div}(\omega)$, canonical divisor on algebraic curves?
not a research-level question in my opinion.
May
27
comment lifting morphism of torsors
Don't you mean $H^1(X,\mathfrak{g}_P\otimes I)$ instead of $H^1(X,\mathfrak{g}_P)$ ?
May
25
comment Do line bundles descend to coarse moduli spaces of Artin stacks with finite inertia?
@David Rydh: thanks for the explanation.
May
25
comment Differentials for algebraic stacks
The twist in $\mathfrak{g}^\vee[1]$ seems wrong to me, I think $\mathfrak{g}^\vee[-1]$ is correct.
May
12
comment Do line bundles descend to coarse moduli spaces of Artin stacks with finite inertia?
Thanks for the answer ! In proof of Lemma 2 : "Since $Z\to X$ is finite, we can trivialize $\mathcal{L}$ étale-locally on $X$". I don't get it - could you elaborate ? Sorry if I am missing something obvious.
May
5
comment Do line bundles descend to coarse moduli spaces of Artin stacks with finite inertia?
@Zsolt Patakfalvi : you are right that Alper's criterion is "the stabilizers act trivially on the fibers" and not stalks. And I was indeed assuming that the moduli space is good.
May
5
comment Do line bundles descend to coarse moduli spaces of Artin stacks with finite inertia?
You can probably answer positively (for S quasi compact) by using Alper's criterion : namely a vector bundle descends to the moduli space iff the stabilizers act trivially on the stalks. See his paper "Good moduli spaces for Artin stacks" Theorem 10.3 .
Apr
25
comment Descending a monomorphism of stacks
You are simply confused, because there is no such new $Q$. That is my point D.
Apr
25
answered Descending a monomorphism of stacks
Apr
21
comment Relative Proj and generation of sections
Your question is hardly understandable to me. What is $Y$ ? Your morphism seems independent from $k$ so what means the condition on $k$ ? What means $\Gamma_*$ ? What is the reference Stacks (Ex. 21.2) ?
Apr
18
answered Exact sequences of groups and Tannakian formalism
Mar
30
comment Representability of morphism of stacks
@user74230 : "see this spaces as sheaves" means to a space associate its functor of points ; this is sufficient since this Yoneda-like operation is fully faithful. Are you sure of "that is very different from stalks as in the SP reference" ?
Mar
29
comment Representability of morphism of stacks
Yes, it is enough to verify the property on geometric points : you need to check that the kernel say $K/U$ a is trivial, that is that the unit section $U\to K$ is an isomorphism. You can see these spaces as sheaves, and then use the fact that the geometric points form "a conservative family of points" see stacks.math.columbia.edu/tag/00YJ .
Mar
29
answered Representability of morphism of stacks
Mar
26
comment Do algebraic stacks satisfy fpqc descent?
@Denis as I already mentioned in the first comment "algebraic stacks with quasi-affine diagonal are fpqc sheaves" is misleading since this can be understood as : a sheaf of sets. That is what most algebraic geometers will understand when you claim a stack is a sheaf. If the answer uses a different terminology than the question, there should be a warning.
Mar
26
comment Do algebraic stacks satisfy fpqc descent?
@Denis the question was formulated with a clear reference to the stacks project and to algebraic stacks, so I don't understand the point of your remark.