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Nov
20 |
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Definition of étale (etc) for non-representable morphisms of algebraic stacks?
@Qfwfq your root stack is a gerbe, banded by $\mathbf \mu_r$. For gerbes there seem to be specific conventions (the influence of Tannaka theory ?) for instance the gerbe has P if the band has. In this case you probably have to assume that $r$ is invertible. |
Nov
20 |
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Definition of étale (etc) for non-representable morphisms of algebraic stacks?
@Ariyan Javanpeykar : for finite morphisms, I doubt it. For instance they are defined as quasi-finite and proper in the reference text : Vistoli, Angelo Intersection theory on algebraic stacks and on their moduli spaces.ams.org/mathscinet-getitem?mr=1005008 definition 1.8 |
Nov
19 |
answered | How do we get the quotient $Ext^1(N,M)/Hom(N,M)$? |
Jul
25 |
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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 |
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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 |
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Intuition for thinking about $R$-module of Kähler differentials, universal receptacles, derivations?
not a research level question in my opinion. |
Jun
6 |
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Kähler differentials, intuition behind $\text{div}(\omega)$, canonical divisor on algebraic curves?
not a research-level question in my opinion. |
May
27 |
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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 |
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Do line bundles descend to coarse moduli spaces of Artin stacks with finite inertia?
@David Rydh: thanks for the explanation. |
May
25 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 . |