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14h
comment lifting morphism of torsors
Don't you mean $H^1(X,\mathfrak{g}_P\otimes I)$ instead of $H^1(X,\mathfrak{g}_P)$ ?
2d
comment Do line bundles descend to coarse moduli spaces of Artin stacks with finite inertia?
@David Rydh: thanks for the explanation.
2d
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.
Mar
25
comment Do algebraic stacks satisfy fpqc descent?
This is not the classical definition, for instance not the definition in the stacks project. What you call sheaf is simply misleading with the current terminology.
Mar
25
comment Do algebraic stacks satisfy fpqc descent?
"analogous statement" has a least two meanings 1 algebraic (Artin) stacks are sheaves in the fpqc topology and better 2 algebraic (Artin) stacks are stacks in the fpqc topology could you clarify ?
Mar
25
comment Do algebraic stacks satisfy fpqc descent?
Both the question and your answer are ambiguous. Usually by sheaf one means sheaf of sets, whereas here you obviously mean fpqc "sheaf of groupoids", usually called fpqc stack.
Mar
2
comment Deformation with fixed ramification
for curves, if you read french, you may find something relevant there : arxiv.org/abs/math/0701680 p.47 §5.2.3 Déformations, versus déformations du diviseur de branchement.
Jan
27
comment line bundle descends?
The statement "By a result of Kempf a line bundle $L$ on $\mathbb{P}^3$ descends to the quotient if and only if the stabilizer $S_x$ of each point $x\in \mathbb{P}^3$ acts trivially on the fiber $L_x$." is not very precise. You probably mean : an equivariant line bundle, else the action on the fiber is not even defined. Could you give a reference ?