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19h
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 ?
Jan
19
comment Stack of curves and universal deformations
Deformation theory from the point of view of fibered categories Mattia Talpo, Angelo Vistoli arxiv.org/abs/1006.0497
Dec
19
awarded  Yearling
Nov
15
comment Etale fundamental group of a curve in characteristic $p$
@jacob: for an affine curve, we just know if a given finite group $G$ arises as a quotient of $\pi_1$ or not and nothing more. In particular the cardinality of the set of $G$-covers is not known for general $G$. And this data would be probably insufficient to recover the $\pi_1$ anyway. To sum up: no the $\pi_1$ is not known for any curve (complete or not) over any field of positive characteristic except for the trivial cases of $\mathbb P^1$ and elliptic curves.
Nov
13
comment Etale fundamental group of a curve in characteristic $p$
@jacob I am not sure I understand your third sentence. About the last sentence : the $\pi_1$ of a proper curve is invariant by algebraically closed field extension. So fixing a specific base field does not change the question for proper curves. And the answer is no, the $\pi_1$ is not known. By specialization theory we just know it is topologically of finite type. For affine curves the situation is even worse since the $\pi_1$ is not topologically of finite type.
Nov
13
comment Etale fundamental group of a curve in characteristic $p$
@Joël I agree "completely wrong" was a bit strong :) but I disagree with your interpretation of the question. Clearly the question is : "is the $\pi_1$ known up to isomorphism ?". An answer to your weak version wouldn't answer the following question : "given a fixed finite group $G$, what is the cardinality of the set of $G$-covers" ?
Nov
13
comment Etale fundamental group of a curve in characteristic $p$
@jacob I think I remember you can find an example in Fried-Jarden Field Arithmetic.
Nov
13
comment Etale fundamental group of a curve in characteristic $p$
@Joël : what you wrote is completely wrong. Abyankhar's conjecture describes the set of finite quotients of the $\pi_1$, but since in the affine case the $\pi_1$ is not topologically of finite type, this is not enough information to get back the $\pi_1$. For instance, the $\pi_1$ of $\mathbb A^1$ is not known in positive characteristic, even if one knows that its finite quotients are the quasi-$p$ groups. In fact to my knowledge there is no example of an hyperbolic curve whose $\pi_1$ is "known" in positive characteristic.
Nov
2
comment Finite etale atlas for Deligne-Mumford stacks
About your first comment: it seems better to me to give the simplest possible example than a class including pointless complications such as non-generic stabilizers. About your second comment : I was careful about what you mention. Your quotient stack is interesting but is definitely not a stack of roots.
Oct
31
answered Finite etale atlas for Deligne-Mumford stacks