All Questions
18
questions
3
votes
1
answer
144
views
Extending Tannakian "dictionary" to gerbes
The following is Proposition 2.21 in Deligne and Milne's "Tannakian Categories".
Let $f: G \to G'$ be a homomorphism of affine group schemes over a field $k$ and let $\omega^f$ be the ...
- 323
8
votes
0
answers
282
views
Does Borel fixed-point theorem hold for Deligne-Mumford stacks?
Let $X$ be a proper Deligne-Mumford stack over $\mathbb{C}$ with an action by a complex torus $T$. Let $X^T$ denote the fixed locus.
Question: Is the following statement true?
...
5
votes
1
answer
525
views
When quotient stacks (for nonsmooth group) are algebraic and related questions
Let $k$ be a field. Consider a group $k$-scheme $G$ and let $X$ be a $k$-scheme equipped with an action of $G$. Then one can define the quotient stack $[X/G]$. Objects of $[X/G]$ over $k$-scheme $T$ ...
- 532
0
votes
1
answer
346
views
basic question on quotient stacks
Let $X$ be a scheme over $S$, and $G$ be an affine group scheme over $S$ acting on $X$. This Wikipedia article (or also this related MO question) defines a quotient stack $[X/G]$ as a category of ...
5
votes
0
answers
406
views
Limit of quotient stacks
Let $k$ be a field (we can set it to be either perfect or algebraically closed if necessary), let $G$ be a (split) reductive group over $k$. Let $(X_i)$ be a filtered projective system of finite type $...
- 51
3
votes
1
answer
250
views
Reduction of structure group for stacks
Consider an action of a smooth linear algebraic group $G$ on a variety $X$ over an arbitrary field $k$, and the quotient stack $[X/G]$. Let $p$ be a $k$-point of $X$. If the action is transitive (i.e. ...
- 95
6
votes
1
answer
554
views
Commutative group algebraic spaces
Is the category of commutative group algebraic spaces (commutative group objects in algebraic spaces) locally of finite type over a field, an abelian category?
I would benefit from a reference
user119470
1
vote
0
answers
238
views
Open/closed immersion and quotient stacks
I'm quite new to stacks, so this might be very easy. In particular, if there is a canonical reference I can consult for these things, please feel free to point it out.
Let $f:X\to Y$ be a $G$-...
- 2,001
4
votes
1
answer
250
views
Smooth algebraic stacks with precisely two $\mathbb C$-objects
In my quest of "understanding" stacks, I recently tried to figure out the structure of a smooth algebraic stack of finite type $\mathcal X$ over $\mathbb C$ with affine diagonal and precisely one $\...
- 193
6
votes
1
answer
840
views
Are Picard stacks group objects in the category of algebraic stacks
I've been wondering about what a "group algebraic stack" should be, and ran into the notion of a Picard stack.
I'm slightly confused by the terminology here.
Given an algebraic stack $\mathcal X$ ...
- 193
5
votes
1
answer
305
views
The stack of group algebraic spaces
The fibred category $\mathcal A$ of algebraic spaces over a scheme $S$ is a stack (over the category of affine schemes with the etale topology). This is proved in Laumon and Moret-Bailly's book (see (...
2
votes
1
answer
980
views
Picard group of classifying stack
Suppose $S$ is a scheme, and $G$ a smooth $S$-group scheme.
Then there exists an algebraic stack BG called the classifying stack of $G$, defined as the quotient stack $[S/G]$ where $G$ acts trivially ...
- 845
6
votes
1
answer
582
views
Pulling back quasi-coherent sheaves from a quotient stack
In a problem I am trying to solve, the following situation occurs. $X$ is a smooth variety and $G$ is a reductive group acting transitively on $X$. We have the stack $X/G$ and a morphism $\pi : X \to ...
- 3,770
8
votes
1
answer
1k
views
Quasi-coherent sheaves on classifying stacks
Let $G$ be a smooth group scheme over some base $S$. Then we have the $S$-stack $BG$ whose $T$-points are the $G$-torsors on $T$. Under which conditions do we have $\mathsf{Qcoh}(BG) \simeq \mathrm{...
- 62.1k
1
vote
1
answer
238
views
Smooth map to the stack of G-bundles
Let $G$ a semisimple group and $B$ a Borel subgroup.
We denote by $Bun_{G}$ the stack of G-bundles.
Is it true that a certain open subset $Bun_{B,r}$ maps smoothly to $Bun_{G}$?
My question comes ...
- 3,452