Questions tagged [algebraic-stacks]
Use this tag for questions related to algebraic stacks, which are a stacks in groupoids X over the etale site such that the diagonal map of X is representable, and there exists a smooth surjection from a scheme to X.
159 questions
1
vote
0
answers
22
views
When do coequalizers of algebraic stacks exist?
This question might need some work to actually get a "good" answer.
Here's the background motivation: the $2$-category of algebraic stacks has fibre products and products and so has ...
- 427
0
votes
0
answers
38
views
Automorphism of root gerbe/root stack
Consider a smooth DM stacks (or even a smooth scheme) $X$. Let $L$ be a line bundle and let $s$ be a section. We can consider either the root gerbe or the root stack by considering the fibre product ...
- 2,750
0
votes
0
answers
48
views
A question on relative spectrum $\mathrm{Spec}_X(\mathcal{A})$
Let $X$ be an algebraic stack over a scheme $S$, and let $\mathcal{A}$ denoted a quasi - coherent sheaf of algebra on $X$. In the definition of stack $\mathrm{Spec}_X(\mathcal{A})$, a morphism between ...
- 121
4
votes
0
answers
70
views
A question about abelian cone $\mathcal{S}pec_X(Sym(\mathcal{F}))$
This is a question when I read this paper, page 7.
$X$ is a Deligne Mumford stack, $\mathcal{F}$ is a quasi coherent $\mathcal{O}_X$-module. Consider the stack $X':=\mathcal{S}pec_X(Sym(\mathcal{F}))$....
- 121
1
vote
0
answers
99
views
What is relative spectrum $\mathcal{S}pec_X(\mathcal{A})$ when $X$ a DM stack?
Let $X$ be a DM stack, $\mathcal{A}$ be a quasi coherent $\mathcal{O}_X$-algebra. I see the a definition of relative spectrum $\pi :\mathcal{S}pec_X(\mathcal{A})\rightarrow X$, which is a stack whose ...
- 121
2
votes
1
answer
123
views
What is the stack $[C/E]$ of a stacky quotient?
$X$ is a Deligne Mumford stack, $E$ is a vector bundle over $X$ and $C$ is a $E$ cone (This definition comes from Behrend's paper The intrinsic normal cone). What does the stack $[C/E]$ mean? Is it a ...
- 121
1
vote
0
answers
56
views
Fibers of morphisms of classifying stacks of group schemes
Given an exact sequence of topological / simplicial groups:
\begin{align*}
1\longrightarrow K\longrightarrow G\longrightarrow H\longrightarrow1
\end{align*}
we have a fibration of classifying spaces:
\...
4
votes
1
answer
184
views
The definition of ringed topoi
I have a question on the notation for ringed topoi given in the Stacks project, where
a ringed topos is defined as a pair $(\mathrm{Sh}(\mathcal{C}),\mathcal{O})$ where $\mathcal{C}$ is a site and $\...
- 76
7
votes
1
answer
154
views
Try to write a $\mu_{n}$-gerbe as a quotient stack
Consider the real number field $\mathbb{R}$. Apply the following short exact sequence
$$
0 \to \mu_{2} \to \mathbb{G}_{m} \xrightarrow{(\cdot)^2} \mathbb{G}_{m} \to 0
$$
to $X = \operatorname{Spec} \...
- 1,219
2
votes
0
answers
88
views
Associated bundle via quotient stack
Let $G$ be an affine smooth group scheme over a scheme $S$ and $X$ an $S$-scheme with a $G$-action. Then one can form the quotient stack $[X/G]$ over $S$. Its $T$-valued points (for a $S$-scheme $T$) ...
- 683
0
votes
0
answers
35
views
Finite full exceptional collection
Let $X$ be an algebraic stack which has infinite stabilisers for some points. Can $X$ have a finite full exceptional collection?
My idea is that if we consider a point $x \in X$ with an infinite ...
- 2,750
4
votes
1
answer
114
views
Isom Sheaf as Stacky Fibre Product
Given an algebraic stack $\mathcal{X}$ (definitions from Olsson's "Algebraic Spaces and Stacks") over a category $C$ (e.g. $\operatorname{Sch}/S$) and let $U$ be an object of $C$.
Pick two ...
- 10.1k
2
votes
0
answers
96
views
Why do bundles on a de Rham stack $X_{dR}$ correspond to bundles on $X$ with a connection?
Let $X$ be smooth scheme of finite type over a field $k$. Let $(X\times X)^{\widehat{ \ \ }}$ be the formal scheme obtained by completing $X\times X$ along the diagonal $\Delta: X\to X\times X$. The ...
- 315
5
votes
0
answers
135
views
Is quotient stack $[\mathrm{Spec}(k^{\mathrm{sep}})/G_k]$ representable by $\mathrm{Spec}(k)$ in general?
Let $k$ be a field and $k^{\mathrm{sep}}$ one of its separable closures. Let $G_k$ be the absolute Galois group of $k$.
Now by abuse of notation, we define a group scheme on $\mathbf{\mathrm{Sch}}_k$ ...
- 650
0
votes
0
answers
72
views
Morphism from a scheme to algebraic space is determined by points
Suppose $X$ is a smooth algebraic space of finite type over an algebraically closed field $k$ and $U$ is a smooth $k$-scheme of finite type. Let $\phi, \psi:U \to X$ be two morphisms such that the ...
- 335