Questions tagged [stacks]
In mathematics a stack or 2-sheaf is a sheaf that takes values in categories rather than sets.
473
questions
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votes
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answer
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2-completeness of stacks
I am looking for a reference which discusses the 2-categorical properties of the 2-category $St(C,J)$ of stacks and the stackification $\dashv$ inclusion of presheaves 2-adjunction.
My stacks are ...
- 745
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votes
1
answer
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Is there a ring stacky approach to $\ell$-adic or rigid cohomology?
Ever since Simpson's paper [Sim], it was observed that many different cohomology theories arise in the following way: we begin with our space $X$, we associate to it a stack $X_\text{stk}$ (which ...
- 980
2
votes
1
answer
241
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Proving Zariski descent
I want to understand why the functor $\mathscr{D}$
sending an affine scheme to its associated derived
$\infty$-category satisfies Zariski descent. My understanding
is that one has to show that given a ...
- 143
2
votes
0
answers
146
views
Picard group of a DM-stack
I wonder if the followings are true. Let $k$ be a field, $G/k$ is a finite constant group scheme acting on an affine $k$-scheme $\mathrm{Spec}(A)$. Then a line bundle on the quotient DM stack $[\...
- 363
1
vote
0
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139
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Moduli of morphisms between varieties
Let $\mathcal{M}, \mathcal{N}$ be two "well-behaved" (i.e. representable by an algebraic stack parametrising flat, proper, surjective, finitely presented morphisms with geometric fibres ...
- 113
5
votes
0
answers
70
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Stacks v.s. Stratifolds?
Both stacks and stratifolds can model spaces with singularities (say, singular quotient space for example). In my little experience, stacks are widely used in moduli problems and stratifolds are often ...
- 93
6
votes
0
answers
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Local structure of smooth morphisms of stacks
Let $\varphi:X \to Y$ be a smooth morphism of schemes.
There is a well-known local structure theorem: Zariski locally $\varphi$ is given by the composition of an etale morphism and an affine space (...
- 19.4k
3
votes
0
answers
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When a stack quotient coincides with GIT quotient?
Let $G$ be a reductive group over $\mathbb{C}$, and $H=H_r\ltimes H_u$ be a subgroup of $G$. Here, $H_u$ is unipotent and $H_r$ is reductive.
Question: Is it true that when $G/H$ is open in its affine ...
- 31
1
vote
0
answers
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Computing the equivariant Chern character
Suppose I know the Chern character of an object $F \in D^b(X)$, where $X$ is some smooth complex projective variety with a finite group $G = \mathbb{Z}/m$ acting on it. In $D^b([X/G]) \simeq D^b(X)^G$ ...
- 51
2
votes
1
answer
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Maps from $\mathbb A^1/ \mathbb G_m$ to Coherent sheaves
I am reading this paper https://arxiv.org/abs/1608.04797
Let $\Theta$ be the stack $\mathbb A^1/{\mathbb G_m}$. Let $X$ be a smooth projective curve of genus $g$ over a field $k$. Let $Coh_P$ be the ...
1
vote
0
answers
128
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The stack $\operatorname{GL}_2/B$
Let $F$ be the functor from the category of affinoid Tate algebras over $\mathbb{Q}_p$ to the category $\mathrm{Sets}$, which maps an affinoid $\operatorname{Spm} R$ to the set of orbits $\...
- 685
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votes
0
answers
116
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Moduli stack of doubly periodic complexes?
Let $\mathcal{A}$ be an abelian category.
In HAG II Toen and Vessozi built a higher derived stack $X$ whose category of perfect complexes is $\text{Perf}(X)\simeq D^b(\mathcal{A})$. So $X$ is a good ...
- 5,122
7
votes
1
answer
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views
Faithfully flat descent in complex analytic geometry
A common technique for constructing objects (sheaves) and morphisms in algebraic geometry is faithfully flat descent. Roughly speaking this consists on constructing an object or a morphism "...
- 441
5
votes
1
answer
237
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How the automorphism group of an elliptic curve acts at the localization of the stack $\mathcal{M}_{1, 1, k}$ at the corresponding point
I am studying the enlightening article "The Picard Group of $\mathcal{M}_{1, 1, S}$", written by Fulton and Olsson, but I have some problems with a proof.
Setting
Let $\mathcal{M}_{1, 1, k}$ ...
- 53
2
votes
0
answers
191
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Understanding the universal family $\mathcal{M}_{g,1}\to\mathcal{M}_g$
I am trying to understand why the morphism of prestacks $\mathcal{M}_{g,1}\to\mathcal{M}_g$ is the "universal family". Here $\mathcal{M}_g$ is the moduli stack of curves of genus $g$, its ...
- 747
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votes
1
answer
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Completion of the classifying stack $BG$ at a point
With the classifying stack $BG$ I have come across "the formal completion of $BG$ at point", which is denoted $\widehat{BG}$, for instance on page 7 of https://arxiv.org/pdf/1703.08578.pdf, ...
3
votes
0
answers
148
views
Linear deformations of a morphism between stacks
Given smooth algebraic stacks $\mathcal{X}$, $\mathcal{Y}$ what are the linear deformations $\operatorname{Def}^1(f: \mathcal{X} \to \mathcal{Y})$ of a morphism $f:\mathcal{X} \to \mathcal{Y}$?
In ...
1
vote
0
answers
238
views
Deformation theory of stacks and the tangent complex
On a smooth stack X one can construct the two-term tangent complex $T_X \in D(X)$, as in Sam Raskin's notes (https://people.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept22(Dmodstack1).pdf).
I ...
6
votes
1
answer
312
views
Fibers of the coarse moduli space map
Let $\mathcal{X}$ be a Deligne-Mumford stack over a field $k$ which admits a coarse scheme $c : \mathcal{X}\rightarrow X$. This will be the case if $\mathcal{X}$ is separated and locally of finite ...
- 6,253
1
vote
0
answers
96
views
Extending Grothendieck topology from analytic manifolds to spaces?
Let $k\text{-Man}$ denote the Euclidean site of $k$-analytic manifolds where $k=\mathbb{R}, \mathbb{C}$. In words, $k\text{-Man}$ is the usual category of real/complex analytic manifolds considered ...
- 2,840
0
votes
0
answers
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What happens if you apply $\operatorname{QCoh}$ to $(X,\underline{\mathbb{C}}_X)$?
$\DeclareMathOperator\QCoh{QCoh}
\newcommand\numC{\mathbb{C}}
\newcommand\numQ{\mathbb{Q}}$Let $X$ be a topological space (possibly a scheme). What happens if you apply $\QCoh$ to the prestack $(X,\...
- 295
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votes
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answers
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Moduli stacks of algebraic surfaces—obstructions to existence?
The moduli stack $\mathcal{M}_g$ of genus $g$ curves is one of the deepest objects in mathematics, so of course you wonder to what extent you can construct an (Artin?) stack parametrising algebraic ...
- 5,122
3
votes
1
answer
146
views
Compact groupoid presentations for closed 2-orbifolds (or finite graphs of finite groups)?
A result of Noohi says that if $\mathsf{X}$ and $\mathsf{Y}$ are topological stacks and $\mathsf{Y}$ admits a groupoid presentation $\mathcal{G}$ in which both $\mathcal{G}_0$ and $\mathcal{G}_1$ are ...
- 1,741
3
votes
1
answer
132
views
Studying a connection between Iwasawa theory and the $K(n)$-local Picard group by some geometry on the Lubin-Tate stack
Let $Pic_n^0$ denote the even part of the $K(n)$-local Picard group, and let $Pic_n^*$ denote $Hom(Pic_n^0, W(\mathbb{F}_{p^n})^x)$. Denote by $L$ the profinite group ring $\mathbb{Z}_p[[Pic_n^*]] $. ...
- 428
3
votes
0
answers
165
views
Base change and quotient stacks
Let $X$ be an algebraic space over a scheme $S$, let $G$ be a smooth group scheme over $S$, acting on $X$, and let $T\to S$ be a morphism of schemes. Is it true that there is an isomorphism
$$[X/G]\...
1
vote
0
answers
319
views
In what sense is an orbifold a DM stack?
My advisor mentioned in passing that orbifolds are Deligne-Mumford stacks, and I'd like to know in which sense this is true. The only reference I can find is this article (https://arxiv.org/abs/0806....
- 181
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0
answers
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Is there an inverse image functor for sheaves on stacks?
I'm interested specifically in an inverse image functor between differentiable stacks, ie. stacks coming from Lie groupoids. Specifically, if I have a morphism of Lie groupoids $H\to G$ and I have a ...
- 885
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votes
2
answers
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Quotient of a quotient stack: interesting examples?
Let $X$ be a scheme acted on by an algebraic group $G$. Also, let $H$ be an algebraic group acting on the quotient stack $X/G$, for the definition of "act", see Romagny - Group Actions on ...
- 5,122
4
votes
1
answer
221
views
Sheafification of presheaf of trivial vector bundles is the stack of vector bundles
This is a deliberately vague question, possibly obvious to experts. Let $F$ be a field. Over the (say, fpqc) site of $F$-schemes, we may define a presheaf $T^{\textrm{pre}}$ that takes a scheme $S$ ...
- 141
1
vote
0
answers
97
views
Geometric quotients of DM stacks by group actions
Let $G$ be a finite group acting on a DM stack $X$ and $Y$ the quotient stack. I.e., $Y \to BG$ has fiber $X$. Is there a geometric quotient $X/G$ in some sense? I want automorphism groups of points ...
- 1,004
6
votes
0
answers
356
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Examples of descent in basic algebraic geometry
I'm studying descent theory and I recall that there were multiple instances before where I heard something like "we can prove this as follows, but this is just descent applied to [...]". ...
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4
votes
0
answers
197
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K theoretic pushforward along gerbes
I have a nontrivial gerbe $\pi : \mathscr{G} \to X$ banded by a cyclic group $G = \mathbb{Z}/r$. I'm working over $\mathbb{C}$. I want to describe $\pi_\ast$ and relate the fundamental class $[\...
- 1,004
2
votes
1
answer
130
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Necessary and sufficient conditions for a Lie groupoid to present a stack
Let $\mathcal{G} = G_1 \rightrightarrows G_0$ be a Lie Groupoid (although I am also interested in groupoids internal to other sites), the stack associated to $\mathcal{G}$, which is sometimes denoted $...
1
vote
2
answers
294
views
What is the pull-back of a polarization of abelian schemes over different bases?
The following came up when reading the definition of the moduli stack of principally polarized abelian varieties in [1].
Let $\pi_1:A_1 \to S_1$ and $\pi_2: A_2 \to S_2$ be abelian schemes over $S_i$, ...
- 875
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votes
0
answers
259
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?
...
4
votes
1
answer
145
views
Internal principal $G$-bundles
Let $(C, J)$ be a small site and let $\mathsf{Sh}_{(2, 1)}(C, J)$ be the $(2, 1)$-sheaf topos of sheaves of (small) groupoids on $(C, J)$. Let $G$ be a sheaf of groups on $(C, J)$, and let $\mathbf{...
- 1,393
1
vote
1
answer
309
views
Universal principal bundle on stack
I am studying the notes of Sorger concerning the moduli problem of principal bundles over curve https://inis.iaea.org/collection/NCLCollectionStore/_Public/38/005/38005695.pdf and there is something I ...
- 439
1
vote
0
answers
73
views
Canonical covering stack of a flop
In section 6 of Kawamata's paper https://arxiv.org/abs/math/0205287, he defines the canonical covering stack. In the proof of theorem 6.5, he considered a flop $$X\xrightarrow{\phi}W\xleftarrow{\psi}Y$...
- 253
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votes
0
answers
137
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Torsors for nonabelian groups and maps to contracted products
$\newcommand\op{\mathrm{op}}$My question concerns torsors for a sheaf of groups $G$ that is not commutative, and left/right are messing me up. A left $G$-torsor is equivalent to a right $G^{\op}$-...
- 1,004
1
vote
1
answer
286
views
Image of a projective variety is closed
Let $X$ be a projective variety and $Y$ an Artin stack. Suppose that $f:X\to Y$ is a morphism of Artin stacks. Is $f(X)$ necessarily a closed substack of $Y$?
This seems like it should be true and ...
- 143
3
votes
1
answer
260
views
Integral locus of Hitchin morphism
Let $\Sigma$ be a Riemann surface of genus $g$. To it, we can associated $M_{Dol}$ be the Higgs moduli space of rank $n$ and degree $d$. Fo simplicity let us take $(n,d)=1$. This quasiprojective ...
11
votes
0
answers
127
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Stack completions in realizability toposes
An internal category $\mathbb A$ in an elementary topos $\mathcal E$ is called an (intrinsic) stack if the indexed category that it represents, $(X\in \mathcal E) \mapsto \mathcal{E}(X,\mathbb A) \in \...
- 62.8k
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votes
1
answer
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Irreducible components of an algebraic stack
Let $\mathcal{X}$ be an algebraic stack of finite type over a (separably closed) field $ k$. Let's say that $\mathcal{X}$ has finite dimension $d \in \mathbb{Z}$. Is it still true that the number of ...
7
votes
0
answers
528
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Understanding the higher stack of perfect complexes
One of the most famous examples of higher Artin stacks is the stack of perfect complexes. I recall here the basic stuff:
We fix a function $b: \mathbb{Z} \rightarrow
\mathbb{N}$ which is zero ...
- 757
3
votes
1
answer
246
views
Universal bundles over algebraic stacks
$\DeclareMathOperator\el{ell}$In the topological case, given a group $G$, we can define the classifying space $BG$ for principal $G$-bundles and there is a universal $G$-principal bundle $EG \to BG$ ...
- 457
3
votes
0
answers
142
views
Do stacks with non-algebraic automorphism groups arise in nature?
Do stacks parametrizing objects with non-algebraic automorphism groups happen in nature e.g. in complex analysis or number theory?
For example can the automorphism groups be non-projective complex ...
user178109
3
votes
0
answers
190
views
Interesting stacks with affine space as coarse moduli
I am looking for examples of Deligne-Mumford stacks whose coarse moduli space is $\mathbb{A}^n$ or at least an open subscheme of $\mathbb{A}^n$ whose complement has codimension $2$. (Thus the whole ...
- 11.8k
3
votes
1
answer
219
views
Hom-space between Picard stacks
This is from Deligne, La formule de dualité globale, SGA 4, tome 3, Expose XVIII, and I am confused about how the hom-space between Picard stacks is again a Picard stack.
A quick rewind. For a site $S$...
- 33
3
votes
0
answers
220
views
Sheaf $\operatorname{Isom}(x,y)$ isomorphic to fibered product $U \times_{(x,y),X \times X, \Delta} X$
$\DeclareMathOperator\Mor{Mor}\DeclareMathOperator\Isom{Isom}$Let $S$ be a scheme and $C$ be the category $(Sch/S)$. Let moreover $p:X \to C$ be an algebraic stack over $C$. Consider an arbitrary ...
- 4,994
10
votes
1
answer
392
views
Is ${\bf Set}$ the terminal autological topos
An autological topos is a type of topos defined by Mike Shulman in his paper on stack semantics; specifically, they are toposes satisfying an additional topos theoretic axiom schema expressed in their ...
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