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Questions tagged [stacks]

In mathematics a stack or 2-sheaf is a sheaf that takes values in categories rather than sets.

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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 ...
Nico's user avatar
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19 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 ...
Gabriel's user avatar
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2 votes
1 answer
241 views

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 ...
user141099's user avatar
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 $[\...
user837898's user avatar
1 vote
0 answers
139 views

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 ...
Matthias's user avatar
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5 votes
0 answers
70 views

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 ...
Ruizhi liu's user avatar
6 votes
0 answers
178 views

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 (...
Daniel Loughran's user avatar
3 votes
0 answers
122 views

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 ...
Ruotao Yang's user avatar
1 vote
0 answers
86 views

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 $D^b([X/G]) \simeq D^b(X)^G$ ...
alg_et_geom's user avatar
2 votes
1 answer
206 views

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 ...
angry_math_person's user avatar
1 vote
0 answers
128 views

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 $\...
kindasorta's user avatar
2 votes
0 answers
116 views

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 ...
Pulcinella's user avatar
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7 votes
1 answer
254 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 "...
G. Gallego's user avatar
5 votes
1 answer
237 views

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 $\mathcal{M}_{1, 1, k}$ ...
PIELEO13's user avatar
2 votes
0 answers
191 views

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 ...
IDC's user avatar
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6 votes
1 answer
361 views

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, ...
Robert Hanson's user avatar
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 ...
Robert Hanson's user avatar
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 ...
Robert Hanson's user avatar
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 ...
stupid_question_bot's user avatar
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 ...
Andy Sanders's user avatar
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0 votes
0 answers
180 views

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,\...
Gaussler's user avatar
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11 votes
0 answers
336 views

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 ...
Pulcinella's user avatar
  • 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 ...
Rylee Lyman's user avatar
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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^*]] $. ...
KingVon's user avatar
  • 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]\...
Andrés Ibáñez Núñez's user avatar
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....
EJAS's user avatar
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1 vote
0 answers
55 views

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 ...
Josh Lackman's user avatar
5 votes
2 answers
437 views

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 ...
Pulcinella's user avatar
  • 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 $S$ ...
lagicol123015's user avatar
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 ...
Leo Herr's user avatar
  • 1,004
6 votes
0 answers
356 views

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 [...]". ...
Gabriel's user avatar
  • 980
4 votes
0 answers
197 views

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 $[\...
Leo Herr's user avatar
  • 1,004
2 votes
1 answer
130 views

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 $...
Emilio Minichiello's user avatar
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 $S_i$, ...
red_trumpet's user avatar
8 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? ...
Chi Hong Chow's user avatar
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{...
Dat Minh Ha's user avatar
  • 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 ...
Samantha Smith's user avatar
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$...
Peter Liu's user avatar
  • 253
2 votes
0 answers
137 views

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 $G^{\op}$-...
Leo Herr's user avatar
  • 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 ...
LAGC's user avatar
  • 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 ...
Tommaso Scognamiglio's user avatar
11 votes
0 answers
127 views

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 \...
Mike Shulman's user avatar
  • 62.8k
6 votes
1 answer
401 views

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 ...
Tommaso Scognamiglio's user avatar
7 votes
0 answers
528 views

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 ...
Martin Hurtado's user avatar
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 $EG \to BG$ ...
E. KOW's user avatar
  • 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 ...
user avatar
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 ...
Lennart Meier's user avatar
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 $S$...
Frid Fu's user avatar
  • 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 ...
user267839's user avatar
  • 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 ...
Alec Rhea's user avatar
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