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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Good introductory references on algebraic stacks?

Are there any good introductory texts on algebraic stacks? I have found some readable half-finsished texts on the net, but the authors always seem to give up before they are finished. I have also ...
Daniel Bergh's user avatar
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52 votes
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D-modules, deRham spaces and microlocalization

Given a variety (or scheme, or stack, or presheaf on the category of rings), some geometers, myself included, like to study D-modules. The usual definition of a D-module is as sheaves of modules over ...
Ben Webster's user avatar
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40 votes
10 answers
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Phenomena of gerbes

What is your favourite example of Gerbes? I would like to know Where do we find Gerbes in "nature"? The examples could vary from String theory to Galois theory. For example my favourite examples of ...
tttbase's user avatar
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37 votes
1 answer
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What about stacks of categories in algebraic geometry?

Stacks qua moduli spaces were introduced to keep track of nontrivial automorphisms of the objects they parameterize. In essence they are groupoids of objects with some form geometric cohesion. The ...
David Roberts's user avatar
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36 votes
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Conjectures in Grothendieck's "Pursuing stacks"

I read on the nLab that in "Pursuing stacks" Grothendieck made several interesting conjectures, some of which have been proved since then. For example, as David Roberts wrote in answer to ...
AAK's user avatar
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31 votes
7 answers
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Categorical construction of the category of schemes?

The answer to the following question is probably well known or the question itself is well known not to have a reasonable answer. In the latter case could you please let me know what the "right" ...
algori's user avatar
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28 votes
2 answers
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Seeing stacks in the Calculus of Functors

Recently I was told (by an algebraic geometer) that when algebraic geometers look at the Calculus of Functors, they think of stacks. When I look at the Calculus of Functors, I see a categorification ...
B. Bischof's user avatar
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28 votes
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morphisms representable by algebraic spaces vs morphisms representable by schemes

So I've been working with moduli stacks in algebraic geometry for a while now, with no formal training in the technicalities of the theory of algebraic stacks (ie, I've read a few articles and I learn ...
stupid_question_bot's user avatar
25 votes
3 answers
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Stacks and sheaves

I'm a bit confused by the double role which sheaves play in the theory of stacks. On the one hand, sheaves on a site are the obvious generalization of a sheaf on a topological space. On the other ...
Andrea Ferretti's user avatar
25 votes
4 answers
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algebraic group G vs. algebraic stack BG

I've gathered that it's "common knowledge" (at least among people who think about such things) that studying a (smooth) algebraic group G, as an algebraic group, is in some sense the same as studying ...
Anton Geraschenko's user avatar
24 votes
2 answers
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Different interpretations of moduli stacks

I'm taking my first steps in the language of stacks, and would like something cleared up. The intuitive idea of moduli spaces is that each point corresponds to an object of what we're trying to ...
Randy Brown's user avatar
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23 votes
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What can we do with a coarse moduli space that we can't do with a DM moduli stack?

A couple weeks ago I attended a talk about the Keel-Mori theorem regarding existence of coarse moduli spaces for Deligne-Mumford stacks with finite inertia. Here are some questions that I have been ...
Kevin H. Lin's user avatar
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22 votes
1 answer
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fpqc covers of stacks

Artin has a theorem (10.1 in Laumon, Moret-Bailly) that if $X$ is a stack which has separated, quasi-compact, representable diagonal and an fppf cover by a scheme, then $X$ is algebraic. Is there a ...
Matt Satriano's user avatar
22 votes
3 answers
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Applications of topological and diferentiable stacks

What are some examples of theorems about topology or differential geometry that have been proven using topological/differentiable stacks, or, some examples of proofs made easier by them? I'm well ...
David Carchedi's user avatar
22 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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21 votes
3 answers
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stacks as Morita equivalence classes

I have often encountered definitions of the kind "stacks are equivalence classes of groupoids under Morita equivalence" in topological or differentiable context, with the notion of Morita equivalence ...
Dima Sustretov's user avatar
20 votes
7 answers
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What are the occurrences of stacks outside algebraic geometry, differential geometry, and general topology?

What are the occurrences of the notion of a stack outside algebraic geometry, differential geometry, and general topology? In most of the references, the introduction of the notion of a stack takes ...
Praphulla Koushik's user avatar
20 votes
2 answers
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Cohomologically trivial stacks

The following theorem of Serre is well-known: A noetherian scheme $X$ is affine if and only if $H^i(X; \mathcal{F}) = 0$ for all quasi-coherent sheaves $\mathcal{F}$ on $X$ and all $i>0$. (...
Lennart Meier's user avatar
19 votes
4 answers
3k views

Moduli stack of principally polarized abelian varieties

I'm looking for an accessible reference for the fact that the moduli stack of principally polarized abelian varieties is in fact an algebraic stack. Faltings/Chai sketch two possible proofs in their ...
ppav's user avatar
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19 votes
3 answers
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Stacks in modern number theory/arithmetic geometry

Stacks, of varying kinds, appear in algebraic geometry whenever we have moduli problems, most famously the stacks of (marked) curves. But these seem to be to be very geometric in motivation, so I was ...
19 votes
2 answers
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Why do gerbes live in H^2?

Line bundles on a scheme $X$ live in $H^1(X,O_X^*)$, where $O_{X}^{*}$ is the sheaf of invertible functions. If $X$ is noetherian separated, then we can think of this $H^1$ to be Čech cohmology w.r.t....
Qfwfq's user avatar
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19 votes
1 answer
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Algebraic spaces as locally ringed spaces

Let $S$ be a scheme (although I am more than happy to have $S=\text{Spec}(k)$ for a field $k$) and $\mathsf{AlgSp}/S$ the category of algebraic spaces over $S$. Does there exist an embedding $\...
Alex Youcis's user avatar
19 votes
1 answer
1k views

How difficult is Morse theory on stacks?

The title is a little tongue-in-cheek, since I have a very particular question, but I don't know how to condense it into a pithy title. If you have suggestions, let me know. Suppose I have a Lie ...
Theo Johnson-Freyd's user avatar
19 votes
2 answers
1k views

Algebraic versus Analytic Brauer Group

Let $X$ be a smooth projective algebraic variety over $\mathbb{C}$. Then I think that someone (Serre?) showed that the Cohomological Etale Brauer Group agrees with the torsion part of the Analytic ...
Oren Ben-Bassat's user avatar
19 votes
0 answers
602 views

Coarse moduli spaces of stacks for which every atlas is a scheme

Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth quasi-projective scheme $P$ by the action of a smooth (finite type separated) reductive ...
Ariyan Javanpeykar's user avatar
18 votes
3 answers
2k views

The quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$

Consider the affine space $\mathbb{A}^n$ (over some base scheme) with the usual $\mathrm{GL}_n$-action. What does the quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$ classify? If $n=1$, then we get $[\...
Martin Brandenburg's user avatar
18 votes
0 answers
601 views

The Grothendieck Ring of Higher Stacks

The Grothendieck ring of varieties is defined to be the free abelian group spanned by isomorphism classes of varieties modulo the cut & paste (or scissor) relations, which say that $[X] = [U] + [Y]...
Jacob Bell's user avatar
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17 votes
2 answers
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Are curves with `fractional points' uniquely determined by their residual gerbes?

One makes precise the vague notion of "curve with a fractional point removed" (see for instance these slides) using stacks -- one should really consider Deligne-Mumford stacks whose coarse spaces are ...
David Zureick-Brown's user avatar
17 votes
2 answers
1k views

Applications of Stacks

I've been aware of stacks since grad school, and I can usually follow in rough lines a discussion about stacks, but I've often wondered what particular (purely!) scheme-theoretic argument or theorem ...
17 votes
1 answer
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In what topology DM stacks are stacks

Background/motivation One of the main reason to introduce (algebraic) stacks is build "fine moduli spaces" for functors which, strictly speaking, are not representable. The yoga is more or less as ...
Andrea Ferretti's user avatar
17 votes
1 answer
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Homotopy theory of topological stacks/orbifolds

Motivation $\newcommand{\T}{\mathscr{T}}$ I have many times found myself saying some variant of the following. Let $\T_g$ be the Teichmüller space of a surface of genus $g$, and $\Gamma_g$ its ...
Dan Petersen's user avatar
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17 votes
2 answers
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The different types of stacks

This question is very naive, but it will help me a lot in getting in to the vast literature about stacks. The question is this: there are many kinds of stacks (algebraic spaces, DM, algebraic stacks, ...
16 votes
6 answers
4k views

Representation of Groupoids

The title is vague, my actuall question is the following: Has the representations of groupoids been systematically studied? Is there any new phenomenon, compare with the representation of groups? (...
Yuhao Huang's user avatar
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16 votes
2 answers
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Is the Torelli map an immersion?

The Torelli map $\tau\colon M_g \to A_g$ sends a curve C to its Jacobian (along with the canonical principal polarization associated to C); see this question for a description which works for families....
David Zureick-Brown's user avatar
16 votes
2 answers
2k views

Understanding the definition of stacks

First of all I should apologies if this question does not count as a research level one. I asked the same question on MathUnderflow and didn't receive any answer. Let me cross post (copy and paste) it ...
Bumblebee's user avatar
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16 votes
3 answers
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Is every algebraic space the quotient of a scheme by a finite group?

In this MO question it is claimed that a catchphrase for "algebraic spaces" could be that they are "the result of looking at the orbit space of the action of a finite group on a scheme". Hence my ...
Qfwfq's user avatar
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16 votes
1 answer
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GAGA for stacks

I am curious about stacky generalizations of the following GAGA theorem: If $X, U$ are complex algebraic varieties of finite type, $X$ is proper and $f:X\to U$ is an analytic map then $f$ is ...
Dmitry Vaintrob's user avatar
15 votes
2 answers
2k views

Which definition of "proper" is better?

It is well known that topology and algebraic geometry assign different meanings to the word "proper". Let us recall the relevant definitions from topology (and we work in the context of topological ...
John Pardon's user avatar
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15 votes
1 answer
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Qcoh(-) algebraic stack?

The $2$-functor $\text{Qcoh} : \text{Sch}^{op} \to \text{Cat}$, which sends a scheme to its category of quasi-coherent modules, is a stack by Descent Theory. Is it actually an algebraic stack? If not, ...
Martin Brandenburg's user avatar
15 votes
2 answers
5k views

Understanding the definition of the quotient stack $[X/G]$

I'm trying to understand the definition of the quotient stack $[X/G]$ as defined in Frank Neumann's Algebraic Stacks and Moduli of Vector Bundles. Explicitly, let $G$ be an affine smooth group $S$-...
Brian Fitzpatrick's user avatar
15 votes
2 answers
2k views

Why is lack of functoriality of the Lisse-Etale topology specific to the Lisse-Etale topology?

I'm trying to follow the explanation given in Olsson's "Sheaves on Artin stacks" for the lack of functoriality for lisse-étale topology: Let $f:Y \to X$ be a morphism of algebraic stacks. The functor $...
ykm's user avatar
  • 702
14 votes
3 answers
3k views

what is $\mathrm{Bun}(G)$?

I don't even know where to begin. There's a discussion of stacks and they talk about $\mathrm{Bun}(G)$. I don't know what it is, or what it's elements are or why it is important. Google and ...
john mangual's user avatar
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14 votes
2 answers
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When does sheaf cohomology commute with arbitrary direct sums?

It is well known and more or less proven in Hartshorne's 'Algebraic Geometry' (p. 209) that for every noetherian scheme $X$ and every collection of abelian sheaves $\mathcal{F}_i$ the canonical map $$...
Lennart Meier's user avatar
14 votes
6 answers
2k views

Does every morphism BG-->BH come from a homomorphism G-->H?

Given a homomorphism f:G→H between smooth algebraic groups, we get an induced homomorphism of algebraic stacks Bf:BG→BH, given by sending a G-torsor P over a scheme X to the H-torsor PxGH, ...
Anton Geraschenko's user avatar
14 votes
1 answer
740 views

Hodge to de Rham spectral sequence for stacks

For some work I'm doing, I need a version of the Hodge to de Rham spectral sequence for stacks. I am not at all an expert on stacks, so please excuse me if I make minor technical mistakes in stating ...
Sarah's user avatar
  • 143
14 votes
2 answers
2k views

Ordinary cohomology of stacks

Let $\mathbf{X}$ be a stack over $Top$ (a lax sheaf of groupoids, or some such thing). If it admits a surjective representable map $F \to \mathbf{X}$ then one can form the iterated fibre product to ...
Oscar Randal-Williams's user avatar
14 votes
1 answer
630 views

Does a degeneration always have a larger-dimensional automorphism group?

Suppose $\newcommand{\X}{\mathcal{X}}\X$ is an algebraic stack over a field $k$, $\xi$ is a $k$-point which has another $k$-point $x$ in its closure ($x$ is an isotrivial degeneration of $\xi$). Must ...
Anton Geraschenko's user avatar
13 votes
2 answers
1k views

Are non-algebraic stacks useful in algebraic geometry?

The title is a bit vague. What I want to know is if there is any geometric application of non-algebraic stacks. I know e.g. the category of coherent sheaves is an example. But I want to ask if people ...
36min's user avatar
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13 votes
4 answers
2k views

Motivation for definition of Quotient stack

I am reading "Some notes on Differentiable stacks" by J. Heinloth. In that paper, the notion of quotient stack is defined as follows. Let $G$ be a Lie group action on a manifold $X$ (left ...
Praphulla Koushik's user avatar
13 votes
2 answers
2k views

Can a singular Deligne-Mumford stack have a smooth coarse space?

Let XX be a Deligne-Mumford stack and let XX \to X be a coarse moduli space. Suppose that X is smooth. Is XX smooth? If not, what is an example? What if XX is of finite type over C (the complex ...
David Zureick-Brown's user avatar

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