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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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 ...
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601
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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]...
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Stack of Tannakian categories? Galois descent?
I'm having trouble finding a reference for something that I'm guessing the experts worked out long ago. Let's take a local or global field $F$ for this post, and fix a separable algebraic closure $\...
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Other examples of the algebro-geometric Ran space
First off, sorry if this seems vague.
Let's recall some definition. Let $X$ be a curve over a field $k$ and $G$ an algebraic group, then the space $Ran_G(X)$ as defined by Lurie in his Tamagawa ...
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Stacks in the fpqc topology
This is related to Matt Satriano's earlier question about an analog of Artin's theorem for stacks with an fpqc cover by a scheme.
Suppose one developed the theory of stacks in the fpqc topology and ...
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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 ...
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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 \...
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Visualization and new geometry in higher stacks
I am trying to develop a geometrical intuition for "higher spaces", i.e. both in the sense of higher dimensional spaces (more than three dimensions) and in the sense of abstractions beyond ...
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What is there in the book Cohomologie non abélienne by Jean Giraud
These days I am trying to understand about stacks and gerbes.
Most of the articles that has something to do with gerbes cite this work Cohomologie non abélienne by Jean Giraud.
I do not read the ...
11
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Atiyah-Bott from Beauville-Laszlo
This is a question about the cohomology groups of the stack of vector bundles (with fixed discrete invariants) on an algebraic curve. Explicit formulas for these cohomology groups are known, and they ...
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What is a derived Kähler manifold?
From what I understand, there exists a notion of derived $\mathbb{C}$-analytic space.
Let $T_{an}$ be the pregeometry in the sense of Lurie whose underlying $\infty$-category is the category of open ...
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Segal-Freed-Hopkins-Teleman = Atiyah-Hirzebruch/Leray-Serre?
Freed-Hopkins-Teleman (section 3.7) generalise Segal's (Proposition 5.3) spectral sequence for equivariant K-theory to more general local quotient groupoids (that is, topological groupoids locally ...
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520
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Deformations of some simple quotient stacks.
I am interested in stacks of vector bundles on varieties and how deformations of the variety (including non-commutative ones) reflect themselves in deformations of the stack of vector bundles.
I will ...
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598
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Two questions on algebraic stacks
Question 1: The main reference on algebraic stacks (Laumon and Moret--Bailly) defines a separable algebraic stack as one having universally closed diagonal. For schemes separability is simply defined ...
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What is the signficance of the existence of a moduli stack to a moduli problem?
This is a question in pretty unfamiliar territory for me, so if I have conceptual mistakes please correct me.
Let's say we begin with a naive moduli problem: we want a moduli space (whatever space ...
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Grothendieck purity for Brauer groups of stacks
Let $X$ be a smooth variety over a field $k$ (for the sake of simplicity of characteristic $0$) and $\operatorname{Br}(X) := H^2_{\text{ét}}(X, \mathbb{G}_m)$ its (cohomological) Brauer group (...
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Geometric stacks, groupoids and étendues
If $(C, \tau)$ is a site with pullbacks and $\tau$ subcanonical, it is well known that these things are essentially equivalent:
Groupoids $s,t: U_1 \to U_0$ where $s,t$ are covering for the $\tau$-...
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Does the category of $G$-equivariant sheaves have enough injectives?
The question is related to this one.
Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.
Let $G$ be a topological group which ...
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Do isomorphisms spread out under suitable assumptions?
I'm still trying to wrap my head around the "pathological" algebraic space $\mathbb{A}^1/\mathbb{Z}$; see Questions about the algebraic space $\mathbb{A}^1/\mathbb{Z}$ and Why is this not an algebraic ...
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derived schemes and perfect obstruction theories
In a survey article of Toen's it is claimed that that there is forgetful $\infty$-functor between the $\infty$-category of derived schemes locally of finite presentation over a field $k$ and the $\...
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How does the machinery of left-exact comonads generalize from sheaves to stacks?
Suppose that we have two Grothendieck sites, their associated sheaves $\mathcal{E}=\rm{Sh}(\bf{C},J)$ and $\mathcal{F}=\rm{Sh}(\bf{D},K)$ and a geometric surjection $f:\mathcal{E}\to\mathcal{F}$. This ...
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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?
...
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What are the Newton groupoids from Drinfeld's paper on the Grinberg-Kazhdan theorem?
The paper the Grinberg-Kazhdan formal arc theorem and the Newton groupoids by Drinfeld seems to contain many interesting things which are beyond me. For now, I am trying to get some intuition for the ...
8
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173
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Smooth sub-orbifolds in the language of stacks
In most geometric categories, "monomorphism" is too general to describe useful notions of "embedding". This is the case e.g. for schemes, complex manifolds, and differentiable manifolds.
So "embedding"...
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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 ...
7
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291
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Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?
We are taught that, in general:
A type of objects that has nontrivial automorphisms cannot have a fine moduli space.
The proof generally goes along the lines of:
Take an object $X$ with a non-...
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Is the quotient of a scheme by the free action of an elliptic curve an algebraic space?
Let $X$ be a scheme (I'm happy to assume that $X$ is of finite type, separated, and over $\mathbb{C}$) and let $E$ be an elliptic curve which acts freely on $X$. Does the quotient stack $[X/E]$ have ...
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Groupoid cardinality of DM stack and point counting on coarse moduli spaces
Let $X$ be a finite type DM stack over a finite field $k$ with a coarse moduli space $X_c$. (We only assume $X_c$ is an algebraic space and $X$ might have infinite inertia stack.)
Under which ...
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Derived Functors, Projection Formula and Base Change in the Derived $\infty$-Category
Let $X$ be a smooth stack and $\mathcal O_X$ the ring of smooth functions on $X$, i.e. for any smooth $M \to X$, $\mathcal O_X(M \to X) = C^\infty(M)$.
In HigherAlgebra, the derived category $\...
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Quasi-coherent sheaves on $M_{FG}$ and the exact functor theorem
I'm struggling with these notes, and one of the things I don't really understand is the following. The notes consider the stack $M_{FG}$ of formal groups; this is the stack associated to the prestack ...
7
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Is the category of quasi-coherent sheaves on a concentrated stack locally finitely presentable?
Let's call an Artin stack $X$ concentrated iff it is quasi-compact and quasi-separated (the latter usually being included in the definition of an Artin stack). The category of quasi-coherent sheaves $\...
6
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Non-invertible map between principal bundles only locally trivial in a supercanonical Grothendieck topology?
It is a truth universally acknowledged that a map between principal bundles must be in want of an inverse.
However, the construction of said inverse in the context of a more general site $(S,J)$ ...
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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 (...
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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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Is there a Geometric/Smooth version of Homotopy Hypothesis using the path $\infty$-Groupoid of a Smooth Space?
A version of Homotopy Hypothesis says that the Fundamental $n$-grupoids model Homotopy $n$-types... and if we continue upto $\infty$, then the Fundamental $\infty$- groupoids or Kan Complexes model ...
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closed substack of quotient stack
The question concerns quotient stacks. I am not very comfortable with stacks, so feel free to edit the question if I am saying nonsense. It is also the reason why I may be spelling in too much detail ...
6
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Dense (∞,1)-subsites
So if $C$ is a 1-site and $D$ is a subsite (with the induced coverage), there are some conditions that ensure that the pre-composition and right Kan extension functors yield an equivalence of ...
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How does one define the complete local ring of an algebraic stack at a (geometric) point?
Question
How does one define the complete local ring of an algebraic stack at a (geometric) point?
Including what the right definition might be, this is all I'm asking.
I can do this for schemes ...
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When stacks don't help
Say I have a map between two coarse moduli spaces $f:M\to M'$, which is very clearly described (by performing some operation that works well for families). I want to prove that $f$ is formally ...
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How to realize the descent data of Qcoh as a (pseudo)-limit in Cat?
It is well-know that $Qcoh$ is a fibered category on $Sch$. In more details let $\mathcal{C}$ be the category $(Sch/S)$ of schemes over a fixed base scheme S. For each scheme $U$ we define $Qcoh(U)$ ...
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Is the stack of varieties with a big line bundle algebraic
In Starr's paper https://www.math.stonybrook.edu/~jstarr/papers/moduli4.pdf the folk result that the fibred category of pairs $(X\to S, L)$, where $S$ is an affine scheme, $X\to S$ is flat proper ...
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Nowhere vanishing section of vector bundles over varieties as connectivity of morphism of stacks
The following is, amongst others, a Hartshorne exercise:
Let $V$ be a $k$-variety of dimension $n$ and $\mathcal{E}$ a vector bundle of rank greater than $n$, then, generically, a generating section ...
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Description of pull-back of coherent sheaves under a smooth morphism of Artin stacks
I am new to these formalisms, so pardon me if the question is basic. Let $\mathscr{X}$ be an Artin stack (you can take it to be Deligne-Mumford stack if it helps). By a coherent sheaf on $\mathscr{X}$ ...
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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 ...
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Étale stack on $\text{Spec}(k)$
Let $\mathcal{F}:\text{Ét}(\text{Spec}(k))^{\mathrm{op}}\rightarrow \text{Set}$ be an étale presheaf. The étale sheaf condition for the cover $\text{Spec}(l)\rightarrow \text{Spec}(k)$ is
$$\mathcal{F}...
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What is the motive of $\operatorname{Bun}_G(X)$?
$\DeclareMathOperator\Bun{Bun}$Let $X$ be a scheme over algebraically closed field $k$, $G$ a reductive group and $\Bun_G(X)$ the stack of $G$ bundles on $X$. Write $[\Bun_G(X)]\in K_\text{st}$ for ...
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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 $...
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Compactly supported cohomology of a topological DM stack
Consider a separated topological Deligne-Mumford stack $\mathfrak X$, i.e., a topological stack which is presentable by a proper etale topological groupoid (equivalently, $\mathfrak X$ is locally ...
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"Strict" homotopy theory of topological stacks/orbifolds
If we fix a finite group $G$, there are two different useful homotopy theories on the set of $G$-equivariant topological spaces (which are CW complexes, say). One, the "weak" homotopy theory, is given ...
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Is the analytification of the coarse space equal to the coarse moduli space of the analytification?
If $X$ is a smooth finite type separated DM algebraic stack over $\mathbb C$ with coarse space $X^c$, then do we know whether the analytification of $X^c$ is the coarse space of the analytification of ...