Questions tagged [stacks]
In mathematics a stack or 2-sheaf is a sheaf that takes values in categories rather than sets.
513 questions
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Definition of derived category of a stack
In their book, Bernstein an Lunts define the equivariant derived category in several ways. One can be expressed as follows:
Let $X$ be a say complex variety with an action by an algebraic group $G$.
...
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Twisting an object P by an H-Torsor I
I am reading Brylinski's Loop Spaces, Characteristic Classes, and Geometric Quantization.
The Statement
Let $C$ be a gerbe on a space $X$ with "abelian" band $H$, $f: Y \to X$ a local ...
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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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How to specify a finite group up to inner automorphism?
I want some finite set of data to which I can canoically associate a "group up to inner automorphism", and which can be constructed canoically from a "group up to inner automorphism". I have a few ...
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examples of moduli functors for which coarse moduli space does not exists
Well, the title almost says it all. I would like to list as many examples as possible of moduli functors, for which a coarse moduli space does not exist (and maybe explain why). So, examples such as $[...
- 3,779
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universal property of blow up for stacks?
I will use as a reference Hartshorne Prop. II.7.14, the universal property of blow-up. $\tilde{X}$ is the blow up of $X$ along a sheaf of ideals. $Z\to X$ is the morphism that is to be lifted to $\...
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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
$$...
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On the local structure of stacks
1) Is it true that any Deligne-Mumford stack is locally a quotient stack $[X/G]$ with a finite group $G$?
2) Is it true that any Deligne-Mumford stack can be globally presented as a quotient stack $[...
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On the coarse moduli space of a stack
Consider a stack $\mathcal{X}$ over $\mathbb{C}$ as a category fibred in groupoids over the category of schemes. Let $\mathcal{X}^s$ be the $\pi_0$ of this category, i.e. objects of $\mathcal{X}^s$ ...
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Enumerativity of Gromov-Witten invariants of orbifolds
For smooth Deligne-Mumford stacks, there is a well-defined Gromov-Witten theory, see http://arxiv.org/pdf/math/0103156.pdf and http://arxiv.org/pdf/math/0603151.pdf.
Is there some sense, or some ...
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Teaching stacks to differential geometry students
Does anyone have any experience teaching stacks over the category of manifolds to students whose background is, say, a semester-long course on manifolds? Does anyone know of any publicly available ...
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A question on $Isom(p_1^*E,p_2^*E) \rightrightarrows X$
I am learning the moduli stacks of vector bundles and have trouble understanding some definitions. Let $E$ be a rank $n$ vector bundle over the scheme $X$. We denote by $p_i$ the $i$th projection $p_i:...
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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 ...
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How does descent theory imply a sheaf is a scheme?
I've noticed that often authors will comment that "descent theory" shows that some sheaf in the étale topology is actually a scheme. I was wondering what result in descent theory actually implies ...
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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$. (...
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The quotient of a scheme by a proper equivalence relation
Let $X$ be a scheme and $R$ be a proper equivalence relation on $X$.
What can be said about the geometric structure of the quotient $X/R$? Is it representable by a stack, for example?
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337
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family of gerbes over smooth and proper algebraic varieties
Let $X$ be a smooth and proper variety over $\mathbb{C}$. Let $F$ be an $\mathbb{A}^1$ family of $\mathbb{G}_m$ gerbes over $X$. Suppose the fibers over every point away from 0 in $\mathbb{A}^1$ are ...
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stackification commutes with finite limits?
Suppose we work on the Grothendieck site $\mathcal{C}$ of all schemes in the fpqc topology. If it helps it is also fine with me to work only over affine schemes.
Let us denote the category of stacks ...
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Recovering classical Tannaka duality from Lurie's version for geometric stacks
In Lurie's paper Tannaka Duality for Geometric Stacks, it is essentially shown that specifying a morphism of geometric objects
$$ f \colon X \to Y$$
is equivalent to giving a corresponding pullback ...
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Fpqc sheafification and localisation
I am slightly confused about sheafification at the moment.
I first learned sheaves defined as a subcategory of presheaves, then I was told that sheaves are also a localisation of presheaves, then I ...
- 1,273
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Can stabilizer groups in an orbifold have global twisting?
Can stabilizer groups in an orbifold have global twisting?
For example, consider the two groups $\mathbb Z/3\times\mathbb Z$ and $\mathbb Z/3\rtimes\mathbb Z$ (where $\mathbb Z\to\operatorname{Aut}(\...
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Are quotients of stacks flat?
Let $\cal X$ be a DM stack of finite type over a field (if necessary, I will assume that $k=\mathbb{C}$ and $\cal X$ is a scheme, or even a variety) and $G$ be a finite group. Then we have a quotient ...
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Divisorial contraction: when is the image an algebraic space or a stack?
Let $X$ be a smooth projective surface (in the category of varieties, or schemes), and let $C\subset X$ be a curve (a priori not irreducible, but the irreducible case in itself is already interesting)....
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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 ...
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what exactly is the moduli functor for classifying elliptic curves with (full) level N structure?
So, when people say, "the moduli problem of classifying elliptic curves over $\mathbb{C}$ with level $N$ structure", there are usually two associated functors I've seen:
$P_N : \textbf{Ell}\...
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If $X$ is a smooth and proper stack, does it admit a smooth and proper atlas?
Fix a ground scheme $S$ (a field say).
By atlas for an algebraic stack I mean a smooth and surjective morphism $Y \to X$ from a scheme (or algebraic space or affine scheme) $Y$.
If the stack $X$ is ...
- 1,273
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Irreducibility of monodromy of eigenspaces of families of cyclic coverings
In the article "La conjecture de Weil", Deligne proves that for the primitive cohomology of a universal family $f:X \rightarrow S$ for $M_{d,n}$ the moduli stack of hypersurfaces of degree $d$ in $\...
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Chern classes of vector bundles on a stack
Is there literature about chern classes of vector bundles on DM-stacks? I had a look at a lot of different papers about intersection theory on stacks and related stuff and this seems to be known, but ...
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stack quotient question
Hi,
I have the following question:
let $k$ a field with $char(k)= p>0$, which we can assume to be perfect, $W(k)$ the ring of Witt vector, and $a,b$ positive integers.
Consider the ring $R=W(k)[x,...
- 11
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Representability of Hom-sheaves of various moduli spaces
(May be a poor title, happy to update)
Recall that for a stack $\mathcal{X} \to Sch$ on schemes (e.g. fppf site) and a pair of morphisms $x,y\colon U\to \mathcal{X}$ ($U$ a representable stack) there ...
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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 ...
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Cohomology of line bundles on smooth projective toric Deligne-Mumford stacks
Let $\mathcal{X}$ be a smooth projective toric Deligne-Mumford stack with coarse moduli scheme $\pi\colon \mathcal{X}\rightarrow X.$
Let $[D]$ be a Cartier divisor on $\mathcal{X}$ such that $[D']=\...
user17778
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how to construct a $C^\infty$ stack from a holomorphic stack
Given a complex manifold, you can `weaken' its structure to give a smooth manifold. Is there an analogous construction that constructs a stack over the category of smooth manifolds from a stack over ...
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Does every stack with a connection admit an atlas with a connection?
Dear all,
Let $S$ be a scheme in characteristic $0$,
and let $\mathscr{X}/S$ mean a crystal in Artin stacks over $S$ in the sense of this handout, page 4, Definition 0.5, where we replace the scheme $...
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Reference request: an algebraic stack whose closed points have no automorphisms is an algebraic space
The question is stated in the title. I think BCnrd states in a comment here
Is every (Artin/DM) algebraic stack fibered in sets an algebraic space?
that while the answer is not found in Laumon & ...
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Why are sheaves not preserved in this case?
Suppose that $C$ is a Grothendieck site, and $\mathscr{X}$ is a stack over $C$ (which is NOT equivalent to a sheaf). Let $$\pi_{\mathscr{X}}:\int_{C} \mathscr{X}\to C$$ denote the associated fibered ...
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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 ...
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Separation condition for higher Deligne-Mumford stacks
Let $X$ be a stack of $n$-groupoids on the site of affine schemes over a fixed base, with the etale topology. If $n=1$ then for $X$ to be Deligne-Mumford, aside from having an etale atlas from an ...
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what's the coherent sheaf and sheaf cohomology on Deligne-Mumford stack ?
hello, everyone, I am reading Deligne-Mumford's paper, and the section 4 of this paper, they given some properties of Deligne-Mumford stack without proof. I can't understand the coherent sheaf and ...
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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 $\...
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Classification of principal G-bundles over a differentiable stack
According to "Notes on differentiable stacks" by Heinloth,
the classifying stack will also classify $G$-bundles on stacks. (Remark 2.13)
(Here $G$ is a Lie group.) My questions are:
(1) What is ...
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answers
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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 ...
- 1,180
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818
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To what extent does Poincare duality hold on moduli stacks?
Poincare duality gives us, for a smooth orientable $n$-manifold, an isomorphism $H^k(M) \to H_{n-k}(M)$ given by $\gamma \mapsto \gamma \frown [M]$ where $[M]$ is the fundamental class of the manifold,...
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Properties of the Zariski-Riemann topology on the set of valuations
One can classify all valuations on a function field $K$ of transcendence degree $2$ over $\mathbf{C}$. Let's consider the set $S_K$ of all valuations on $K$ endowed with the Zariski-Riemann topology.
...
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Reference for the derived category of $X$, $[X/G]$ and $X/G$
I'm trying to learn about derived categories of algebraic stacks. To be honest, as of now, I don't need anything fancy nor deep. In my setup I have a scheme $X$ (well, a smooth and projective variety ...
- 1,889
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Does Hom(X,Spec R) = Hom(R, O(X)) hold for algebraic stacks?
For an affine scheme $Spec R$ and a scheme $X$ we know that $Hom(X,Spec R) = Hom(R,\Gamma(X,\mathcal{O}_X))$.
Does it still hold when we replace $X$ with an algebraic stack?
My guess is yes as $Hom(-...
- 1,889
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surjective morphism of schemes or epimorphism of sheaves?
I have a technical question coming from reading Toen's master course on stacks.
If we view schemes as locally ringed spaces then there we could define a morphism to be surjective if it the underlying ...
- 1,889
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0
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239
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properness of stack
Hi,
assume we have an algebraic stack $A$ over $Sch(\mathbb{Z})$ which is quasi-compact and with separated diagonal. Assume that I have a stack $B$ which is obtained by rigidifying $A$ by a subgroup ...
- 39
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2
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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 ...
- 3,806
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320
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Derived pullback of quasi-coherent complexes between algebraic stacks
Let $X$ and $Y$ be Artin algebraic stacks, and $f:X\to Y$ a morphism. I am interested in a pullback morphism
$Lf^\ast : D^-_{qcoh}(Y)\to D^-_{qcoh}(X)$.
In his paper Sheaves on Artin stacks, ...
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