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Questions tagged [algebraic-stacks]

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16
votes
3answers
795 views

Are there prominent examples of operads in schemes?

There is an abundance of examples of operads in topological spaces, chain complexes, and simplicial sets. However, there are very few (if any) examples of operads in algebraic geometric objects, even ...
4
votes
1answer
231 views

Stack being represented by a scheme/manifold

On page $10$ of the survey article Algebraic stacks, by T. Gomez (arXiv:math/9911199), we have following result If a stack has an object with an automorphism other than the identity, then the ...
2
votes
0answers
108 views

English translation of G.Laumon, L.Moret-Bailly book Champs algébriques

Is there an English translation of G.Laumon, L.Moret-Bailly book Champs algébriques. Most questions on this site on stacks received this book as reference in comments/answers. So, I want to ask if ...
2
votes
0answers
108 views

Is this construction with stacks a blow-up?

Let $X$ be the stack of rank $1$ degree $b$ coherent sheaves $E$ with torsion of length at most 1 on an elliptic curve $C$. Let $Y$ be the stack of pairs $E^{'} \subset E$ such that $E \in X$ and $E/E^...
1
vote
0answers
91 views

Atlas for a stack of sheaves of rank 1 with torsion

I would like to construct an atlas for the stack of sheaves E of rank 1 and degree b on an elliptic curve C such that E has torsion of length at most 1. Am I allowed to fix both the determinant L of ...
2
votes
1answer
157 views

Is the stack of stable curves with no rational component algebraic?

Let $g\geq 2$ be an integer and let $\overline{\mathcal{M}}_g$ be the (smooth proper Deligne-Mumford) algebraic stack of stable curves of genus $g$. Let $\mathcal{M}_g^{nr}$ be the substack of ...
3
votes
0answers
199 views

from functor of points to stacks (or almost)

I want to have a better "working knowledge" of stacks. However every time that I approach the topic, I finish hitting a wall of technical details. They are essential, but I don't need them for now. ...
0
votes
0answers
133 views

Singular cohomology of an Artin algebraic stack

I am trying to understand: how can we make sense of Singular cohomology of an Artin algebraic stack? A good introductory reference will be very helpful.
3
votes
0answers
143 views

A naive question about representations of group stacks

For an algebraic group $G$ over a field $k$, the abelian category $ Rep_k(G)$ of $k$-linear locally finite representations of $G$ can be identified $QCoh(BG_{lis-et})$. Suppose now I replace $G$ with ...
1
vote
0answers
125 views

On presentations of algebraic stacks

Every algebraic stack $X$ admits a presentation as quotient stack $[U/R]$ of a groupoid in algebraic spaces ($U$ is an algebraic space of objects, $R$ is an algebraic space of arrows, and there are ...
7
votes
2answers
389 views

Understand the difference between two stacks

Let us work over $\mathbb{C}$. Let $G$ be a finite group, acting on $\mathbb{A}^1$ via a character, and let $H$ be the kernel of the action. Assume that $\mathbb{A}^1$ is the coarse moduli space of ...
2
votes
0answers
189 views

Tangent and cotangent bundle of a smooth algebraic stack

Are there any good notion of tangent and cotangent bundle (and stacks) of a smooth algebraic stack, similar to the notion of tangent and cotangent bundles (and spaces) of smooth schemes? I am ...
1
vote
1answer
123 views

Properties inherited by the coarse moduli space

Let $\mathcal{X}$ be a regular, seperated Deligne-Mumford stack and $X$ be the coarse moduli scheme associated with $\mathcal{X}$. Then, is $X$ regular? I guess this is a basic fact but am not able to ...
4
votes
0answers
172 views

Quotient of a motive by a finite group

Given a smooth scheme $X$ over a field $k$, we can consider its motive $M(X)$. It is an object in Voevodsky's triangulated category of motives $DM(k,\Lambda)$ where $\Lambda$ is the ring of ...
2
votes
0answers
113 views

Quotient of a finite morphism by an action of a reductive group is still finite?

Let $X, Y$ be two quasi-affine schemes over $\mathbb{C}$. Let $G$ be a reductive algebraic group. Suppose that we are given an action of $G$ on $X,Y$ and a $G$-equivariant finite morphism $X \...
5
votes
0answers
152 views

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 ...
3
votes
0answers
108 views

Skyscraper sheaf on a stack associated to a singular surface

Suppose $X$ is a normal projective surface with a du Val singularity. In this case, we know a crepant resolution $Y$ exists, and results of Kawamata (https://arxiv.org/abs/0804.3150, Corollary 3.5) ...
3
votes
0answers
152 views

What properties does the inertia stack $I_X$ of an algebraic stack $X$ inherit from $X$?

Let $S$ be a noetherian scheme and let $X$ be a "nice" algebraic stack over $S$. For instance, let's say $X$ is a finitely presented algebraic stack over $S$, or that $X$ is a finite type separated DM-...
4
votes
1answer
206 views

moduli stack of double covers of $\mathbb{P}^1$ with one marked point

I am trying to improve my moderate knowledge of moduli spaces/stacks by examining the moduli stack of stable double covers of $\mathbb{P}^1$ with one marked point. My idea is to ignore the stack ...
5
votes
0answers
116 views

Tame Moduli Space of a stack without finite inertia

Let $\mathcal{X}$ be an Artin stack over $S$. In the paper by Jarod Alper, Good moduli spaces for Artin stacks, Ann. Inst. Fourier 63 (2013) 2349-2402, a tame moduli space is defined as a morphism $\...
5
votes
0answers
117 views

Presentations of algebraic stacks that are surjective on k points

Let $\mathfrak X$ be an algebraic stack defined over a finite field $k$. Is the following known: there exists a presentation $X \to \mathfrak X$ such that (*) for any finite extension $F/k$, the ...
2
votes
0answers
213 views

Vanishing of space of first order infinitesimal deformations for irreducible algebraic stack

This question has a few bits, and apologies if some questions are phrased poorly since I am not knowledgeable on the language of stacks or deformation theory. Suppose $\mathscr{X}$ is an algebraic ...
3
votes
0answers
242 views

Stacks algebraic over a given stack

Given a base site $S$ of schemes, consider a stack $\mathcal{C}$ on $S$ not assumed to be algebraic. Then in principle, given another stack $\mathcal{E}$ on $S$, together with a map of stacks $\...
1
vote
1answer
140 views

Modifications and dilatations for stacks

In the paper "Algebraization of formal moduli II" Artin proves the existence of "modification" and "dilatation". Are these opertaions valid on algebraic stacks?? I am attaching the particular page in ...
11
votes
1answer
447 views

Is $\mathscr{M}_{1,1,\mathbb{Z}}$ isomorphic to a quotient stack by a finite group?

Let $\mathscr{M}_{1,1,\mathbb{Z}}$ denote the moduli stack of elliptic curves. Does there exist a scheme $X$ and a finite group $G$ acting on $X$ such that $\mathscr{M}_{1,1,\mathbb{Z}}$ is ...
9
votes
1answer
371 views

Universal homeomorphism of stacks and etale sites

A morphism between schemes is a universal homeomorphism if it is integral, surjective, universally injective. For morphism between algebraic stacks, this notion also make sense. It is well know that ...
4
votes
0answers
140 views

Sheaf of rational functions on algebraic stacks

I am trying to understand sheaf of rational functions of an algebraic stack. As given in nLab https://ncatlab.org/nlab/show/sheaf+of+rational+functions, the definition holds in particular for ...
1
vote
0answers
118 views

Dimension of artin stacks

I was reading the article of Laumon 1988: Un analogue global du cone nilpotent (I am sorry but I could not find an available link to share). He fixes a curve $X$ (over $\mathbb{C}$) and considers ...
1
vote
0answers
92 views

Localization of a 2-category

I am looking for a basic reference about localization of 2-categories, possibly avoiding the full formalism of n-categories.
4
votes
1answer
211 views

Residual gerbe and field of moduli

I am studying residual gerbes from Laumon Moret-Bailly and I would like to know if the residue field of the residue gerbe has the following property. I am a beginner in this subject so I find ...
3
votes
0answers
246 views

Linear $\infty$-categories $\mathrm{QC(X)}$ and $\mathrm{Perf(X)}$ of a “derived” stack $\mathrm{X}$

For each scheme or algebraic stack their $\infty$-category of quasicoherent sheaves (resp., perfect complexes) on it is $k$-linear for a commutative ring $k$. That is (by a recent result of L.Cohn), ...
3
votes
0answers
166 views

Generic points of algebraic stacks

I am aware that this is not a esearch question, but I don't know where else to ask. I have come across the fact that the stack of bundles of rank r and degree d over a curve of genus g with a ...
1
vote
0answers
92 views

Open/closed immersion and quotient stacks

I'm quite new to stacks, so this might be very easy. In particular, if there is a canonical reference I can consult for these things, please feel free to point it out. Let $f:X\to Y$ be a $G$-...
3
votes
0answers
218 views

Understanding sheaves on lisse-etale site of an algebraic stack

I was trying to understand the lisse-etale toplogy on Artin stacks when I came across this problem. First, let $\mathcal{X}$ be an algebraic stack with say the small smooth site structure on it. And ...
2
votes
0answers
99 views

Noetherian property in the small fppf site of an algebraic stack

Let $X \longrightarrow Y$ be a flat and finitely presented morphism where $X$ is a scheme and $Y$ is a Noetherian (or locally Noetherian) algebraic stack, then will this imply $X$ is also Noetherian (...
8
votes
1answer
506 views

Derived noncommutative geometry includes derived, or spectral algebraic geometry?

Let $k$ be a commutative ring. In derived noncommutative (algebraic) geometry a "noncommutative space over $k$" is a $k$-linear $\mathrm{DG}$-category. This is motivated by the fact that homological ...
13
votes
2answers
676 views

Quiver representations and coherent sheaves

I've heard that under certain assumptions on an algebraic variety $X$ there exist a quiver $Q$ for which there is an equivalence $$D^b(\mathsf{Coh}(X))\simeq D^b(\mathsf{Rep}(Q))$$ between the ...
3
votes
1answer
416 views

Reference for etale cohomology on stacks

Is there good reference for general theory.of etale cohomology on stacks and more advanced topics? Thanks
4
votes
0answers
140 views

Topological invariance of periodic cyclic homology of stacks

Goodwillie proved (in Cyclic homology, derivations, and the free loopspace) that the periodic cyclic homology of a connective dg algebra is that of its reduced classical ring. Preygel proved (in Ind-...
5
votes
0answers
161 views

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 ...
2
votes
1answer
198 views

Does every tame Deligne-Mumford stack over a perfect field have a dense substack which is smooth?

Does every tame Deligne-Mumford stack over a perfect field have a open substack which is smooth ? If we assume the stack is reduced ?
10
votes
0answers
246 views

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 $\...
3
votes
0answers
128 views

“standard limit arguments” involved in showing that roughly every DM stack is locally a quotient stack

I am trying to understand proposition 3.6 of this paper (perhaps I am in over my head): https://arxiv.org/pdf/math/0703310.pdf If we denote the stack $\mathcal{M}$ and its coarse moduli space as $M$ ...
4
votes
1answer
185 views

Zero locus of a family of morphisms of vector bundles

Let $X$ be a variety and $\varphi : F_1 \to F_2$ be a morphism of vector bundles over $X$. Then it is easy to check that the locus on $X$ for which $\varphi$ vanishes is a closed subscheme of $X$. ...
2
votes
0answers
173 views

Is there a formal local criterion of finiteness?

Vague question: Is there a criterion to deduce that a morphism between algebraic stacks is finite based on the local deformation functors? For sure this is not enough, so let me be more specific. ...
5
votes
0answers
235 views

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 ...
1
vote
1answer
263 views

Elementary question: Sheaf on quotient is locally free

I'm sorry for the following elementary question. Things are algebraic/holomorphic over $\mathbb{C}$. I'm reading the book "Vector bundles on Complex Projective Spaces" by Okonek et al. and the ...
3
votes
0answers
177 views

Exterior tensor of derived categories of coherent sheaves

Let $X, Y$ be Noetherian perfect derived stacks over $S$ a regular perfect derived stack. Consider the exterior tensor functor $$\text{DCoh}(X) \otimes_{\text{DCoh}(S)} \text{DCoh}(Y) \rightarrow \...
1
vote
0answers
225 views

How to think about the quotient field of an integral stack?

This is the definition given in Vistoli's paper. Let $F$ be an integral stack. A rational function of $F$ is a morphism $G \rightarrow A^1_S$ defined on a nonempty open substack $G$ of $F$. ...
8
votes
0answers
404 views

Orbits of groups acting on algebraic stacks. What are they?

Let $X$ be a smooth finite type integral algebraic stack over $\mathbb C$ with affine diagonal and trivial generic stabilizer. Let $G$ be a smooth affine connected group scheme over $\mathbb C$ acting ...