Questions tagged [algebraic-stacks]

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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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28 votes
2 answers
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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
20 votes
7 answers
3k views

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
19 votes
4 answers
1k 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 ...
Patrick Elliott's user avatar
19 votes
2 answers
2k 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 ...
user avatar
19 votes
1 answer
882 views

What is $Aut(Ell)$?

Consider the stack $Ell$ (of groupoids) of elliptic curves. I'm interested in the autoequivalence 2-group of $Ell$, the objects of which consists of transformations $Ell \Rightarrow Ell: Ring \to Gpd$ ...
David Roberts's user avatar
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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
17 votes
4 answers
4k views

Soft question: beginners reference to moduli spaces

What is a geometrically intuitive yet reasonably general first introduction to the theory of Moduli spaces? (Possibly introducing stacks also)? I'm looking for something which really gets the pictures ...
ABIM's user avatar
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17 votes
2 answers
3k views

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, ...
14 votes
2 answers
1k views

Is $\mathcal{D} \bigl( \mathrm{QCoh}(\mathfrak{X}) \bigr)$ compactly generated?

An object $E$ in a triangulated category $\mathcal{T}$ with (small) coproducts is called compact if the functor $\mathrm{Hom}_{\mathcal{T}}(E,-)$ commutes with arbitrary coproducts or, equivalently, ...
Tobias Sitte's user avatar
14 votes
0 answers
320 views

Do connected algebraic stacks have a smooth cover by a connected scheme?

An algebraic stack $X$ has an induced topological space $|X|$ given by equivalence classes of fields mapping to $X$ as outlined in the stacks project. If $|X|$ is connected, does that imply there ...
Leo Herr's user avatar
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13 votes
1 answer
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Several simple questions on the geometry of higher stacks

I'm trying to understand definition/work out some examples. So, there are some simple questions about higher stacks. For the simplicity assume that we are working with higher DM (Deligne-Mumford) ...
Klim Puhov's user avatar
12 votes
1 answer
330 views

Are coarse spaces of 1-dimensional smooth proper Artin stacks smooth?

Let $\mathcal{X}$ be a regular proper 1-dimensional Artin stack with finite diagonal, with coarse space morphism $\mathcal{X} \to X$. Question: Is $X$ regular? Some comments: I'm happy to assume ...
David Zureick-Brown's user avatar
12 votes
1 answer
424 views

Verifying that $\epsilon^!$ is indeed the right adjoint of $\epsilon_*$ in the context of algebraic stacks

The question is about the last paragraph of Remark 12.4.3 in the book on algebraic stacks by Laumon and Moret-Bailly. Let $S$ be a (quasi-separated) scheme and let $\mathscr{X}$ be an algebraic stack ...
O-Ren Ishii's user avatar
12 votes
0 answers
277 views

birational geometry of moduli spaces: why work on the coarse space?

In studying the birational geometry of $\overline{\mathcal{M}}_g$, it seems standard to work with the coarse space $\overline{M}_g$ rather than the smooth stack $\overline{\mathcal{M}}_g$. Why is this?...
Hans Sachs's user avatar
11 votes
1 answer
3k views

What is the intuition behind the inertia orbifold (or stack)?

I am studying orbifolds with view towards Chen-Ruan cohomology. I have been struggling with inertia orbifolds but have no intuition about them at this point. I would appreciate your motivating me by ...
Kim's user avatar
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11 votes
1 answer
749 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 ...
Minseon Shin's user avatar
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11 votes
1 answer
1k views

coarse moduli space and $\pi_0$

I've been reading this really nice paper by Alper http://math.columbia.edu/~jarod/good_moduli_spaces.pdf, and there's a question that doesn't seem to be answered (perhaps it's not relevant). Any ...
Yosemite Sam's user avatar
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11 votes
0 answers
361 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
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10 votes
1 answer
447 views

Residue field of point on an algebraic stack

$\DeclareMathOperator{\Spec}{Spec}$ Let $X$ be an algebraic stack. Is there is a well-defined notion of the residue field of a point $x \in |X|$? Attempts: Recall that a point on a stack is an ...
Daniel Loughran's user avatar
10 votes
1 answer
611 views

Ramification of the map from the stack of elliptic curves to the $j$-line

Let $\mathcal{M}_{1, 1}$ be the stack of elliptic curves. Its coarse moduli space is $\mathbb{A}^1_{\mathbb{Z}}$ with the map $\mathcal{M}_{1, 1} \rightarrow \mathbb{A}^1_{\mathbb{Z}}$ given by the $j$...
O-Ren Ishii's user avatar
9 votes
2 answers
726 views

Does inclusion from n-stacks into (n+1)-stacks preserve the sheaf condition?

I'm going to describe two situations that seem to contradict each other, and I'm interested to know precisely what's wrong with this reasoning. Let $M$ be a manifold, and consider the presheaf $C^*(-,...
David Corwin's user avatar
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9 votes
3 answers
2k views

Definition of étale (etc) for non-representable morphisms of algebraic stacks?

I've stumbled upon the statement that the morphism $\pi$ from a root stack of the form $\sqrt[r]{\mathscr{L}/\mathscr{Y}}$ (i.e. the "generic" version, not the one concentrated along a divisor) to its ...
Qfwfq's user avatar
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9 votes
1 answer
677 views

Representable diagonal map $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ for DM-Stacks/algebraic spaces

Following the standard definitions of a algebraic space or Deligne–Mumford stack one imposed condition is that the diagonal morphism $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ has to be ...
user267839's user avatar
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9 votes
1 answer
806 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 ...
Jingren Chi's user avatar
9 votes
0 answers
238 views

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 (...
Tim Santens's user avatar
9 votes
0 answers
472 views

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 $\tau$-...
Damien Robert's user avatar
9 votes
0 answers
281 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 $\...
Fred's user avatar
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8 votes
1 answer
796 views

Inverse galois problem and étale homotopy

Is there any relation between étale homotopy theory (Grothendieck-Galois theory) and the inverse Galois problem?...I mean...in classical homotopy theory, every finite group $G$ realizes as a "Galois ...
user avatar
8 votes
1 answer
663 views

Milnor excision for algebraic stacks

Recall that a commutative square of commutative rings $$\begin{matrix} A&\to&B\\ \downarrow &&\downarrow\\ A^\prime&\to&B^\prime\end{matrix}$$ is called a Milnor square if the ...
Harry Gindi's user avatar
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8 votes
1 answer
2k views

Is every module the colimit of its finitely generated submodules? (for algebraic spaces or stacks)

For (quasi-compact and quasi-separated) schemes there is a categorical way to characterise quasi-coherent sheaves of finite type using purely the abelian category $\operatorname{QCoh}(X)$. In an ...
John Salvatierrez's user avatar
8 votes
2 answers
284 views

Residual gerbes and coarse moduli space of a DM stack

Let $X$ be a nice DM stack (Noetherian, separated), and $X_0$ its coarse moduli space which exist by the Keel-Mori theorem. (I like the exposition in D. Rydh. “Existence and properties of geometric ...
RandomMathUser's user avatar
8 votes
1 answer
1k 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 ...
Doelt_k's user avatar
  • 419
8 votes
2 answers
289 views

Smallest atlas for algebraic stack

Let $X$ be an algebraic stack of finite type over a field. Is there an intrinsic way to calculate the minimum of the dimensions of all atlases of $X$? By intrinsic here I mean using constructions such ...
NZK's user avatar
  • 345
8 votes
1 answer
326 views

Objects of a category of homological dimension 1 is a smooth stack?

I heard a reference to a statement like: Suppose $A$ is an (Abelian?) category of homological dimension one, then the stack of objects of $A$ is smooth. (I am not really sure what the stack of ...
Elle Najt's user avatar
  • 1,422
8 votes
1 answer
690 views

on a Deformation long exact sequence of moduli space of stable maps

I am reading the book "mirror symmetry" by Hori,Katz,Klemm,etc. And I want to understand the following Deformation long exact sequence \begin{align} 0 & \to Aut(Σ, p_1, . . . , p_n, f)\to Aut(Σ, ...
Xiaobo Zhuang's user avatar
8 votes
0 answers
287 views

$\mathbb G_{\mathrm{m}}$-gerbes are to (derived) Azumaya algebras as $G$-gerbes are to …?

Let $X$ be a quasicompact quasiseparated scheme over a field $k$. The connection between Azumaya algebras over $X$ and $\mathbb G_{\mathrm{m}}$-gerbes over $X$ is well-known: there exists an injection ...
W. Rether's user avatar
  • 395
8 votes
0 answers
188 views

Closed immersion → Pro-open immersion factorization for residual gerbes

Let $X$ be a quasi-separated algebraic stack. Then it is a theorem of Rydh that every point $x$ in $X$ admits a residual gerbe. More or less, the construction proceeds by first taking the closure ...
Harry Gindi's user avatar
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8 votes
0 answers
1k views

Visualization of an algebraic stack

As the visuallization of an algebraic stack is virtually impossible I warn about this is a soft question. I am interested in thinking visually about algebraic stacks (also higher and derived stacks, ...
Martin Hurtado's user avatar
7 votes
2 answers
435 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 ...
Arcilan's user avatar
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7 votes
2 answers
2k views

What about stacks of categories in algebraic geometry? II

I've made this a new question, rather than expanding the first one. Torsten gives a good answer, and it partially illustrates in practice the 'second approach' I outlined in my other question. (You ...
David Roberts's user avatar
  • 33.4k
7 votes
2 answers
602 views

Is a Deligne-Mumford curve defined over Qbar if and only if its coarse moduli space is

Let $\mathcal X$ be a smooth proper finite type Deligne-Mumford stack over $\mathbb C$ that is generically a scheme. Let $X$ be its coarse moduli space. If $\mathcal X$ can be defined over $\overline{...
Ariyan Javanpeykar's user avatar
7 votes
2 answers
979 views

Differentials for algebraic stacks

Let $S$ be a base scheme. For which algebraic stacks $X$ over $S$ can we define a sheaf of differentials $\Omega^1_{X/S}$ (classifying derivations)? Probably it works when $X$ is Deligne Mumford ...
Martin Brandenburg's user avatar
7 votes
1 answer
882 views

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 ...
Jacob Bell's user avatar
  • 1,275
7 votes
1 answer
2k views

The fibre product of two quotient stacks

My question is to know whether the fibre product of $[X/G]$ by $[Y/H]$ over a base scheme is $S$ is $[X \times_S Y/G \times H]$? And if yes, do you have any reference for it? Thank you.
Kimra's user avatar
  • 131
7 votes
1 answer
932 views

Representability of morphism of stacks

A morphism of Artin stacks $f:X\to Y$ over $\mathbb Q$ is representable by algebraic spaces if and only if its geometric fibres are algebraic spaces. I would like to know if one can use this to prove ...
user234's user avatar
  • 71
7 votes
1 answer
366 views

Fiberwise criterion for a stack to be a gerbe

Let $f:X\to Y$ be a morphism of algebraic stacks. If the geometric fibres of $f$ are algebraic spaces, then $f$ is representable by algebraic spaces. I'm wondering about analogues of this fiberwise ...
Marci Senf's user avatar
7 votes
1 answer
464 views

Does the compactified Torelli map extend to a proper map of stacks?

Let $M_g^{ct}$ denote the moduli stack of compact type genus $g$ stable curves and $A_g$ the moduli stack of principally polarized $g$-dimensional abelian varieties. Can someone provide a reference ...
Aaron Landesman's user avatar
7 votes
1 answer
744 views

Degree formalism for line bundles on Deligne-Mumford stacks

Let $k$ be an algebraically closed field and let $\mathcal{C}$ be proper, Cohen-Macaulay, purely $1$-dimensional Deligne-Mumford stack over $k$. From looking at section 4.3 on page 135 of the paper "...
O-Ren Ishii's user avatar
7 votes
0 answers
324 views

Character stack and character variety

Let $\Sigma$ be a Riemann surface of genus $g$. We can consider two different type of objects associated to it parametrising representations of its fundamental groups. On one side we have the ...
Tommaso Scognamiglio's user avatar

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