Questions tagged [algebraic-stacks]
The algebraic-stacks tag has no usage guidance.
280
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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 ...
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 ...
20
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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 ...
19
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4
answers
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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 ...
19
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2
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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 ...
19
votes
1
answer
882
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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 $Ell \Rightarrow Ell: Ring \to Gpd$ ...
19
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0
answers
602
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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 ...
17
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4
answers
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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 ...
17
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2
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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, ...
14
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2
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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, ...
14
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0
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320
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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 ...
13
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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) ...
12
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1
answer
330
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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 ...
12
votes
1
answer
424
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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 ...
12
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0
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277
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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?...
11
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1
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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 ...
11
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1
answer
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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 ...
11
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1
answer
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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 ...
11
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0
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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 ...
10
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1
answer
447
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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 ...
10
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1
answer
611
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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 $j$...
9
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2
answers
726
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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^*(-,...
9
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3
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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 ...
9
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1
answer
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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 ...
9
votes
1
answer
806
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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 ...
9
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0
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238
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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 (...
9
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0
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472
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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 $\tau$-...
9
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0
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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 $\...
8
votes
1
answer
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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 ...
8
votes
1
answer
663
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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 ...
8
votes
1
answer
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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 ...
8
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2
answers
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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 ...
8
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1
answer
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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 ...
8
votes
2
answers
289
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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 ...
8
votes
1
answer
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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 ...
8
votes
1
answer
690
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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, ...
8
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0
answers
287
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$\mathbb G_{\mathrm{m}}$-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 ...
8
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0
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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 ...
8
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0
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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, ...
7
votes
2
answers
435
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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 ...
7
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2
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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 ...
7
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2
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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{...
7
votes
2
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979
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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 ...
7
votes
1
answer
882
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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 ...
7
votes
1
answer
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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.
7
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1
answer
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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 ...
7
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1
answer
366
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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 ...
7
votes
1
answer
464
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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 ...
7
votes
1
answer
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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 "...
7
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0
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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 ...