# Tagged Questions

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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 ...
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### Help for reference of moduli stack of fake elliptic curves

I see everywhere the following: Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, $\mathcal{O}_B$ be a maximal order, $N$ be an positive integer coprime to $D$. ...
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### representing base changes of the unit section

Let $S$ be a scheme and $G$ be a sheaf in groups on the big étale site over $S$. Let $e:S\rightarrow G$ be the unit section. Is it true that given an algebraic space in groups $H$, étale over $S$, and ...
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### The non-existence of the fine moduli scheme of vector bundles. Why?

The reference I am using is this one. The question is about the moduli space of vector bundles. I am trying to understand why the fine moduli scheme does not exist. Let $C$ a projective curve. Let $S$ ...
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### representability of a certain extension of group algebraic spaces

Let S be a scheme. Suppose we have sheaves in abelian groups $A,B,C$ over the big étale site of $S$. Suppose that $A$ and $C$ are representable by algebraic spaces in groups locally of finite type ...
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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 $j$...
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### Do equivariant morphisms induce representable maps of quotient stacks?

Let $f: X \to Y$ be a $G$-equivariant map between schemes $X$, $Y$ with action of a flat group scheme $G$. Then why is the induced map of algebraic stacks $[X/G] \to [Y/G]$ representable?
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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 "...
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### Are Picard stacks group objects in the category of algebraic stacks

I've been wondering about what a "group algebraic stack" should be, and ran into the notion of a Picard stack. I'm slightly confused by the terminology here. Given an algebraic stack $\mathcal X$ ...
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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 ...
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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 ...
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### Testing the vanishing of cohomology fiberwise for a proper morphism from an Artin stack

Let $S$ be a Noetherian scheme, let $f\colon\mathscr{X} \rightarrow S$ be a proper morphism with $\mathscr{X}$ an algebraic stack, and let $\mathscr{F}$ be an $S$-flat coherent sheaf on $\mathscr{X}$. ...
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### Question about canonical DM stacks

Let $\mathcal{X}$ be a canonical stack (edit: I forgot to say I also want $\mathcal{X}$ smooth), and $\pi : \mathcal{X}\to X$ its coarse moduli space morphism. Let also $D$ be a prime divisor (i.e. ...
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### Vanishing theorems on toric DM stacks

In chapter 9 of the book Toric varieties by Cox-Little-Schenck several cohomology vanishing theorems for toric varieties are proved or mentioned. In this question I am interested in references for ...
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### On an example by Romagny about fixed point stack not commuting with coarse moduli space

This is to understand better Example 3.9 on page 221 of Group actions on stacks and applications by M.Romagny. For an action of an algebraic group (scheme) $G$ on an algebraic stack $\mathcal{M}$, ...
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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 ...
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### pushing out families of curves

Let $f:X\rightarrow Y$ be a morphism of schemes with smooth curves as fibers. Let $g:X\rightarrow Z$ be a family of smooth or nodal curves with $Z$ a regular scheme. Does the push-out $Z\coprod_X Y$ ...
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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 projective scheme $P$ by the action of a smooth (finite type separated) reductive group ...
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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 ...
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### Sheaf of isogenies representable?

It is well-known that "the" stack of elliptic curves (allow me to be vague as to singular curves, compactifications etc) has a presentation by a groupoid in schemes. One of the things that needs to be ...
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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, ...
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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 ...
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 $\overline{... 1answer 279 views ### Grothendieck duality for stacks Let$\mathcal{X}$be a smooth, proper and separated Deligne-Mumford stack and let$\pi:\mathcal{X}\rightarrow X$be its coarse moduli space. Does Grothendieck duality hold for the morphism$\pi$? In ... 1answer 402 views ### The singularity of the algebraic stack and the singularity of the coarse moduli space It is possible that an algebraic stack is smooth while the coarse moduli space is not smooth. I want to know what is relationship between the singularity of the algebraic stack and that of its coarse ... 0answers 141 views ### Quasi-finite morphisms of stacks Let$f:X\to Y$be a morphism of nice" stacks over$\mathbf C$such that the induced morphism on coarse moduli spaces is quasi-finite. Is$f$quasi-finite? By a "nice" stack I mean a smooth finite ... 1answer 647 views ### Several simple questions on the geometry of higher stacks I'm trying to understand definition/work out some examples. So, there are some simple questions about higher stacks. For the simplicity assume that we are working with higher DM (Deligne-Mumford) ... 0answers 219 views ### Homology of stack points (from math.stackexchange) (This question is duplicated here) This is a very basic question about how definitions in homology carry over to the easiest example of stacks. Let$G$be a finite cyclic group. Consider the ... 1answer 246 views ### Finite-type Artin Stack over$\mathbb C$Suppose I have an Artin stack$\mathfrak M$locally of finite-type over$\mathbb C$with presentation$M\rightarrow \mathfrak M$. Suppose further that$\mathfrak M$"represents" (in the stack sense) ... 1answer 537 views ### Is “stackiness” transitive? (and a couple other basic questions about stacks) Say,$B$is a category fibered in groupoids over some category$C$, and$A$is a category fibered in groupoids over$B$. Suppose$A$is a stack (over whatever site) over$B$, and$B$is a stack over$...
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.