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2 votes
0 answers
139 views

What are the categories of IND and PRO schemes?

below is a mathexchange question with no answers so I drop it here. I have some difficulties to figure out what the category of IND-schemes and PRO-schemes are, in particualer the relations with ...
Marsault Chabat's user avatar
1 vote
0 answers
263 views

An algebraic stack is an algebraic space if and only if it has the trivial stabilizer group

Let $G\to S$ be a smooth affine group scheme over a scheme. Let $U$ be a scheme over $S$ with an action of $G$. Let $[U/G]$ be the quotient stack. In Alper's note: Stacks and Moduli, there is a result ...
Yuen's user avatar
  • 11
8 votes
0 answers
325 views

Moduli stacks and representability of diagonal by schemes

The answer to my question might very well be standard, but I have had trouble finding the right keywords to search for it, so I apologize if this is something well-known to the experts. I am learning ...
Wojowu's user avatar
  • 28.2k
3 votes
0 answers
291 views

Generalizations of Artin–Verdier duality?

Constructible étale abelian sheafs on $Spec\ O_\mathbb K$, for number fields $\mathbb K$, satisfy Artin-Verdier duality. Are there known any algebraic schemes or algebraic stacks, other than $Spec\ O_\...
Adam's user avatar
  • 2,390
9 votes
1 answer
835 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
  • 6,028
3 votes
1 answer
212 views

Algebraic spaces in the étale topology (proof from Stacks project)

I have a question about the proof of Lemma 78.12.1 from Stacks Project. The aim of the last paragraph of the proof is to verify that the map of sheaves in the étale topology $F \to U/R$ is an ...
user267839's user avatar
  • 6,028
2 votes
0 answers
127 views

$\mathscr Coh_{X|S} $ is algebraic and of finite type

Let $S$ be a Noetherian scheme and $X$ a projective $S$-scheme. Reading Laumon-Moret--Bailly's "Champs Algebriques", Theorem 4.6.2.1: $ \mathscr Coh_{X|S} $ and $ \mathscr Fib_{X|S,r} $ are ...
Conjecture's user avatar
5 votes
1 answer
307 views

The stack of group algebraic spaces

The fibred category $\mathcal A$ of algebraic spaces over a scheme $S$ is a stack (over the category of affine schemes with the etale topology). This is proved in Laumon and Moret-Bailly's book (see (...
Stacky student's user avatar
4 votes
1 answer
301 views

Stacks with a small coarse moduli space

Let $k$ be a field of characteristic zero. Let $X$ be a finite type algebraic stack over $k$ with a coarse (or good) moduli space $M$. Suppose that $M$ is isomorphic to a point, i.e., $M = Spec k$. ...
Fater's user avatar
  • 41
6 votes
0 answers
254 views

Is the stack of varieties with a big line bundle algebraic

In Starr's paper https://www.math.stonybrook.edu/~jstarr/papers/moduli4.pdf the folk result that the fibred category of pairs $(X\to S, L)$, where $S$ is an affine scheme, $X\to S$ is flat proper ...
Pancho's user avatar
  • 171
3 votes
0 answers
303 views

Finiteness of the connected components of a stack

Let $X$ be an algebaic stack over a scheme $S$, for any $S$-scheme $Y$ we can consider the groupoid $X(Y)$ of $Y$-points. Denote by $\pi_0(X(Y))$ the set of isomorphism classes of the groupoid. Are ...
Bear's user avatar
  • 845
3 votes
0 answers
410 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 ...
Ledumdi's user avatar
  • 31