Questions tagged [algebraic-spaces]
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Spaces of fixed points
I am reading the paper Space with $\mathbb{G}_{m}$-action, hyperbolic localization and nearby cycles by Timo Richarz and I am having some troubles in understanding the proof of Lemma 1.10.
The setting ...
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Is a finite morphism of Deligne-Mumford stacks proper?
The situation that I am in is the following. Let $\mathcal{X}$ be a smooth Deligne-Mumford stack over a field $k$. Let $X$ be a $k$-scheme together with a morphism $\pi;\mathcal{X}\rightarrow X$ (you ...
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Fppf or étale extension of group algebraic spaces
Let $S$ be a scheme and let
$$0 \to A \to B \to C \to 0$$
be an exact sequence of abelian sheaves on $(\mathrm{Sch}/S)_\text{fppf}$. Assume that $A$ and $C$ are representable by flat algebraic spaces. ...
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Is this double quotient of $\operatorname{SL}_2$ representable by an algebraic space or a scheme?
$\DeclareMathOperator\SL{SL}$Let $B$ be a Borel subgroup (upper triangular matrices), and let $\Gamma := \langle \sigma\rangle$ be the group generated by a (hyperbolic) element of $\SL_2/\mathbb{Q}_p$ ...
- 2,113
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Regular two-dimensional algebraic spaces
Let $X$ be an algebraic space which is integral, noetherian, separated, two-dimensional and regular. We keep these assumptions throughout.
Question 1. Is $X$ always a scheme?
Question 2. If $X$ is a ...
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Is $\operatorname{Rep}(G,\operatorname{SL}_2)$ representable by an algebraic space?
Let $G$ be a finite group. Consider the category of rigid analytic spaces over $\operatorname{Spf}\mathbb{Q}_p$, and let $\operatorname{Rep}(G, \operatorname{SL}_2)$ be the fibred category above it, ...
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Perfect algebraic spaces on a paper of Xinwen Zhu
I have problem reading Xinwen Zhu's paper Affine Grassmannians and the geometric Satake in mixed characteristic about perfect algebraic spaces in Section A.1.
Let $k$ be a perfect field of ...
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Are the tensor-invertible coherent sheaves on an algebraic space (Zariski) locally free of rank one?
On a scheme, the coherent sheaves that are invertible objects for the tensor product (monoid) operation are precisely the coherent sheaves that are (Zariski) locally free of rank one. Is the same ...
- 4,101
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Quotient stack is an algebraic space when $G$ is finite and acts freely
I have been following Jarod Alper's lecture series on YouTube on Stacks https://youtube.com/playlist?list=PLhFI5R_xInjdhtWuhgYlA8NZGXO-unnl4
From what I understand -
If a smooth affine group scheme $...
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Example of an Algebraic Space ("false" affine line with different tangents at origin)
I have a question about the following example from the Algebraic spaces and quotients by equivalence relation of schemes by Roy Mikael Skjelnes (page 12)
of a presheaf quotient, which
has associated ...
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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 ...
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Is an algebraic space having a monomorphism to an affine scheme a scheme?
Definition
An algebraic space is a functor $X$ from the opposite of the category of commutative rings to the category of sets satisfying the following conditions:
The functor $X$ is a (large) etale ...
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Why diamonds are only defined in characteristic $p$?
I'm trying to read Scholze's article "Etale cohomology of diamonds" (arXiv link) and both in this article and in Berkeley notes, the diamonds are defined as sheaves on the category of ...
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Reference for Grothendieck's theorem on representation of unramified functors
In the Exposé 294 of the Bourbaki Seminar of the year 1964-1965, Murre gives an outline of proof of a theorem of Grothendieck giving necessary and sufficient conditions of representability by a scheme ...
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Algebraic spaces as functors on complete local rings
Let $X$ be an algebraic space locally of finite presentation, and let $\tilde{X}$ denote the restriction of $X$ (as a functor on schemes) to the category of complete local rings. Is it true that the ...
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