# Questions tagged [algebraic-spaces]

The algebraic-spaces tag has no usage guidance.

The algebraic-spaces tag has no usage guidance.

61
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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 ...

9
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1
answer

270
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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 ...

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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 $...

2
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1
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260
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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 ...

7
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0
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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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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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563
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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 ...

5
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209
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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 ...

3
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348
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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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Let $f : X \to Y$ be a finite, surjective étale morphism of algebraic spaces (say, of finite type over some noetherian scheme). Assume that $X$ is a scheme. Does this imply that $Y$ is a scheme? Is $Y$...

3
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192
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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 ...

5
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289
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I think that wikipedia article on Algebraic spaces contains a serious content error in the part on the definition of Algebraic spaces as quotients of schemes and I would like to discuss if it is ...

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188
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According to Wikipedia
there are two common ways to define algebraic spaces:
they can be defined as either quotients of schemes by étale
equivalence relations,
or as sheaves on a big étale site that ...

3
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1
answer

371
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By SGA 4 IX Proposition 2.7, any constructible sheaf $\mathcal{F}$ on a qcqs scheme $X$ can be represented as an equalizer of two etale maps between representable (by schemes) sheaves. This would ...

2
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125
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$\DeclareMathOperator\dim{dim}$I want to apply EGA IV 4, Proposition 17.13.2 to a cartesian diagram
of algebraic spaces over a fixed scheme $S$.
I know the relative dimensions $\dim(\mathfrak{X}'/S)...

5
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1
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488
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I saw the following statement in a paper of Bhatt-Mathew:
Let $X$ be a quasicompact quasiseparated scheme. Then there is an equivalence of categories between constructible étale sheaves (of sets) on ...

3
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0
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140
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Let $\text{Sch}$ be the category of schemes. Let $S$ be an object of $\text{Sch}$. Consider the category $\text{Sch}/S$.
Some interesting topologies on $\text{Sch}/S$ are Zariski, fpqc, étale, fppf.....

3
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1
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287
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Let $S$ be a Noetherian scheme, let $Y$ be a scheme of finite type over $S$, and let $X$ be an algebraic space of finite type over $S$. Suppose that there is a morphism $f:Y \rightarrow X$ which is ...

26
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2
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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 ...

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82
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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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0
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108
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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 ...

0
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2
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353
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Let $k$ be an algebraically closed field, $X$ a connected smooth curve over $k$, $G$ a connected reductive group over $k$, and $B \subset G$ a Borel subgroup.
Given a $G$-torsor $E$ on $X$ in the ...

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0
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245
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it is known from Knutson's work that an algebraic space which is separated and étale over a scheme is a scheme. Let $S$ be a locally noetherian scheme. I am looking for a reference giving an Artin's ...

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173
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Let $\mathcal M$ be a stack representing some moduli problem. Let $\mathcal X\to \mathcal M$ be the corresponding universal family.
What is the difference between the inertia stack $I\to \mathcal M$ ...

5
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1
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631
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Let $S$ be a scheme, and let $f:X\to S$ be a morphism of algebraic spaces.
If $f$ is smooth proper curve of genus at least two, then $X$ is a scheme. (Here I mean that $f$ is a smooth proper morphism ...

12
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1
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646
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This question is in some sense a follow up to a related question Is a normal proper relative curve over a DVR projective?
Let $R$ be a Dedekind domain, let $S := \mathrm{Spec}(R)$, and let $X \...

8
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1
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595
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What does it mean to say that an algebraic space $S$ is excellent? One knows that excellence of a Noetherian ring is not a property that is etale local (that is, excellence cannot be checked over an ...

3
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260
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The question is about Corollary 1.6.2 (b) in the book by Laumon and Moret-Bailly on algebraic stacks.
There we have a scheme $S$ and morphisms $X \xrightarrow{f} Y \xrightarrow{g} Z$ of sheaves on a (...

3
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219
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In Group actions on stacks and applications (Section 4 of part A), M.Romagny gives a definition of $G$-torsor over a scheme $S$ in which the total space need not be a scheme, just an algebraic space. ...

3
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183
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It is said so in Knutson's book 'algebraic sapces' in several places for different topologies on schemes, see Chapt. I, 2.19 for Zariski top, 3.13 for flat top., 4.11 for etale topology.
But this ...

2
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0
answers

195
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How far is it to extend the results of SGA III Exp. VIB from group schemes to group spaces?
In particular, does Corollary 4.4 from SGA III Exp. VIB hold for G/S being merely a group space? Here the ...

4
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2
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634
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It seems to be a known fact that an algebraic space is a scheme if and only if its associated reduced closed subspace is a scheme. For instance, this is used in Chai-Faltings in proving that the dual ...

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1
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195
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Let $X$ be a normal Noetherian algebraic space and $\mathscr{L}$ a line bundle on $X$. If $X$ is a scheme, then there is locally principal Weil divisor on $X$ that gives rise to $\mathscr{L}$. Is the ...

0
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107
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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 $Z\coprod_X Y$ ...

2
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344
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Let $S$ be a locally noetherian scheme and let $G$ be a separated and smooth $S$-group algebraic space of finite presentation. Does there exist an open sub-(group algebraic space) $G^0 \subset G$ ...

19
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585
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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 ...

6
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316
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The question in the title can be reformulated as follows. Let $f : Y \to X$ be a monomorphism of algebraic spaces where $X$ is a scheme. Is it true that $Y$ is a scheme?
If $f$ is locally of finite ...

4
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0
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The sources known to me (Knutson's Algebraic Spaces and Pascual-Gainza's Ampleness criteria for algebraic spaces) define a line bundle $L$ on an algebraic space $X$ (over a base scheme $S$) to be ...

6
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This is a follow-up (of sorts) to this question.
Let $f : X \to T$ be a proper morphism of schemes. Then the notion of a relative ample (or $f$-ample) line bundle can be defined in several ...

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2
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729
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Is an algebraic space over a DVR, whose special fibre (and all its infinitesimal neighborhood) and generic fibre are schemes, actually a scheme?

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I found in the literature many references on the representability of quotients of group schemes but almost nothing about subgroups. For this reason I hope that my question is a silly one and that what ...

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285
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I'd like to ask this question to make sure I understand a very basic thing about supports. Let $X$ be an algebraic space and F a quasi-coherent sheaf on it of finite type.
In here the schematic ...

9
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0
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450
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Does every algebraic space have a Nisnevich cover by a scheme?
(Assume that the algebraic space is quasi-separated, quasi-compact and over a scheme $S$.)
Background:
Every algebraic space has an ...

4
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1
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Q: Is there a simple proof of the fact that the Weil restriction of an abelian scheme along a finite étale morphism is an abelian scheme ?
Details: Let $S$ be a scheme and $f:S'\rightarrow S$ a ...

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1
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Let $X$ be a (reasonable) scheme. I'm curious about constructing the constructing the covering space of a scheme algebraically. The covering space functor $F$ (below) can be represented by a ...

0
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212
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My question concerns morphisms between algebraic spaces. I like the definitions of Artin, but I do not see a simple proof of the fact that the composition of two morphisms is a morphism. Could someone ...

9
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1
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I have a technical question coming from reading Toen's master course on stacks.
If we view schemes as locally ringed spaces then there we could define a morphism to be surjective if it the underlying ...

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0
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Suppose $F:Sch^\text{op}\to Set$ is a sheaf in the fpqc topology, has quasi-compact representable diagonal, and has an fpqc cover by a scheme. Must $F$ be an algebraic space? That is, must $F$ have an ...

16
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3
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In this MO question it is claimed that a catchphrase for "algebraic spaces" could be that they are "the result of looking at the orbit space of the action of a finite group on a scheme".
Hence my ...

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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, ...