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1
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0answers
105 views

Artin's criterion for étale, quasi-separated algebraic spaces

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 ...
3
votes
0answers
104 views

Is there a difference between the inertia stack and the universal automorphism group

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
votes
1answer
244 views

Algebraic spaces which are automatically schemes

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 ...
7
votes
1answer
191 views

Is every proper regular relative algebraic space curve over a Dedekind domain projective?

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
votes
1answer
425 views

What is an excellent algebraic space?

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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0answers
197 views

Is this diagram of sheaves actually Cartesian as claimed?

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 ...
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0answers
113 views

When does an algebraic space that is a torsor over a scheme have to be a scheme?

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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0answers
133 views

Is surjectivity for morphisms of schemes local on the domain?

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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0answers
169 views

How far is it to extend the results of SGA III Exp. VIB from group schemes to group spaces?

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 ...
3
votes
2answers
331 views

A reference for “an algebraic space is a scheme iff its reduction is”?

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 ...
1
vote
1answer
132 views

Does a line bundle on a normal Noetherian algebraic space come from a Weil divisor?

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
votes
0answers
91 views

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 $Z\coprod_X Y$ ...
1
vote
1answer
186 views

Relative identity component for group algebraic spaces

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) $G^0 \subset G$ ...
12
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0answers
293 views

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 ...
3
votes
0answers
307 views

Ample Line Bundles on Algebraic Spaces

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 ...
5
votes
0answers
162 views

(Relative) ampleness on algebraic spaces

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 ...
6
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2answers
482 views

Is an algebraic space over a DVR, whose special fibre and generic fibre are schemes, actually a scheme?

Is an algebraic space over a DVR, whose special fibre (and all its infinitesimal neighborhood) and generic fibre are schemes, actually a scheme?
3
votes
0answers
163 views

Subgroups of a group algebraic space

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 ...
3
votes
0answers
159 views

The support of a finite type module on an algebraic space

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 ...
8
votes
0answers
295 views

Nisnevich covers of algebraic spaces

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
votes
1answer
386 views

Weil restriction of abelian schemes along finite étale (resp. finite locally free) morphisms

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 ...
0
votes
1answer
149 views

Prorepresentable functors repres. by alg. spaces? Covering spaces by alg. spaces.

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
votes
0answers
167 views

morphisms between algebraic spaces

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 ...
7
votes
1answer
1k views

surjective morphism of schemes or epimorphism of sheaves?

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 ...
14
votes
0answers
453 views

Are “fpqc algebraic spaces” algebraic spaces?

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 ...
15
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3answers
2k views

Is every algebraic space the quotient of a scheme by a finite group?

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 ...
5
votes
2answers
737 views

Is the category of affine fppf groups closed under normal quotients?

Let $S$ be a scheme and let $N$, $G$ be affine flat group schemes of finite presentation over $S$. If we assume that $N$ is a closed normal subgroup of $G$, we may form the fppf quotient sheaf $G/N$, ...
11
votes
1answer
813 views

Hodge structures on algebraic spaces

Let $X$ be a proper smooth algebraic space over $\mathbb C$ (which amounts, due to Artin, to giving a Moishezon space: a compact complex manifold whose dimension equals the transcendence degree of its ...
3
votes
1answer
243 views

Semicontinuity and cohomological flatness for algebraic spaces

Let $f \colon X \to S$ be a proper morphism form a seperated algebraic space $X$ to an affine noetherian scheme $S$. Given a coherent sheaf $F$ on $X$, we know from Knutson's book, that the ...
18
votes
6answers
1k views

Categorical construction of the category of schemes?

The answer to the following question is probably well known or the question itself is well known not to have a reasonable answer. In the latter case could you please let me know what the "right" ...
3
votes
0answers
346 views

Does an étale equivalence relation of schemes induce an equivalence relation on points?

Let $R \rightrightarrows U \to X$ be a presentation of an algebraic space by schemes. Does this induce an exact sequence $|R| \rightrightarrows |U| \to |X|$ on underlying points? The reason I ask is ...
3
votes
2answers
469 views

Quasi-separatedness for Algebraic Spaces

I'm reading Knutson's book on algebraic spaces, and I stumbled over the quasi-separatedness axiom in his definition of algebraic spaces (Definition 1.1, Chapter II). He defines an algebraic space A as ...
20
votes
3answers
1k views

Why is this not an algebraic space?

This question is related to the following question which I've just seen which was posted by Anton. His question is whether a algebraic space which is a group object is necessarily a group scheme, and ...
28
votes
7answers
2k views

Is an algebraic space group always a scheme?

Suppose G is a group object in the category of algebraic spaces (over a field, if you like, or even over ℂ if you really want). Is G necessarily a scheme? My feeling is that the answer is "yes" ...
6
votes
2answers
573 views

Do quotients of representable sheaves represent quotients?

Here's the context for the question: Proposition 4.6 of Freitag and Kiehl's book on etale cohomology shows that a sheaf (of sets) $\mathcal{F}$ (on the site Et(X)) is constructible if and only if it ...
9
votes
1answer
352 views

Can an algebraic space fail to have a unviersal map to a scheme?

Let $\mathcal{X}$ be an algebraic space. Can it happen that there does not exist a map $\mathcal{X} \to X$ with $X$ a scheme that is initial for maps from $\mathcal{X}$ to schemes? Are there ...
3
votes
1answer
267 views

Points of a weakly locally separated algebraic space

If X is a quasi-separated algebraic space and Spec k -> X is an etale presentation, then X is isomorphic to Spec k' for a field k'. (This is also true if X is Zariski locally quasi-separated.) The ...