Questions tagged [algebraic-spaces]

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27 votes
3 answers
3k views

Why is this not an algebraic space?

This question is related to the question Is an algebraic space group always a scheme? which I've just seen which was posted by Anton. His question is whether an algebraic space which is a group object ...
Chris Schommer-Pries's user avatar
37 votes
7 answers
4k 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" ...
Anton Geraschenko's user avatar
31 votes
7 answers
3k 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" ...
algori's user avatar
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28 votes
2 answers
2k views

morphisms representable by algebraic spaces vs morphisms representable by schemes

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 ...
stupid_question_bot's user avatar
16 votes
3 answers
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 ...
Qfwfq's user avatar
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12 votes
1 answer
772 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 \...
Lisa S.'s user avatar
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9 votes
2 answers
1k 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 $G/N$, ...
Daniel Bergh's user avatar
  • 1,538
5 votes
1 answer
597 views

Constructible étale sheaves on X are étale algebraic spaces over X

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 ...
Steve's user avatar
  • 453
4 votes
2 answers
667 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 ...
Daniel Bergh's user avatar
  • 1,538
4 votes
2 answers
709 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 ...
Question Mark's user avatar
3 votes
0 answers
149 views

Other interesting notions when we change topology on $\text{Sch}/S$

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.....
Praphulla Koushik's user avatar