Questions tagged [algebraic-spaces]
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71 questions
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 an ...
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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 ...
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The different types of stacks
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, ...
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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 $G/N$, ...
- 1,538
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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 ...
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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 ...
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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" ...
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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 ...
- 1,538
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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 ...
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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 ...
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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" ...
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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 ...
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Can an algebraic space fail to have a universal 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 ...
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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 ...
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