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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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
2
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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) $G^0 \subset G$ ...
- 1,103
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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 ...
- 5,998
2
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1
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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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A schematic representability of an algebraic space with group action
In the book "Néron Models" (BLR), there is a statement as follows (on page 164):
Let $S$ be a locally noetherian scheme and let $G$ be a smooth algebraic group space over $S$ with connected ...
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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 $\SL_2/\mathbb{Q}_p$ ...
- 2,907
2
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174
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Finite étale morphism from a scheme to an algebraic space
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 $Y$...
- 197
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130
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Transversality for algebraic spaces
$\DeclareMathOperator\dim{dim}$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)...
- 91
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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 ...
- 997
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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 ...
- 997
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1
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201
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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 ...
- 1,103
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Section of étale morphism of algebraic spaces
I am sorry in advance if this question is too naive for specialists. I just realized that I need it when doin research and I haven't taken any serious course on algebraic spaces. Let $u \colon U \...
- 883
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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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representability of a certain extension of group algebraic spaces
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 ...
- 13
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272
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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 ...
- 11
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2
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437
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Existence of $B$-reduction of a $G$-torsor on a curve
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 ...
- 5,562
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1
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87
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representing base changes of the unit section
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 ...
- 13
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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 ...
- 3,555
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109
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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
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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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