Questions tagged [torsors]
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30
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Galois action on étale path torsors
TLDR: How is the Galois action on étale path torsors defined?
Let $X$ be a smooth proper scheme, over a field $k$, and let $x,y\in X(k)$ be a pair of points. Let $\pi_1^{\text{ét}}(\overline{X},\...
2
votes
1
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276
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Equivalence between twists of a curve and torsors of its automorphism group
Let $X$ be a curve defined over a number field $K$, and let $G_K$ be the absolute Galois group of $K$. Let $\text{Aut}(X)$ be the group of $\overline{K}$-defined automorphisms of $X$, and consider the ...
2
votes
1
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172
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Can we encode a torsor as a binary function on the isomorphism classes of objects?
Let $G$ be a group object in a topos $\mathcal{T}$. Then we have the notion of a $G$-torsor in $\mathcal{T}$, and the set of isomorphism classes of such objects is denoted $H^1(\mathcal{T};G)$. For ...
6
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1
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Construction of a line bundle from a class $[\alpha] \in H^1(X, \mathcal{O}_X^{\times})$ as $\mathcal{O}_X^{\times}$-Torsor
Let $X$ be a complex compact manifold, and write $\mathcal{O}_X$ for the sheaf of holomorphic functions on $X$. Let $\mathcal{O}_X^{\times}$ be the subsheaf consisting of holomorphic functions. These ...
1
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2
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362
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Pushforward of structure sheaf along a torsor for a finite group
Let $\pi : P \to X$ be a torsor for a discrete, finite group $G$ of size $\#G = N$ on a scheme $X$. I want to compare $\pi_* \mathcal O_P$ with $\mathcal{O}_X$. Locally but not globally, $\pi_* \...
2
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0
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146
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Torsors for nonabelian groups and maps to contracted products
$\newcommand\op{\mathrm{op}}$My question concerns torsors for a sheaf of groups $G$ that is not commutative, and left/right are messing me up. A left $G$-torsor is equivalent to a right $G^{\op}$-...
1
vote
1
answer
222
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Taking quotient of a variety by the additive group
1. Let $X$ be a smooth irreducible $\Bbb C$-variety,
on which the algebraic $\Bbb C$-group $G={\bf G}_{a,{\Bbb C}}$
(the additive group) acts freely on the right:
$$ X\times _{\Bbb C} G\to X,\quad (x,...
2
votes
0
answers
91
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lie algebra bundle and underlying vector bundle
Let $G$ be a connected reductive group over a field $k$. Let $E$ be a $G$-bundle, then we can form the adjoint bundle $ad(E)$ which is a Lie algebra bundle over $k$.
As a vector bundle it is trivial, ...
10
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1
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384
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When is a twisted form coming from a torsor trivial?
Consider a sheaf of groups $G$, equipped with a left torsor $P$ and another left action $G$ on some $X$. Form the contracted product $P \times^G X := (P \times X)/\sim$ where $\sim$ is the ...
3
votes
1
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395
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Pushforward of functions on a frame bundle
Apologies in advance for the long setup and question.
Let $L \to X$ be a line bundle. We may take its frame bundle $p \colon Fr(L) \to X$, a $\mathbb{G}_m$-torsor. We have
$$ p_*\mathcal{O}_{Fr(L)} =...
1
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0
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199
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Killing a Brauer class by a flat projective morphism
Let $X$ be a Noetherian scheme and $\beta \in H^2(X, \mathbb{G}_m)$ that is the image of some $\alpha \in H^1(X, \mathrm{PGL}_{n + 1})$ for some $n \ge 0$, so $\beta$ is a class in the Brauer group of ...
4
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0
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219
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Extra line bundles from torsors
Another math.stackexchange question (here: $\mathbb{G}_m$-torsors and line bundles) goes over a way to construct a line bundle $L$ from a $\mathbb{G}_m$-torsor $T \to B$, by using a decomposition $\...
5
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194
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Does an abelian-scheme-torsor with split generic fiber necessarily split?
Let $k$ be a field. Let $S$ be a smooth $k$-variety, let $G/S$ be an abelian scheme. Let $T/S$ be a $G$-torsor. Suppose the generic fiber of $T/S$ admits a section, does $T$ necessarily admits an $S$-...
5
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1
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706
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Automorphism group of a torsor
Given a site $C$ and an object $U$, let $G$ be a sheaf of groups on this site and let $F$ be $G$-torsor, see the Stacks Project for the general definition.
By restriction on the over category $C/U$ (...
2
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226
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Representabillity of torsors and exact sequence of group schemes
I've seen the following property used at many places, I don't exactly why it is true.
Let $S$ be a site with an initial object $e$. There is an exact sequence of group objects
$$1 \to F \to G \to H \...
3
votes
1
answer
247
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Reduction of structure group for stacks
Consider an action of a smooth linear algebraic group $G$ on a variety $X$ over an arbitrary field $k$, and the quotient stack $[X/G]$. Let $p$ be a $k$-point of $X$. If the action is transitive (i.e. ...
1
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0
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122
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Rational points of torsors over a separable closure
I already asked this question on Math Stack few days ago ( torsors over a separable closure ), but did not receive any answer, so I post it here.
Let $G$ be a smooth linear algebraic group defined ...
6
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2
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497
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Torsors over complete local fields
Let $G$ be a linear algebraic group scheme, and let $R$ be a complete discrete valuation ring, with quotient field $K$ and residue field $k$.
If $T$ is an $R$-torsor, it yields by base change a $k$-...
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3
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942
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Notion of Torsors
I am trying to read this paper by Lawrence Breen.
It starts with the definition of a torsor.
Let $G$ be a bundle of groups on a space $X$. The following definition of a principal space is standard, ...
4
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0
answers
274
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Non Zariski-locally trivial $\mathbb{G}_{\mathrm{a}}$-torsor?
The question is pretty much as in the title: are there $\mathbb{G}_\mathrm{a}$-torsors $\pi:P\to X$ for some Grothendieck topology finer than the Zariski one, over a complex algebraic variety $X$ , ...
8
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288
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How are the left and the right group of a bitorsor related?
This question arose from my answer to To what extent does a torsor determine a group: it turns out that I do not know one thing about it.
Let $G$, $G'$ be groups in some nice enough category (you may ...
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2
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To what extent does a torsor determine a group
Let $k$ be a field, and suppose $G$ is a group-scheme over $k$ (I am happy to assume that $k=\mathbb{Q}$ and that $G$ is affine). A $G$-torsor over $k$ is a non-empty $k$-scheme $T$ equipped with an ...
4
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0
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181
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Universal vectorial bi-extension as a scheme
In 'The universal vectorial Bi-extension and p-adic heights' Coleman works with the pullback of the Poincaré biextension of an abelian variety A to its universal vectorial extension and claims this is ...
3
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1
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383
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Torsors of pushforward group schemes
I'm reading William Waterhouse's "Discriminants of etale algebras and related structures", and he makes a basic claim I'm struggling to justify.
Suppose $S/R$ is etale of rank $n$... and let $\pi$ ...
4
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0
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523
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Questions about the algebraic space $\mathbb{A}^1/\mathbb{Z}$
Let $X = \mathbb{A}^1_{\mathbb{C}}/\mathbb{Z}$, where $\mathbb{Z}$ acts on $\mathbb{A}^1$ via translation. [To clarify, $X$ is an \'etale sheaf with a smooth presentation $\mathbb{A}^1_{\mathbb{C}}\to ...
5
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242
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Torsors for discrete groups in the etale topology
Let $S$ be a smooth variety over $\mathbb C$ or a smooth quasi-projective integral scheme over Spec $\mathbb{Z}$.
Let $G$ be an (abstract) discrete group. For instance, $G =\mathbb{Z}^n$ or $G$ a ...
4
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0
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898
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Contracted product of torsors
Given a group $G$ and a left $G$-set $X$, then we can make $X$ a right $G$-set defining the action as $xg:=g^{-1}x$ , or if you prefer we are considering the opposite group $G^{op}$ to make the left ...
2
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0
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135
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Characterization of torsors which are locally trivial in terms of descent
Let $\mathsf C$ be a category and $G$ an internal group. Suppose $\mathsf C$ is finitely complete, so that $\pi_2:G\times B\to B$ is an internal group in $\mathsf C_{/B}$ for every $B$. A $G$-bundle ...
15
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0
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486
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Zariski vs etale torsors over abelian varieties
Question. Let $A$ be an abelian variety (say, over the complex numbers), $G$ an algebraic group, $c$ a class in $H^1_{\rm et}(A, G)$. Denote the multiplication by $N$ map on A by $m_N:A\to A$. Does ...
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Is $G$ always the automorphism group of the trivial $G$-torsor?
If $G$ is just an ordinary set-theoretic group, then the answer to the question in the title is yes: the automorphisms of $G$ as a (left) $G$-set are all of the form "multiply (on the right) by an ...