Questions tagged [torsors]

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6 votes
1 answer
251 views

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 ...
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1 vote
2 answers
261 views

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_* \...
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2 votes
0 answers
134 views

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 $G^{\op}$-...
  • 994
1 vote
1 answer
213 views

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
77 views

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, ...
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10 votes
1 answer
328 views

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 ...
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3 votes
1 answer
323 views

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 vote
0 answers
176 views

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 votes
0 answers
195 views

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 votes
0 answers
178 views

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 $S$-...
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5 votes
1 answer
536 views

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 $C/U$ (...
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2 votes
0 answers
192 views

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 \...
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3 votes
1 answer
236 views

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. ...
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1 vote
0 answers
102 views

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 ...
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6 votes
2 answers
410 views

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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1 vote
3 answers
906 views

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 votes
0 answers
238 views

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 $X$ , ...
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7 votes
1 answer
255 views

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 ...
12 votes
2 answers
1k views

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 ...
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3 votes
0 answers
157 views

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 ...
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3 votes
1 answer
333 views

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 $\pi$ ...
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4 votes
0 answers
485 views

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 ...
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5 votes
0 answers
234 views

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 ...
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2 votes
0 answers
195 views

Motivation for the definition of push-out for $G$-torsors (as seen in Fukaya-Kato)

In the introductory sections to their paper "A Formulation of Conjectures on $p$-adic Zeta Functions in Non-commutative Iwasawa Theory," Fukaya and Kato describe an explicit construction of ...
4 votes
0 answers
734 views

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 ...
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2 votes
0 answers
130 views

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 ...
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14 votes
0 answers
423 views

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 ...
9 votes
1 answer
562 views

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 ...
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