# Questions tagged [torsors]

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27
questions

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109 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 $G^{\op}$-...

**1**

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**1**answer

193 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**

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59 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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93 views

### The isomorphism classes of lifts of a $\operatorname{PGL}_r$-bundle to $\operatorname{GL}_r$

I want to show the following Lemma,
The set of algebraic isomorphism classes of lifts to $\operatorname{GL}_r$ of an algebraic principal $\operatorname{PGL}_r$ -bundle $P$ on a smooth projective ...

**9**

votes

**1**answer

242 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 ...

**2**

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**1**answer

192 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)} =...

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170 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 ...

**3**

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168 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**

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125 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 $S$-...

**5**

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392 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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165 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 \...

**3**

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**1**answer

234 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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88 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 ...

**6**

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**2**answers

345 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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**3**answers

827 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, ...

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207 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 $X$ , ...

**7**

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**1**answer

231 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**

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**2**answers

965 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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142 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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**1**answer

296 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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464 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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222 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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171 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 ...

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528 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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125 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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363 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**

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**1**answer

500 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 ...