A Tannakian classification of torsors on the projective line

TL;DR

This paper provides a Tannakian proof of Grothendieck-Harder’s theorem, classifying torsors under reductive groups on the projective line over a field, offering a new perspective on this classical result.

Contribution

It introduces a Tannakian approach to prove the classification theorem, offering a novel proof technique compared to traditional methods.

Findings

Tannakian formalism applies to torsor classification.

Provides a new proof of Grothendieck-Harder theorem.

Enhances understanding of reductive group torsors.

Abstract

In this small note we present a Tannakian proof of the theorem of Grothendieck-Harder on the classification of torsors under a reductive group on the projective line over a field.

Equations

$${\mathrm{Hom}}^{\otimes }({\mathrm{Rep}}_k(G),{\mathrm{Rep}}_k({\mathbb{G}}_m))$$

$$\omega :{\mathrm{Rep}}_k(G)\to {\mathrm{Rep}}_k({\mathbb{G}}_m).$$

$${\mathrm{Hom}}^{\otimes }({\mathrm{Rep}}_k(G),{\mathrm{Rep}}_k({\mathbb{G}}_m))\cong H_{\acute{\mathrm{e}}\mathrm{t}}^1({\mathbb{P}}_k^1,G).$$

$$\mathrm{Hom}({\mathbb{G}}_m,G)/G(k)\cong H_{\mathrm{Zar}}^1({\mathbb{P}}_k^1,G).$$

$$\mathrm{Hom}({\mathbb{G}}_m,G)/G(k)\cong X_{\ast }(A{)}_{+}$$

$$H_{\acute{\mathrm{e}}\mathrm{t}}^1({\mathbb{P}}_k^1,\mathcal{E})=0$$

$${\mathrm{Rep}}_k({\mathbb{G}}_m)$$

$$\eta :{\mathbb{A}}_k^2\setminus \{0\}\to {\mathbb{P}}_k^1,(x_0,x_1)\mapsto [x_0:x_1],$$

$$\mathcal{E}(V):={\mathbb{A}}_k^2\setminus \{0\}{\times }^{{\mathbb{G}}_m}V\to {\mathbb{P}}_k^1$$

$${\mathrm{Bun}}_{{\mathbb{P}}_k^1}$$

$$\mathcal{E}(-):{\mathrm{Rep}}_k({\mathbb{G}}_m)\to {\mathrm{Bun}}_{{\mathbb{P}}_k^1}$$

$$\mathcal{E}(V)$$

$$V=\underset{i\in \mathbb{Z}}{⨁}{V}_i$$

$${\mathbb{G}}_m\to {\mathbb{G}}_m,z\mapsto z^{-i}$$

$${\mathrm{fil}}^i(V):=\underset{j\ge i}{⨁}{V}_j$$

$$\dots \subseteq {\mathrm{HN}}^{i+1}(\mathcal{E})\subseteq {\mathrm{HN}}^i(\mathcal{E})\subseteq \dots \subseteq \mathcal{E}.$$

$${\mathrm{HN}}^i(\mathcal{E}):=\mathcal{E}({\mathrm{fil}}^i(V)).$$

$$\mathrm{HN}:{\mathrm{Bun}}_{{\mathbb{P}}_k^1}\to {\mathrm{FilBun}}_{{\mathbb{P}}_k^1}$$

$$0\to {\mathcal{O}}_{{\mathbb{P}}_k^1}(-1)\to {\mathcal{O}}_{{\mathbb{P}}_k^1}\oplus {\mathcal{O}}_{{\mathbb{P}}_k^1}\to {\mathcal{O}}_{{\mathbb{P}}_k^1}(1)\to 0$$

$$\mathrm{gr}(\mathcal{E}):=\underset{i\in \mathbb{Z}}{⨁}{F}^i\mathcal{E}/F^{i+1}\mathcal{E}$$

$$\mathrm{gr}:{\mathrm{FilBun}}_{{\mathbb{P}}_k^1}\to {\mathrm{GrBun}}_{{\mathbb{P}}_k^1}$$

$$H^0({\mathbb{P}}_k^1,{\mathcal{O}}_{{\mathbb{P}}_k^1})\cong k.$$

$${\mathrm{Rep}}_k({\mathbb{G}}_m)\stackrel{\mathcal{E}(-)}{\to }{\mathrm{Bun}}_{{\mathbb{P}}_k^1}\stackrel{\mathrm{HN}}{\to }{\mathrm{FilBun}}_{{\mathbb{P}}_k^1}\stackrel{\mathrm{gr}}{\to }{\mathrm{GrBun}}_{{\mathbb{P}}_k^1}$$

$$\mathcal{E}=\underset{i\in \mathbb{Z}}{⨁}{\mathcal{E}}^i$$

$$\mathcal{E}(-):{\mathrm{Rep}}_k({\mathbb{G}}_m)\to {\mathrm{Bun}}_{{\mathbb{P}}_k^1}$$

$$\omega :{\mathrm{Rep}}_k(G)\to {\mathrm{Bun}}_S,V\mapsto \mathcal{P}{\times }^G(V{\otimes }_k{\mathcal{O}}_S)$$

$$\omega :{\mathrm{Rep}}_k(G)\to {\mathrm{Bun}}_{{\mathbb{P}}_k^1}$$

$$\mathrm{HN}:{\mathrm{Bun}}_{{\mathbb{P}}_k^1}\to {\mathrm{FilBun}}_{{\mathbb{P}}_k^1}$$

$$\mathrm{HN}\circ \omega :{\mathrm{Rep}}_k(G)\to {\mathrm{FilBun}}_{{\mathbb{P}}_k^1}.$$

$$\omega :{\mathrm{Rep}}_k(G)\to {\mathrm{Bun}}_{{\mathbb{P}}_k^1}$$

Full text

1. Introduction

Let $k$ be a field, let $G/k$ be a reductive group and let $\mathbb{P}^{1}_{k}$ be the projective line over $k$. In this small note we present a Tannakian proof of the classification of $G$-torsors on $\mathbb{P}^{1}_{k}$, thereby reproving known results of A. Grothendieck [Gro57] and G. Harder [Har68, Satz 3.4.]. To state our main theorem we denote by

[TABLE]

the set of isomorphism classes of exact tensor functors

[TABLE]

Theorem 1.1** (cf. Theorem 3.3, Corollary 3.5).**

There exists a canonical bijection

[TABLE]

In particular, there exists a canonical bijection

[TABLE]

If $A\subseteq G$ denotes a maximal split torus, then

[TABLE]

is in bijection with the set of dominant cocharacters of $A\subseteq G$, which gives a very concrete description of the set $H^{1}_{\mathrm{Zar}}(\mathbb{P}^{1}_{k},G)$. Using pure inner forms of $G$ over $k$ one can describe similarly the whole set $H^{1}_{\mathrm{\acute{e}t}}(\mathbb{P}^{1}_{k},G)$ (cf. Lemma 3.4).

Our proof of Theorem 1.1, which originated in questions about torsors over the Fargues-Fontaine curve (cf. [Ans]), is based on the Tannakian description of $G$-torsors (cf. Lemma 3.1), the Tannakian theory of filtered fiber functors (cf. [Zie15]), the canonicity of the Harder-Narasimhan filtration (cf. Lemma 2.2) and, most importantly, the good understanding of the category $\mathrm{Bun}_{\mathbb{P}^{1}_{k}}$ of vector bundles on $\mathbb{P}^{1}_{k}$ (cf. Theorem 2.1). In particular, we use crucially the fact that

[TABLE]

for $\mathcal{E}$ a semistable vector bundle on $\mathbb{P}^{1}_{k}$ of slope $\geq 0$.

In a last section we mention applications of Theorem 1.1 to the the computation of the Brauer group of $\mathbb{P}^{1}_{k}$ (avoiding Tsen’s theorem) and to the Birkhoff-Grothendieck decomposition of $G(k((t)))$.

1.2. Acknowledgment

We want to thank Jochen Heinloth for his interest and for answering several questions.

2. Vector bundles on $\mathbb{P}^{1}_{k}$

Let $k$ be an arbitrary field. We recall, in a more canonical form, the classification of vector bundles on the projective line $\mathbb{P}^{1}_{k}$ due to A. Grothendieck (cf. [Gro57]). Let

[TABLE]

be the category of finite dimensional representations of the multiplicative group ${\mathbb{G}_{m}}$ over $k$. More concretely, the category $\mathrm{Rep}_{k}({\mathbb{G}_{m}})$ is equivalent to the Tannakian category of finite dimensional $\mathbb{Z}$-graded vector spaces over $k$.

Over $\mathbb{P}^{1}_{k}$ there is the canonical ${\mathbb{G}_{m}}$-torsor

[TABLE]

also called the “Hopf bundle”. Given a representation $V\in\mathrm{Rep}_{k}({\mathbb{G}_{m}})$ the contracted product

[TABLE]

defines a (geometric) vector bundle over $\mathbb{P}^{1}_{k}$. The well known classification of the category

[TABLE]

of vector bundles on $\mathbb{P}^{1}_{k}$ can now be phrased in the following way.

Theorem 2.1**.**

The functor

[TABLE]

is an exact, faithful tensor functor inducing a bijection on isomorphism classes.

However, the functor $\mathcal{E}(-)$ is not an equivalence. For example, by semi-simplicity of the category $\mathrm{Rep}_{k}({\mathbb{G}_{m}})$ every short exact sequence of ${\mathbb{G}_{m}}$-representations splits, but this is not true for short exact sequences of vector bundles on $\mathbb{P}^{1}_{k}$.

For $V\in\mathrm{Rep}_{k}({\mathbb{G}_{m}})$ the Harder-Narasimhan filtration of the vector bundle

[TABLE]

has a very simple description. Namely, write

[TABLE]

with ${\mathbb{G}_{m}}$ acting on $V_{i}$ by the character111The sign is explained by the fact that the standard represention $z\mapsto z$ of ${\mathbb{G}_{m}}$ is sent by $\mathcal{E}(-)$ to $\mathcal{O}_{\mathbb{P}^{1}_{k}}(-1)$ and not to $\mathcal{O}_{\mathbb{P}^{1}_{k}}(1)$.

[TABLE]

and set

[TABLE]

for $i\in\mathbb{Z}$. Then the Harder-Narasimhan filtration of $\mathcal{E}:=\mathcal{E}(V)$ is given by

[TABLE]

where

[TABLE]

Lemma 2.2**.**

Sending a vector bundle $\mathcal{E}$ to the filtered vector bundle $\mathcal{E}$ with the Harder-Narasimhan filtration $\mathrm{HN}^{\bullet}(\mathcal{E})$ defines a fully faithful tensor functor

[TABLE]

into the exact tensor category of filtered vector bundles (with filtration by locally direct summands) (cf. [Zie15, Chapter 4] for a definition of $\mathrm{FilBun}_{\mathbb{P}^{1}_{k}}$).

Proof.

This is clear from the description of the Harder-Narasimhan filtration. ∎

We remark that the functor $\mathrm{HN}$ is not exact as one sees for example by looking at the Euler sequence

[TABLE]

on $\mathbb{P}^{1}_{k}$.

Sending a filtered vector bundle $(\mathcal{E},F^{\bullet})$ to the associated graded vector bundle

[TABLE]

defines an exact tensor functor

[TABLE]

(cf. [Zie15, Chapter 4]).

The following lemma is immediate from Theorem 2.1, Lemma 2.2 and the fact that

[TABLE]

Lemma 2.3**.**

The composite functor

[TABLE]

is an equivalence of exact categories from $\mathrm{Rep}_{k}({\mathbb{G}_{m}})$ onto its essential image which consists of graded vector bundles

[TABLE]

such that each $\mathcal{E}^{i}$ is semistable of slope $i$.

3. Torsors over $\mathbb{P}^{1}_{k}$

Let $G/k$ be an arbitrary reductive group. In this section we want to classify $G$-torsors on $\mathbb{P}^{1}_{k}$ for the étale topology. For this we keep the notation from the last section. In particular, there is the functor

[TABLE]

from Theorem 2.1

In order to apply the formulations from the previous section we need a more bundle theoretic interpretation of $G$-torsors (for the étale topology). This is achieved by the Tannakian formalism (cf. [Del90])

Lemma 3.1**.**

Let $S$ be a scheme over $k$. Sending a $G$-torsor $\mathcal{P}$ over $S$ to the exact tensor functor

[TABLE]

defines an equivalence from the groupoid of $G$-torsors to the groupoid of exact tensor functors from $\mathrm{Rep}_{k}(G)$ to $\mathrm{Bun}_{S}$. The inverse equivalence sends an exact tensor functor $\omega\colon\mathrm{Rep}_{k}(G)\to\mathrm{Bun}_{S}$ the $G$-torsor $\mathrm{Isom}^{\otimes}(\omega_{\mathrm{can}},\omega)$ of isomorphisms of $\omega$ to the canonical fiber functor $\omega_{\mathrm{can}}\colon\mathrm{Rep}_{k}(G)\to\mathrm{Bun}_{S},\ V\mapsto V\otimes_{k}\mathcal{O}_{S}$.

In fact, for a general affine group scheme over $k$ one has to use the fpqc-topology in Lemma 3.1. However, as $G$ is assumed to be reductive, thus in particular smooth, a theorem of Grothendieck (cf. [Gro68, Théorème 11.7]) allows to reduce to the étale topology.

Composing an exact tensor functor

[TABLE]

with the Harder-Narasimhan functor

[TABLE]

defines a, a priori not necessarily exact, tensor functor

[TABLE]

But using Haboush’s theorem reductivity of $G$ actually implies that the composition $\mathrm{HN}\circ\omega$ is still exact.

Lemma 3.2**.**

Let

[TABLE]

be an exact tensor functor. Then the composition

[TABLE]

is still exact.

Proof.

The crucial observation is that the functors

[TABLE]

are compatible with duals, and exterior resp. symmetric products. This is clear for $\omega$ as $\omega$ is assumed to be exact and follows from Lemma 2.3 for the functor $\mathrm{HN}\circ\mathrm{gr}$. In fact, for a representation $V\in\mathrm{Rep}_{k}({\mathbb{G}_{m}})$ with associated vector bundle

[TABLE]

we can conclude

[TABLE]

by exactness of the functor $\mathcal{E}(-)$. But by Lemma 2.3

[TABLE]

is an exact tensor equivalence of $\mathrm{Rep}_{k}({\mathbb{G}_{m}})$ with a subcategory of $\mathrm{GrBun}_{\mathbb{P}^{1}_{k}}$, which implies the stated compatibility with exterior and symmetric powers. Using this the proof can proceed similarly to [DOR10, Theorem 5.3.1]. We note that for a representation $V$ of $G$ there is a canonical isomorphism

[TABLE]

from the $r$-th symmetric power $\mathrm{Sym}^{r}(V^{\vee})$ of the dual of $V$ to the dual of the module

[TABLE]

of symmetric tensors. In particular, $G$-invariant homogenous polynomials on $V$ define $G$-invariant linear forms on $\mathrm{TS}_{r}(V)^{\vee}$.

Let now $0\to V\xrightarrow{f}V^{\prime}\xrightarrow{g}V^{\prime\prime}\to 0$ be an exact sequence in $\mathrm{Rep}_{k}(G)$. We have to check that the sequence

[TABLE]

with

[TABLE]

is still exact. We claim that $\tilde{\omega}(f)$ is injective. This can be checked after taking the exterior power $\Lambda^{\dim V}$ of $f$ because $\tilde{\omega}$ commutes with exterior powers. In particular, to prove injectivity we can reduce the claim for general $f$ to the case $\dim V=1$. Tensoring with the dual of $V$ reduces further to the case the $V$ is moreover trivial. By Haboush’s theorem (cf. [Hab75]) there exists an $r>0$ and a $G$-invariant homogenous polynomial $f\in\mathrm{Sym}^{r}(V^{\vee})$ such that $f_{|V}\neq 0$. Using the above isomorphism $\mathrm{Sym}^{r}(V^{\vee})\cong\mathrm{TS}_{r}(V)^{\vee}$ this shows that there exists an $r>0$ such that the morphism

[TABLE]

splits. This implies that $\tilde{\omega}(\mathrm{TS}_{r}(f))$ splits and thus that $\tilde{\omega}(f)$ is in particular injective because $\tilde{\omega}$ commutes with the symmetric tensors $TS_{r}$ as it commutes with symmetric powers and duals.

Dualizing yields that $\tilde{\omega}(g)$ is surjective at the generic point of $\mathbb{P}^{1}_{k}$. However, the sequence

[TABLE]

lies in the essential image of the functor $\mathrm{Rep}_{k}({\mathbb{G}_{m}})\to\mathrm{GrBun}_{\mathbb{P}^{1}_{k}}$ from Lemma 2.3. In particular, we see that the cokernel of $\tilde{\omega}(g)$ cannot have torsion, i.e., that it is zero. Finally, exactness in the middle of the sequence follows because

[TABLE]

This finishes the proof. ∎

We briefly recall some results about filtered fiber functors on $\mathrm{Rep}_{k}G$ (cf. [Zie15] and [Cor]). By definition a filtered fiber functor for $\mathrm{Rep}_{k}G$ over a $k$-scheme $S$ is an exact tensor functor

[TABLE]

into the exact tensor category of filtered vector bundles (with filtration by locally direct summands) on $S$. Associated to each filtered fiber functor $\omega$ is an exact tensor functor

[TABLE]

i.e., a graded fiber functor, by mapping a filtered vector bundle to its associated graded. A splitting $\gamma$ of a filtered fiber functor $\omega$ is a graded fiber functor

[TABLE]

such that

[TABLE]

where the exact tensor functor

[TABLE]

sends a graded vector bundle

[TABLE]

to the filtered vector bundle $(\mathcal{E},\mathrm{fil}^{\bullet}\mathcal{E})$ with filtration

[TABLE]

For a scheme $f\colon S^{\prime}\to S$ over $S$ let $\omega_{S^{\prime}}$ be the base change of the filtered fiber functor $\omega$ to $S^{\prime}$, i.e., $\omega_{S^{\prime}}$ is defined as the composition

[TABLE]

which is again a filtered fiber functor. For a filtered fiber functor $\omega$ the presheaf

[TABLE]

on the category of $S$-schemes is represented by an fpqc-torsor for the affine and faithfully flat group scheme

[TABLE]

over $S$ (cf. [Zie15, Lemma 4.20]). In particular, every filtered fiber functor

[TABLE]

admits a splitting fpqc-locally on $S$. The group scheme $U(\omega)$ can be described more explicitely (cf. [Zie15, Theorem 4.40]). Namely there exists a decreasing filtration by normal subgroups

[TABLE]

for $i\geq 1$, which has the property that for $i\geq 1$ the quotient

[TABLE]

is abelian and isomorphic to

[TABLE]

We can now give a proof of our main theorem about the classification of $G$-torsors on $\mathbb{P}^{1}_{k}$. We denote for a scheme $S$ over $k$ by

[TABLE]

the groupoid of exact tensor functors $\omega\colon\mathrm{Rep}_{k}(G)\to\mathrm{Bun}_{S}$ and by

[TABLE]

its set of isomorphism classes. Similarly, we use the notations

[TABLE]

resp.

[TABLE]

for the groupoid resp. the isomorphism classes of exact tensor functors

[TABLE]

Theorem 3.3**.**

Let $G$ be a reductive group over $k$. Then the composition with $\mathcal{E}(-)$ defines faithful functor

[TABLE]

which induces a bijection

[TABLE]

on isomorphism classes.

Proof.

By Lemma 2.3 the composition

[TABLE]

is an equivalence onto its essential image. In particular, the functor

[TABLE]

is faithful and induces an injection on isomorphism classes. Thus we have to prove that every exact tensor functor

[TABLE]

factors as

[TABLE]

for some exact tensor functor

[TABLE]

Let $\tilde{\omega}:=\mathrm{HN}\circ\omega$ be the functor

[TABLE]

By Theorem 3.3 the functor $\tilde{\omega}$ is still exact, i.e., a filtered fiber functor in the terminology of [Zie15], and we can use the results recalled above. We get a $U(\tilde{\omega})$-torsor

[TABLE]

of splittings of $\tilde{\omega}$. But for the filtration

[TABLE]

the graded quotients

[TABLE]

are semistable vector bundles of slope $i\geq 1$. Hence,

[TABLE]

because

[TABLE]

by Theorem 2.1. We can conclude that

[TABLE]

hence the $U(\tilde{\omega})$-torsor

[TABLE]

is in fact trivial, i.e., there exists a splitting

[TABLE]

of $\tilde{\omega}$ already over $\mathbb{P}^{1}_{k}$. As

[TABLE]

the functor $\gamma$ takes its image in the full subcategory

[TABLE]

which by Lemma 2.3 is equivalent to the category $\mathrm{Rep}_{k}{\mathbb{G}_{m}}$ of representations of ${\mathbb{G}_{m}}$. Thus there exists an exact tensor functor

[TABLE]

such that

[TABLE]

by simply setting

[TABLE]

where

[TABLE]

is the the equivalence of Lemma 2.3. ∎

Let

[TABLE]

be the canonical fiber functor of $\mathrm{Rep}_{k}({\mathbb{G}_{m}})$ over $k$. Composing with $\omega_{\mathrm{can}}^{\mathbb{G}_{m}}$ defines a morphism

[TABLE]

of groupoids, where the right hand side denotes the groupoid of exact tensor functors

[TABLE]

which by Lemma 3.1 identifies with the groupoid of $G$-torsors on $\mathrm{Spec}(k)$. Geometrically, the morphism $\Phi$ can be identified on isomorphisms classes with the map

[TABLE]

restricting a $G$-torsor over $\mathbb{P}^{1}_{k}$ to a $G$-torsor over $\mathrm{Spec}(k)$ along a $k$-rational point $x\in\mathbb{P}^{1}_{k}(k)$.

In the following lemma we analyze the fibers of this functor $\Phi$.

Lemma 3.4**.**

Let $\omega\colon\mathrm{Rep}_{k}(G)\to\mathrm{Vec}_{k}$ be an exact tensor functor and let

[TABLE]

be the pure inner form of $G$ defined by $\omega$. Then the fiber

[TABLE]

is equivalent to the quotient groupoid

[TABLE]

of cocharacters of $H$. Moreover, passing to isomorphism classes yields a bijection

[TABLE]

Proof.

The first statement follows from the Tannakian formalism (cf. [Del90]). Namely, $\omega$ defines an equivalence

[TABLE]

and the groupoid of exact tensor functors

[TABLE]

which commute (with a given isomorphism) with the canonical fiber functors on $\mathrm{Rep}_{k}H$ resp. $\mathrm{Rep}_{k}{\mathbb{G}_{m}}$ is equivalent to the quotient groupoid

[TABLE]

with $H(k)$ acting by conjugation. Clearly, for every cocharacter

[TABLE]

the push forward

[TABLE]

is an $H$-torsor, which is locally trivial in the Zariski topology, because this is true for the Hopf bundle

[TABLE]

Let conversely $\mathcal{P}$ be an $H$-torsor over $\mathbb{P}^{1}_{k}$ which is trivial for the Zariski topology and let

[TABLE]

be the induced fiber functor (cf. Lemma 3.1). Let $x\in\mathbb{P}^{1}_{k}(k)$ be a point a $k$-rational point and let $U\subseteq\mathbb{P}^{1}_{k}$ be open subset containing $x\in U$ such that

[TABLE]

is trivial. Then the exact tensor functor

[TABLE]

is isomorphic to the trivial fiber functor. This holds then also true after restricting to $x\in U$. Let

[TABLE]

be an exact tensor functor such that

[TABLE]

We can conclude that $\varphi$ preserves the canonical fiber functors on $\mathrm{Rep}_{k}H$ resp. $\mathrm{Rep}_{k}{\mathbb{G}_{m}}$ because the composition

[TABLE]

is the canonical fiber functor. In particular, there exists a cocharacter

[TABLE]

such that $\mathcal{P}$ is obtained via pushout along $\chi$ of the Hopf bundle

[TABLE]

∎

Note that we have actually shown that a $G$-torsor $\mathcal{P}$ is already locally trivial for the Zariski topology if there exists some open $U\subseteq\mathbb{P}^{1}_{k}$ containing a $k$-rational point, such that $\mathcal{P}_{|U}$ is trivial. The classification results of Grothendieck and Harder on torsors on $\mathbb{P}^{1}_{k}$ (cf. [Gro57] resp. [Har68]) are most concretely stated in the collowing form.

Corollary 3.5**.**

Let $k$ be a field and let $G/k$ be a reductive group with maximal split subtorus $A\subseteq G$. Then there exist canonical bijections

[TABLE]

where $X_{\ast}(A)_{+}$ denotes the set of dominant cocharacters of $A\subseteq G$.

Proof.

By Lemma 3.4 it suffices to show

[TABLE]

But this follows from the fact that all maximal split tori in $G$ are conjugated over $k$ and that the set of dominant cocharacters form a system of representatives for the action of the normalizer $N_{G}(A)$ of $A$ in $G$ on the group $X_{\ast}(A)$ of cocharacters for $A$. ∎

A description of $H^{1}_{\mathrm{\acute{e}t}}(\mathbb{P}^{1}_{k},G)$, similar to the one of us, can be found in [Gil02].

4. Applications

In this section we present some applications of the classification of torsors (following (cf. [Far], which discusses analogous applications to the Fargues-Fontaine curve).

The first application is the computation of the Brauer group of $\mathbb{P}^{1}_{k}$. For this we recall the theorem of Steinberg (cf. [Ser02, Chapter 3.2.3]). If $k$ is a field of cohomological dimension $\mathrm{cd}(k)\leq 1$, then Steinberg’s theorem states that

[TABLE]

for every smooth connected affine algebraic group $G/k$. In particular, the Brauer group

[TABLE]

of such fields vanishes. For example, separably closed or finite fields are of cohomological dimension $\leq 1$.

Theorem 4.1**.**

If $k$ is of cohomological dimension $\mathrm{cd}(k)\leq 1$, then the Brauer group

[TABLE]

vanishes.

Proof.

By [Gro95, Corollaire 2.2.] there is an isomorphism

[TABLE]

of the Brauer group $\mathrm{Br}(\mathbb{P}^{1}_{k})$ parametrizing equivalence classes of Azumaya algebras over $\mathcal{O}_{\mathbb{P}^{1}_{k}}$ with the cohomological Brauer group $H^{2}_{\mathrm{\acute{e}t}}(\mathbb{P}^{1}_{k},{\mathbb{G}_{m}})$. It suffices to show that for every $n\geq 0$ the canonical map

[TABLE]

arising as a boundary map of the short exact sequence

[TABLE]

is trivial. Because $k$ is of cohomological dimension $\leq 1$, there exists using Steinberg’s theorem in the case $G=\mathrm{GL}_{n}$ or $G=\mathrm{PGL}_{n}$ and Theorem 3.3 together with Lemma 3.4 a commutative diagram

[TABLE]

It suffices to show that the top horizontal arrow, or equivalently the lower horizontal arrow, is surjective. But every cocharacter

[TABLE]

can be lifted to $\mathrm{GL}_{n}$ because for the standard torus $T\cong\mathbb{G}_{m}^{n}\subseteq\mathrm{GL}_{n}$ there is a split exact sequence

[TABLE]

on cocharacter groups where $T/{\mathbb{G}_{m}}$ is a maximal torus of $\mathrm{PGL}_{n}$. ∎

For a general field $k$, i.e., $k$ not necessarily of cohomological dimension $\leq 1$, the Brauer group of $\mathbb{P}^{1}_{k}$ is given by

[TABLE]

as can be calculated from Theorem 4.1 using the spectral sequence

[TABLE]

where $\bar{k}$ denotes a separable closure of $k$.

The next application we give is to the uniformization of $G$-torsors.

Theorem 4.2**.**

Let $k$ be a field and let $G$ be reductive group over $k$. If $x\in\mathbb{P}^{1}_{k}(k)$ is $k$-rational point, then every $G$-torsor

[TABLE]

which is locally trivial for the Zariski topology becomes trivial on $\mathbb{P}^{1}_{k}\setminus\{x\}$.

Proof.

By Corollary 3.5 we know that every such $G$-torsor $\mathcal{P}$ is isomorphic to the pushout

[TABLE]

along a cocharacter

[TABLE]

of the canonical ${\mathbb{G}_{m}}$-torsor

[TABLE]

corresponding to the line bundle $\mathcal{O}_{\mathbb{P}^{1}_{k}}(-1)$ on $\mathbb{P}^{1}_{k}$. But

[TABLE]

is trivial because $\mathbb{P}^{1}_{k}\setminus\{x\}\cong\mathbb{A}^{1}_{k}$. This shows the claim. ∎

Finally, we reprove the Birkhoff-Grothendieck decomposition of $G(k((t))$ for a reductive group $G$ over $k$ (cf. [Fal03, Lemma 4]).

Theorem 4.3**.**

Let $A\subseteq G$ be a maximal split torus in $G$. Then there exists a canonical bijection

[TABLE]

where $X_{\ast}(A)_{+}$ denotes the set of dominant cocharacters of $A\subseteq G$.

Proof.

Let $x\in\mathbb{P}^{1}_{k}(k)$ be a $k$-rational point. By Beauville-Laszlo [BL95] and Lemma 3.1 there is an injective map

[TABLE]

by glueing the trivial $G$-torsor on $\mathbb{P}^{1}_{k}\setminus\{x\}$ with the trivial $G$-torsor on the formal completion

[TABLE]

along an isomorphism on $\mathrm{Spec}(\mathrm{Frac}(\widehat{\mathcal{O}}_{\mathbb{P}^{1}_{k},x}))$. Note that $\widehat{\mathcal{O}}_{\mathbb{P}^{1}_{k},x}\cong k[[t]]$. From the remark following Lemma 3.4 we can conclude that the $G$-torsors obtained in this way are actually locally trivial for the Zariski topology. By Theorem 4.2 we can conversely see that the image of $\gamma$ contains the set $H^{1}_{\mathrm{Zar}}(\mathbb{P}^{1}_{k},G)$. Using Corollary 3.5 we can conclude that

[TABLE]

∎