---
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.'
author:
- Johannes Anschütz
title: A Tannakian classification of torsors on the projective line
---

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 [@grothendieck_sur_la_classification_des_fibres] and G. Harder [@harder_halbeinfache_gruppenschemata_ueber_vollstaendigen_kurven Satz 3.4.]. To state our main theorem we denote by $$\mathrm{Hom}^\otimes({\mathrm{Rep}}_k(G),{\mathrm{Rep}}_k({{\mathbb{G}_m}}))$$ the set of isomorphism classes of exact tensor functors $$\omega\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Rep}}_k({{\mathbb{G}_m}}).$$

\[theorem: main theorem introduction\] There exists a canonical bijection $$\mathrm{Hom}^\otimes({\mathrm{Rep}}_k(G),{\mathrm{Rep}}_k({{\mathbb{G}_m}}))\cong H^1_{{\mathrm{\acute{e}t}}}({\mathbb{P}}^1_k,G).$$ In particular, there exists a canonical bijection $${\mathrm{Hom}}({{\mathbb{G}_m}},G)/{G(k)}\cong H^1_{\mathrm{Zar}}({\mathbb{P}}^1_k,G).$$

If $A\subseteq G$ denotes a maximal split torus, then $${\mathrm{Hom}}({{\mathbb{G}_m}},G)/{G(k)}\cong X_\ast(A)_+$$ 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. ).

Our proof of , which originated in questions about torsors over the Fargues-Fontaine curve (cf. [@anschuetz_reductive_group_schemes_over_the_fargues_fontaine_curve]), is based on the Tannakian description of $G$-torsors (cf. ), the Tannakian theory of filtered fiber functors (cf. [@ziegler_graded_and_filtered_fiber_functors]), the canonicity of the Harder-Narasimhan filtration (cf. ) and, most importantly, the good understanding of the category ${\mathrm{Bun}}_{{\mathbb{P}}^1_k}$ of vector bundles on ${\mathbb{P}}^1_k$ (cf. ). In particular, we use crucially the fact that $$H^1_{{\mathrm{\acute{e}t}}}({\mathbb{P}}^1_k,\mathcal{E})=0$$ for $\mathcal{E}$ a semistable vector bundle on ${\mathbb{P}}^1_k$ of slope $\geq 0$.

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

Acknowledgment
--------------

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

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. [@grothendieck_sur_la_classification_des_fibres]). Let $${\mathrm{Rep}}_k({{\mathbb{G}_m}})$$ 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 $$\eta\colon \mathbb{A}^2_k\setminus\{0\}\to {\mathbb{P}}^1_k,\ (x_0,x_1)\mapsto [x_0:x_1],$$ also called the “Hopf bundle”. Given a representation $V\in {\mathrm{Rep}}_k({{\mathbb{G}_m}})$ the contracted product $$\mathcal{E}(V):=\mathbb{A}^2_k\setminus\{0\}\times^{{\mathbb{G}_m}}V\to {\mathbb{P}}^1_k$$ defines a (geometric) vector bundle over ${\mathbb{P}}^1_k$. The well known classification of the category $${\mathrm{Bun}}_{{\mathbb{P}}^1_k}$$ of vector bundles on ${\mathbb{P}}^1_k$ can now be phrased in the following way.

\[theorem: classification of vector bundles\] The functor $$\mathcal{E}(-)\colon {\mathrm{Rep}}_k({{\mathbb{G}_m}})\to {\mathrm{Bun}}_{{\mathbb{P}}^1_k}$$ 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 $$\mathcal{E}(V)$$ has a very simple description. Namely, write $$V=\bigoplus\limits_{i\in {\mathbb{Z}}} V_i$$ with ${{\mathbb{G}_m}}$ acting on $V_i$ by the character[^1] $${{\mathbb{G}_m}}\to {{\mathbb{G}_m}},\ z\mapsto z^{-i}$$ and set $$\mathrm{fil}^i(V):=\bigoplus\limits_{j\geq i} V_j$$ for $i\in {\mathbb{Z}}$. Then the Harder-Narasimhan filtration of $\mathcal{E}:=\mathcal{E}(V)$ is given by $$\ldots \subseteq \mathrm{HN}^{i+1}(\mathcal{E})\subseteq \mathrm{HN}^{i}(\mathcal{E})\subseteq\ldots \subseteq \mathcal{E}.$$ where $$\mathrm{HN}^{i}(\mathcal{E}):=\mathcal{E}(\mathrm{fil}^{i}(V)).$$

\[lemma: harder-narasimhan filtration tensor functor\] 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 $$\mathrm{HN}\colon {\mathrm{Bun}}_{{\mathbb{P}}^1_k}\to \mathrm{FilBun}_{{\mathbb{P}}^1_k}$$ into the exact tensor category of filtered vector bundles (with filtration by locally direct summands) (cf. [@ziegler_graded_and_filtered_fiber_functors Chapter 4] for a definition of $\mathrm{FilBun}_{{\mathbb{P}}^1_k}$).

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

Sending a filtered vector bundle $(\mathcal{E}, F^\bullet)$ to the associated graded vector bundle $$\mathrm{gr}(\mathcal{E}):=\bigoplus_{i\in {\mathbb{Z}}} F^i\mathcal{E}/F^{i+1}\mathcal{E}$$ defines an exact tensor functor $$\mathrm{gr}\colon \mathrm{FilBun}_{{\mathbb{P}}^1_k}\to \mathrm{GrBun}_{{\mathbb{P}}^1_k}$$ (cf. [@ziegler_graded_and_filtered_fiber_functors Chapter 4]).

The following lemma is immediate from , and the fact that $$H^0({\mathbb{P}}^1_k,\mathcal{O}_{{\mathbb{P}}^1_k})\cong k.$$

\[lemma: representations of gm as graded vector bundles\] The composite functor $${\mathrm{Rep}}_k({{\mathbb{G}_m}})\xrightarrow{\mathcal{E}(-)} {\mathrm{Bun}}_{{\mathbb{P}}^1_k}\xrightarrow{\mathrm{HN}} \mathrm{FilBun}_{{\mathbb{P}}^1_k}\xrightarrow{\mathrm{gr}} \mathrm{GrBun}_{{\mathbb{P}}^1_k}$$ is an equivalence of exact categories from ${\mathrm{Rep}}_k({{\mathbb{G}_m}})$ onto its essential image which consists of graded vector bundles $$\mathcal{E}=\bigoplus\limits_{i\in {\mathbb{Z}}} \mathcal{E}^i$$ such that each $\mathcal{E}^i$ is semistable of slope $i$.

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

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. [@deligne_categories_tannakiennes])

\[lemma: tannakian description of torsors\] Let $S$ be a scheme over $k$. Sending a $G$-torsor $\mathcal{P}$ over $S$ to the exact tensor functor $$\omega\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Bun}}_{S},\ V\mapsto \mathcal{P}\times^{G}(V\otimes_k \mathcal{O}_S)$$ 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 . However, as $G$ is assumed to be reductive, thus in particular smooth, a theorem of Grothendieck (cf. [@grothendieck_le_group_de_brauer_III Théorème 11.7]) allows to reduce to the étale topology.

Composing an exact tensor functor $$\omega\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Bun}}_{{\mathbb{P}}^1_k}$$ with the Harder-Narasimhan functor $$\mathrm{HN}\colon {\mathrm{Bun}}_{{\mathbb{P}}^1_k}\to \mathrm{FilBun}_{{\mathbb{P}}^1_k}$$ defines a, a priori not necessarily exact, tensor functor $$\mathrm{HN}\circ \omega\colon {\mathrm{Rep}}_k(G)\to \mathrm{FilBun}_{{\mathbb{P}}^1_k}.$$

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

\[lemma: composition with harder-narasimhan filtration still exact\] Let $$\omega\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Bun}}_{{\mathbb{P}}^1_k}$$ be an exact tensor functor. Then the composition $$\mathrm{HN}\circ \omega\colon {\mathrm{Rep}}_k(G)\to \mathrm{FilBun}_{{\mathbb{P}}^1_k}$$ is still exact.

The crucial observation is that the functors $$\omega,\ \mathrm{gr}\circ \mathrm{HN}$$ are compatible with duals, and exterior resp. symmetric products. This is clear for $\omega$ as $\omega$ is assumed to be exact and follows from 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 $$\mathcal{E}:=\mathcal{E}(V)$$ we can conclude $$\Lambda^r(\mathcal{E})\cong \mathcal{E}(\Lambda^r(V)) \textrm{ resp.\ } \mathrm{Sym}^r(\mathcal{E})\cong \mathcal{E}(\mathrm{Sym}^r(V))$$ by exactness of the functor $\mathcal{E}(-)$. But by $$\mathrm{gr}\circ \mathrm{HN}\circ \mathcal{E}(-)$$ 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 [@dat_orlik_rapoport_period_domains Theorem 5.3.1]. We note that for a representation $V$ of $G$ there is a canonical isomorphism $$\mathrm{Sym}^r(V^\vee)\cong \mathrm{TS}_r(V)^\vee$$ from the $r$-th symmetric power $\mathrm{Sym}^r(V^\vee)$ of the dual of $V$ to the dual of the module $$\mathrm{TS}_r(V)=(V^{\otimes r})^{S_r}\subseteq V^{\otimes r}$$ 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 $$0\to \tilde{\omega}(V)\xrightarrow{\tilde{\omega}(f)} \tilde{\omega}(V^\prime)\xrightarrow{\tilde{\omega}(g)} \tilde{\omega}(V^{\prime\prime})\to 0$$ with $$\tilde{\omega}:=\mathrm{gr}\circ \mathrm{HN}\circ \omega$$ 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. [@haboush_reductive_groups_are_geometrically_reductive]) 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 $$V\cong \mathrm{TS}_r(V)\xrightarrow{\mathrm{TS}_r(f)}\mathrm{TS}_r(V^\prime)$$ 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 $$0\to \tilde{\omega}(V)\xrightarrow{\tilde{\omega}(f)} \tilde{\omega}(V^\prime)\xrightarrow{\tilde{\omega}(g)} \tilde{\omega}(V^{\prime\prime})\to 0$$ lies in the essential image of the functor ${\mathrm{Rep}}_k({{\mathbb{G}_m}})\to \mathrm{GrBun}_{{\mathbb{P}}^1_k}$ from . 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 $$\mathrm{rk}(\tilde{\omega}(V^\prime))=\mathrm{rk}(V^\prime)=\mathrm{rk}(V)+\mathrm{rk}(V^{\prime\prime})=\mathrm{rk}(\tilde{\omega}(V))+\mathrm{rk}(\tilde{\omega}(V^{\prime\prime})).$$ This finishes the proof.

We briefly recall some results about filtered fiber functors on ${\mathrm{Rep}}_kG$ (cf. [@ziegler_graded_and_filtered_fiber_functors] and [@cornut_filtrations_and_buildings]). By definition a filtered fiber functor for ${\mathrm{Rep}}_kG$ over a $k$-scheme $S$ is an exact tensor functor $$\omega\colon {\mathrm{Rep}}_k G\to \mathrm{FilBun}_S$$ 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 $$\mathrm{gr}\circ \omega\colon {\mathrm{Rep}}_k G\to \mathrm{GrBun}_S,$$ 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 $$\gamma\colon {\mathrm{Rep}}_kG \to \mathrm{GrBun}_S$$ such that $$\omega=\mathrm{fil}\circ \gamma$$ where the exact tensor functor $$\mathrm{fil}\colon \mathrm{GrBun}_S\to \mathrm{FilBun}_S$$ sends a graded vector bundle $$\mathcal{E}=\bigoplus\limits_{i\in {\mathbb{Z}}}\mathcal{E}^i$$ to the filtered vector bundle $(\mathcal{E}, \mathrm{fil}^\bullet \mathcal{E})$ with filtration $$\mathrm{fil}^i\mathcal{E}=\bigoplus\limits_{j\geq i}\mathcal{E}^j.$$ 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 $${\mathrm{Rep}}_k G\xrightarrow{\omega} \mathrm{FilBun}_{S}\xrightarrow{f^\ast} \mathrm{FilBun}_{S^\prime},$$ which is again a filtered fiber functor. For a filtered fiber functor $\omega$ the presheaf $$\mathrm{Spl}(\omega)(S^\prime):=\{ \textrm{ set of splittings of }  \omega_{S^\prime}\}$$ on the category of $S$-schemes is represented by an fpqc-torsor for the affine and faithfully flat group scheme $$U(\omega):={\textrm{Ker}}({\mathrm{Aut}}^\otimes(\omega)\to {\mathrm{Aut}}^\otimes(\mathrm{gr}\circ \omega))$$ over $S$ (cf. [@ziegler_graded_and_filtered_fiber_functors Lemma 4.20]). In particular, every filtered fiber functor $$\omega\colon {\mathrm{Rep}}_k G\to \mathrm{FilBun}_S$$ admits a splitting fpqc-locally on $S$. The group scheme $U(\omega)$ can be described more explicitely (cf. [@ziegler_graded_and_filtered_fiber_functors Theorem 4.40]). Namely there exists a decreasing filtration by normal subgroups $$U(\omega)=U_1(\omega)\supseteq\ldots \supseteq U_i(\omega)\supseteq \ldots$$ for $i\geq 1$, which has the property that for $i\geq 1$ the quotient $$\mathrm{gr}^iU(\omega):= U_i(\omega)/U_{i+1}(\omega)$$ is abelian and isomorphic to $$\mathrm{gr}^iU(\omega)\cong \mathrm{Lie}(\mathrm{gr}^iU(\omega))\cong \mathrm{gr}^i\omega(\mathrm{Lie}(G)),\ i\geq 1.$$ 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 $$\underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_k(G), {\mathrm{Bun}}_S)$$ the groupoid of exact tensor functors $\omega\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Bun}}_S$ and by $${\mathrm{Hom}}^\otimes({\mathrm{Rep}}_k(G), {\mathrm{Bun}}_S)$$ its set of isomorphism classes. Similarly, we use the notations $$\underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_k(G), {\mathrm{Rep}}_k({{\mathbb{G}_m}}))$$ resp.$${\mathrm{Hom}}^\otimes({\mathrm{Rep}}_k(G), {\mathrm{Rep}}_k({{\mathbb{G}_m}}))$$ for the groupoid resp. the isomorphism classes of exact tensor functors $$\omega\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Rep}}_k({{\mathbb{G}_m}}).$$

\[theorem: main theorem\] Let $G$ be a reductive group over $k$. Then the composition with $\mathcal{E}(-)$ defines faithful functor $$\Phi\colon \underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_k(G),{\mathrm{Rep}}_k({{\mathbb{G}_m}}))\to \underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_k(G),{\mathrm{Bun}}_{{\mathbb{P}}^1_k})$$ which induces a bijection $${\mathrm{Hom}}^\otimes({\mathrm{Rep}}_k(G),{\mathrm{Rep}}_k({{\mathbb{G}_m}}))\cong H^1_{{\mathrm{\acute{e}t}}}({\mathbb{P}}^1_k,G).$$ on isomorphism classes.

By the composition $${\mathrm{Rep}}_k({{\mathbb{G}_m}})\xrightarrow{\mathcal{E}(-)} {\mathrm{Bun}}_{{\mathbb{P}}^1_k}\xrightarrow{\mathrm{HN}} \mathrm{FilBun}_{{\mathbb{P}}^1_k}\xrightarrow{\mathrm{gr}} \mathrm{GrBun}_{{\mathbb{P}}^1_k}$$ is an equivalence onto its essential image. In particular, the functor $$\Phi\colon \underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_k(G),{\mathrm{Rep}}_k({{\mathbb{G}_m}}))\to \underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_k(G),{\mathrm{Bun}}_{{\mathbb{P}}^1_k})$$ is faithful and induces an injection on isomorphism classes. Thus we have to prove that every exact tensor functor $$\omega\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Bun}}_{{\mathbb{P}}^1_k}$$ factors as $$\omega\cong \mathcal{E}(-)\circ \omega^\prime$$ for some exact tensor functor $$\omega^\prime\colon {\mathrm{Rep}}_k(G)\to {\mathrm{Rep}}_k({{\mathbb{G}_m}}).$$ Let $\tilde{\omega}:=\mathrm{HN}\circ \omega$ be the functor $$\tilde{\omega}\colon {\mathrm{Rep}}_k(G)\xrightarrow{\omega}{\mathrm{Bun}}_{{\mathbb{P}}^1_k}\xrightarrow{\mathrm{HN}} \mathrm{FilBun}_{{\mathbb{P}}^1_k}.$$ By the functor $\tilde{\omega}$ is still exact, i.e., a filtered fiber functor in the terminology of [@ziegler_graded_and_filtered_fiber_functors], and we can use the results recalled above. We get a $U(\tilde{\omega})$-torsor $$\mathrm{Spl}(\tilde{\omega})$$ of splittings of $\tilde{\omega}$. But for the filtration $$U(\tilde{\omega})\supseteq U_{2}(\tilde{\omega})\supseteq \ldots$$ the graded quotients $$\mathrm{gr}^iU(\tilde{\omega})\cong \mathrm{gr}^i\tilde{\omega}(\mathrm{Lie}(G))$$ are semistable vector bundles of slope $i\geq 1$. Hence, $$H^1_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_k,\mathrm{gr}^iU(\tilde{\omega}))=0$$ because $$\mathrm{gr}^iU(\tilde{\omega})\cong  \mathcal{O}_{{\mathbb{P}}^1_k}(i)^{\oplus n}$$ by . We can conclude that $$H^1_{{\mathrm{\acute{e}t}}}({\mathbb{P}}^1_k, U(\tilde{\omega}))=1,$$ hence the $U(\tilde{\omega})$-torsor $$\mathrm{Spl}(\tilde{\omega})$$ is in fact trivial, i.e., there exists a splitting $$\gamma\colon {\mathrm{Rep}}_k G\to \mathrm{GrBun}_{{\mathbb{P}}^1_k}$$ of $\tilde{\omega}$ already over ${\mathbb{P}}^1_k$. As $$\gamma\cong \mathrm{gr}\circ \tilde{\omega}$$ the functor $\gamma$ takes its image in the full subcategory $$\{\ \mathcal{E}=\bigoplus\limits_{i\in {\mathbb{Z}}} \mathcal{E}^i\in \mathrm{GrBun}_{{\mathbb{P}}^1}\ |\ \mathcal{E}^i \textrm{ semistable of slope }i\},$$ which by is equivalent to the category ${\mathrm{Rep}}_k {{\mathbb{G}_m}}$ of representations of ${{\mathbb{G}_m}}$. Thus there exists an exact tensor functor $$\omega^\prime\colon {\mathrm{Rep}}_k G\to {\mathrm{Rep}}_k {{\mathbb{G}_m}}$$ such that $$\omega\cong \mathcal{E}(-)\circ \omega^\prime,$$ by simply setting $$\omega^\prime:=\mathcal{E}_{\mathrm{gr}}(-)^{-1}\circ \mathrm{gr}\circ \tilde{\omega}$$ where $$\mathcal{E}_{\mathrm{gr}}(-)\colon {\mathrm{Rep}}_k {{\mathbb{G}_m}}\to \{\ \mathcal{E}=\bigoplus\limits_{i\in {\mathbb{Z}}} \mathcal{E}^i\in \mathrm{GrBun}_{{\mathbb{P}}^1}\ |\ \mathcal{E}^i \textrm{ semistable of slope }i\},$$ is the the equivalence of .

Let $$\omega_{\mathrm{can}}^{{\mathbb{G}_m}}\colon {\mathrm{Rep}}_k({{\mathbb{G}_m}})\to \mathrm{Vec}_k,\ V\mapsto V$$ 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 $$\Phi\colon \underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_k(G),{\mathrm{Rep}}_k({{\mathbb{G}_m}}))\to \underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_k(G),\mathrm{Vec}_k)$$ of groupoids, where the right hand side denotes the groupoid of exact tensor functors $${\mathrm{Rep}}_k(G)\to \mathrm{Vec}_k,$$ which by identifies with the groupoid of $G$-torsors on ${\mathrm{Spec}}(k)$. Geometrically, the morphism $\Phi$ can be identified on isomorphisms classes with the map $$i^\ast\colon H^1_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_k, G)\to H^1_{{\mathrm{\acute{e}t}}}({\mathrm{Spec}}(k),G)$$ 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 $\Phi$.

\[lemma: fibers are zariski torsors\] Let $\omega\colon {\mathrm{Rep}}_k(G)\to \mathrm{Vec}_k$ be an exact tensor functor and let $$H:={\mathrm{Aut}}^\otimes(\omega)$$ be the pure inner form of $G$ defined by $\omega$. Then the fiber $$\Phi^{-1}(\omega)\subseteq \underline{{\mathrm{Hom}}}^\otimes({\mathrm{Rep}}_kG,{\mathrm{Rep}}_k {{\mathbb{G}_m}})$$ is equivalent to the quotient groupoid $$[{\mathrm{Hom}}({{\mathbb{G}_m}},H)/{H(k)}]$$ of cocharacters of $H$. Moreover, passing to isomorphism classes yields a bijection $${\mathrm{Hom}}({{\mathbb{G}_m}},H)/{H(k)}\cong H^1_{\mathrm{Zar}}({\mathbb{P}}^1_k,H).$$

The first statement follows from the Tannakian formalism (cf. [@deligne_categories_tannakiennes]). Namely, $\omega$ defines an equivalence $${\mathrm{Rep}}_k(G)\cong {\mathrm{Rep}}_k(H),\ V\mapsto \omega(V)$$ and the groupoid of exact tensor functors $${\mathrm{Rep}}_k H\to {\mathrm{Rep}}_k {{\mathbb{G}_m}}$$ 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 $$[{\mathrm{Hom}}({{\mathbb{G}_m}},H)/{H(k)}].$$ with $H(k)$ acting by conjugation. Clearly, for every cocharacter $$\chi\colon {{\mathbb{G}_m}}\to H$$ the push forward $$\eta\times^{{{\mathbb{G}_m}}}H$$ is an $H$-torsor, which is locally trivial in the Zariski topology, because this is true for the Hopf bundle $$\eta\colon \mathbb{A}^2_k\setminus\{0\}\to {\mathbb{P}}^1_k.$$ Let conversely $\mathcal{P}$ be an $H$-torsor over ${\mathbb{P}}^1_k$ which is trivial for the Zariski topology and let $$\omega_{\mathcal{P}}\colon {\mathrm{Rep}}_k H\to {\mathrm{Bun}}_{{\mathbb{P}}^1_k}, V\mapsto \mathcal{P}\times^{H}(V\otimes_k \mathcal{O}_{{\mathbb{P}}^1_k})$$ be the induced fiber functor (cf. ). 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 $$\mathcal{P}_{|U}$$ is trivial. Then the exact tensor functor $${\mathrm{Rep}}_k H\xrightarrow{\omega_{\mathcal{P}}}{\mathrm{Bun}}_{{\mathbb{P}}^1_k}\xrightarrow{\mathrm{res}}{\mathrm{Bun}}_U$$ is isomorphic to the trivial fiber functor. This holds then also true after restricting to $x\in U$. Let $$\varphi\colon {\mathrm{Rep}}_k H\to {\mathrm{Rep}}_k {{\mathbb{G}_m}}$$ be an exact tensor functor such that $$\mathcal{E}(-)\circ \varphi \cong \omega_{\mathcal{P}}.$$ 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 $${\mathrm{Rep}}_k {{\mathbb{G}_m}}\xrightarrow{\mathcal{E}(-)} {\mathrm{Bun}}_{{\mathbb{P}}^1_k}\xrightarrow{\mathrm{res}}{\mathrm{Bun}}_{x}\cong \mathrm{Vec}_k$$ is the canonical fiber functor. In particular, there exists a cocharacter $$\chi\colon {{\mathbb{G}_m}}\to H$$ such that $\mathcal{P}$ is obtained via pushout along $\chi$ of the Hopf bundle $$\eta\colon \mathbb{A}^2_k\setminus\{0\}\to {\mathbb{P}}^1_k.$$

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. [@grothendieck_sur_la_classification_des_fibres] resp. [@harder_halbeinfache_gruppenschemata_ueber_vollstaendigen_kurven]) are most concretely stated in the collowing form.

\[corollary: zariski torsors\] 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 $$X_\ast(A)_+\cong \mathrm{Hom}({{\mathbb{G}_m}},G)/G(k)\cong H^1_{\mathrm{Zar}}({\mathbb{P}}^1_k,G),$$ where $X_\ast(A)_+$ denotes the set of dominant cocharacters of $A\subseteq G$.

By it suffices to show $$X_\ast(A)_+\cong \mathrm{Hom}({{\mathbb{G}_m}},G)/G(k).$$ 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 [@gille_torseurs_sur_la_droite_affine].

Applications
============

In this section we present some applications of the classification of torsors (following (cf. [@fargues_g_torseurs_en_theorie_de_hodge_p_adique], 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. [@serre_galois_cohomology Chapter 3.2.3]). If $k$ is a field of cohomological dimension $\mathrm{cd}(k)\leq 1$, then Steinberg’s theorem states that $$H^1_{{{\mathrm{\acute{e}t}}}}({\mathrm{Spec}}(k),G)=1$$ for every smooth connected affine algebraic group $G/k$. In particular, the Brauer group $$\mathrm{Br}(k)=0$$ of such fields vanishes. For example, separably closed or finite fields are of cohomological dimension $\leq 1$.

\[theorem: brauer group of p1\] If $k$ is of cohomological dimension $\mathrm{cd}(k)\leq 1$, then the Brauer group $$\mathrm{Br}({\mathbb{P}}^1_k)\cong H^2_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_k,{{\mathbb{G}_m}})=0$$ vanishes.

By [@grothendieck_le_group_de_brauer_II Corollaire 2.2.] there is an isomorphism $$\mathrm{Br}({\mathbb{P}}^1_k)\cong H^2_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_k,{{\mathbb{G}_m}})$$ 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 $$H^1_{{\mathrm{\acute{e}t}}}({\mathbb{P}}^1_k,\mathrm{PGL}_n)\to H^2_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_k,{{\mathbb{G}_m}})$$ arising as a boundary map of the short exact sequence $$1\to {{\mathbb{G}_m}}\to \mathrm{GL}_n\to \mathrm{PGL}_n\to 1$$ 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 together with a commutative diagram $$\xymatrix{
H^1_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_k, \mathrm{GL}_n)\ar[r]\ar[d]^\cong & H^1_{{\mathrm{\acute{e}t}}}({\mathbb{P}}^1_k,\mathrm{PGL}_n)\ar[d]^\cong \\
{\mathrm{Hom}}({{\mathbb{G}_m}},\mathrm{GL}_n)/{\mathrm{GL}_n(k)}\ar[r] & {\mathrm{Hom}}({{\mathbb{G}_m}},\mathrm{PGL}_n)/{\mathrm{PGL}_n(k)}.
}$$ It suffices to show that the top horizontal arrow, or equivalently the lower horizontal arrow, is surjective. But every cocharacter $$\chi\colon {{\mathbb{G}_m}}\to \mathrm{PGL}_n$$ 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 $$0\to X_\ast({{\mathbb{G}_m}})\to X_\ast(T)\to X_\ast(T/{{\mathbb{G}_m}})\to 0$$ 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 $$\mathrm{Br}({\mathrm{Spec}}(k))\cong \mathrm{Br}({\mathbb{P}}^1_k)$$ as can be calculated from using the spectral sequence $$E^{pq}_2=H^p(\mathrm{Gal}(\bar{k}/k),H^q_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_{\bar{k}},{{\mathbb{G}_m}}))\Rightarrow H^{p+q}_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_k,{{\mathbb{G}_m}})$$ where $\bar{k}$ denotes a separable closure of $k$.

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

\[theorem: uniformization\] 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 $$\mathcal{P}\in H^1_{\mathrm{Zar}}({\mathbb{P}}^1_k,G)$$ which is locally trivial for the Zariski topology becomes trivial on ${\mathbb{P}}^1_k\setminus\{x\}$.

By we know that every such $G$-torsor $\mathcal{P}$ is isomorphic to the pushout $$\mathcal{P}\cong \eta\times^{{\mathbb{G}_m}}G$$ along a cocharacter $$\chi\colon {{\mathbb{G}_m}}\to G$$ of the canonical ${{\mathbb{G}_m}}$-torsor $$\eta\colon \mathbb{A}^2_k\setminus\{0\}\to {\mathbb{P}}^1_k$$ corresponding to the line bundle $\mathcal{O}_{{\mathbb{P}}^1_k}(-1)$ on ${\mathbb{P}}^1_k$. But $$\mathcal{O}_{{\mathbb{P}}^1_k}(-1)_{|{\mathbb{P}}^1_k\setminus\{x\}}$$ 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. [@faltings_algebraic_loop_groups_and_moduli_spaces_of_bundles Lemma 4]).

\[theorem: decomposition\] Let $A\subseteq G$ be a maximal split torus in $G$. Then there exists a canonical bijection $$X_\ast(A)_{+}\cong {G(k[t^{-1}])}\backslash G(k((t)))/{G(k[[t]])},$$ where $X_\ast(A)_+$ denotes the set of dominant cocharacters of $A\subseteq G$.

Let $x\in {\mathbb{P}}^1_k(k)$ be a $k$-rational point. By Beauville-Laszlo [@beauville_laszlo_un_lemme_de_descente] and there is an injective map $$\gamma\colon {G(k[t^{-1}])}\backslash G(k((t)))/{G(k[[t]])}\to H^1_{{{\mathrm{\acute{e}t}}}}({\mathbb{P}}^1_k,G)$$ by glueing the trivial $G$-torsor on ${\mathbb{P}}^1_k\setminus\{x\}$ with the trivial $G$-torsor on the formal completion $${\mathrm{Spec}}(\widehat{\mathcal{O}}_{{\mathbb{P}}^1_k,x})$$ 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 we can conclude that the $G$-torsors obtained in this way are actually locally trivial for the Zariski topology. By we can conversely see that the image of $\gamma$ contains the set $H^1_{\mathrm{Zar}}({\mathbb{P}}^1_k,G)$. Using we can conclude that $${G(k[t^{-1}])}\backslash G(k((t)))/{G(k[[t]])}\cong H^1_{\mathrm{Zar}}({\mathbb{P}}^1_k,G)\cong X_{\ast}(A)_+.$$

[DOR10]{}

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[^1]: The 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 $\mathcal{O}_{{\mathbb{P}}^1_k}(1)$.
