Picard group of connected affine algebraic group

TL;DR

This paper establishes a precise relationship between the Picard group of a connected affine algebraic group and the fundamental group of the derived subgroup of its reductive quotient, linking algebraic and topological invariants.

Contribution

It proves an isomorphism between the Picard group of a connected affine algebraic group and the fundamental group of a related reductive subgroup, clarifying their structural connection.

Findings

Picard group is isomorphic to the fundamental group of the derived subgroup

Provides a new link between algebraic and topological invariants of algebraic groups

Clarifies the structure of the Picard group for connected affine algebraic groups

Abstract

We prove that the Picard group of a connected affine algebraic group $G$ is isomorphic to the fundamental group of the derived subgroup of the reductive algebraic group $G/{\mathscr R}_u(G)$, where ${\mathscr R}_u(G)$ is the unipotent radical of $G$.

Equations

$$\mathrm{Pic}(G)\stackrel{{\varrho }^{\ast }}{\leftarrow }\mathrm{Pic}(G/{\mathcal{R}}_u(G))\stackrel{{\iota }^{\ast }}{\to }\mathrm{Pic}(S)\stackrel{{\epsilon }_{\hat{S},\pi (S)}}{\leftarrow }{\mathrm{Hom}}_{\mathrm{alg}}(\pi (S),{\mathbb{G}}_m).$$

$$\begin{array}{rl}\alpha :X\times U& \to X,(x,u)\mapsto x,\\ \beta :X& \to X\times U,x\mapsto (x,u_0),\end{array}$$

$$\begin{array}{rl}\gamma :X\times {\mathbb{A}}^d& \to X,(x,a)\mapsto x,\\ \delta :X\times U& \to X\times {\mathbb{A}}^d,(x,u)\mapsto (x,u).\end{array}$$

$$\gamma \circ \delta \circ \beta ={\mathrm{id}}_X\text{mbox}and\alpha \circ \beta ={\mathrm{id}}_X.$$

$$G\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mu :S\times Z\to G/{\mathcal{R}}_u(G),(s,z)\mapsto sz,$$

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Full text

All algebraic varieties considered below are defined over a basic algebraically closed field $k$. We follow the point of view on algebraic groups accepted in [1] and use the following notation.

If $S$ is a connected semisimple algebraic group, then $\widehat{S}$ is its universal cover, and $\pi(S)$ is the kernel of the canonical isogeny $\widehat{S}\to S$. If $G$ is a connected affine algebraic group and $H$ is its closed subgroup, then $\varepsilon_{G,H}\colon{\rm Hom}_{\rm alg}(H,\mathbb{G}_{m})\to{\rm Pic}(G/H)$ is the homomorphism that maps each character $\chi\colon H\to\mathbb{G}_{m}$ to the class of the one-dimensional homogeneous vector bundle over $G/H$ determined by $\chi$ (see [3, Thm. 4]). If $\varphi\colon X\to Y$ is a morphism of smooth irreducible algebraic varieties, then $\varphi^{*}\colon{\rm Pic}(Y)\to{\rm Pic}(X)$ is the associated with $\varphi$ homomorphism of Picard groups (see [2, Chap. III, §1, Sect. 2]). Recall that the derived subgroup of a connected reductive algebraic group is connected and semisimple (see [1, Sects. I.2.2 and II.14.2]).

The purpose of this note is to prove the following theorem.

Theorem. Let $G$ be a connected affine algebraic group, let ${\mathscr{R}}_{u}(G)$ be its unipotent radical, let $\varrho\colon G\to G/{\mathscr{R}}_{u}(G)$ be the canonical homomorphism, let $S$ be the derived subgroup of the connected reductive group $G/{\mathscr{R}}_{u}(G)$, and let $\iota\colon S\hookrightarrow G/{\mathscr{R}}_{u}(G)$ be the identical embedding. Then the following canonical homomorphisms are isomorphisms:

[TABLE]

Corollary. *The group ${\rm Pic}(G)$ is canonically isomorphic to the group ${\rm Hom}_{\rm alg}(\pi(S),\mathbb{G}_{m})$ and is noncanonically isomorphic to the group $\pi(S)$. *

Example. Let $G={\rm GL}_{n}$. Then the group ${\mathscr{R}}_{u}(G)$ is trivial, and the derived group of the group $G$ is the semisimple group ${\rm SL}_{n}$. The latter is simply connected, so the group $\pi({\rm SL}_{n})$ is trivial. Therefore, by Theorem, the group ${\rm Pic}(G)$ is trivial. This agrees with the fact that the group variety of the group ${\rm GL}_{n}$ is isomorphic to an open subset of $\mathbb{A}^{n^{2}}$.

The following lemma is used in the proof of Theorem.

Lemma. Let $X$ be an irreducible smooth algebraic variety, let $U$ be a nonempty open subset of $\mathbb{A}^{d}$, and let $u_{0}$ be a point of $U$. Then for the morphisms

[TABLE]

*the homomorphisms $\alpha^{*}$ and $\beta^{*}$ are mutually inverse isomorphisms. *

Proof of Lemma. Consider the morphisms

[TABLE]

It follows from (1) and (2) that the following equalities hold:

[TABLE]

As is known, $\gamma^{*}$ is an isomorphism (see [6, Chap. II, Prop. 6.6 and its proof]), and $\delta^{*}$ is a surjection (see [6, Chap. II, Prop. 6.5(a)]). From this and the left equality in (3) it followes that $\beta^{*}$ is an isomorphism. In view of the right equality in (3), this shows that $\alpha^{*}$ is also an isomorphism, and $\alpha^{*}$ and $\beta^{*}$ are mutually inverse.

Proof of Theorem and Corollary. Since the connected affine algebraic group ${\mathscr{R}}_{u}(G)$ is unipotent, it follows from [7, Props. 1, 2] that

(a) the group variety of ${\mathscr{R}}_{u}(G)$ is isomorphic to an affine space,

(b) there is the commutative diagram

[TABLE]

where $\tau$ is an isomorphism of group varieties (but, generally speaking, not of groups) and $\upsilon$ is the natural projection onto the second factor.

In view of Lemma, it follows from (a) and (4) that $\varrho^{*}$ is an isomorphism.

According to [4, Thm. 1], in the group $G/{\mathscr{R}}_{u}(G)$ there exists a torus $Z$ such that the mapping

[TABLE]

is an isomorphism of group varieties (but, in general, not of groups). Consider the commutative diagram

[TABLE]

in which $\nu\colon S\to S\times Z$,  $s\mapsto(s,e)$, where $e$ is the identity element. Since the group variety of the torus $Z$ is isomorphic to an open subset of the affine space, (5) and Lemma imply that $\iota^{*}$ is an isomorphism.

In view of the semisimplicity of the group $\widehat{S}$ the group ${\rm Hom}_{\rm alg}(\widehat{S},\mathbb{G}_{m})$ is trivial, and since the group $\widehat{S}$ is simply connected, the group ${\rm Pic}(\widehat{S})$ is trivial (see [3, Prop. 1]). According to [3, Thm. 4], it follows from this that $\varepsilon_{\widehat{S},\pi(S)}$ is an isomorphism. This completes the proof of Theorem.

The first part of Corollary follows directly from Theorem, while the second part follows from the fact that $\pi(S)$ is a finite abelian group.

Remark. The above Theorem corrects Theorem 6 of [3]. The latter asserts that the group ${\rm Pic}(G)$ is isomorphic to \pi\big{(}G/{\mathscr{R}}(G)\big{)}, where ${\mathscr{R}}(G)$ is the solvable radical of $G$. If the group extension $1\to{\mathscr{R}}(G)\to G\to G/{\mathscr{R}}(G)\to 1$ splits, then the group $G/{\mathscr{R}}(G)$ is isomorphic to the derived group of $G/{\mathscr{R}}_{u}(G)$, and so the formulated assertion is true in view of Theorem proved above. But in general this is not the case, as Example above shows: in it, the group $G/{\mathscr{R}}(G)$ is isomorphic to ${\rm PGL}_{n}$, and $\pi({\rm PGL}_{n})$ is isomorphic to the group of all $n$-th roots of $1$ in the field $k$. This latter group is nontrivial if $n$ is not a power of the characteristic of the field $k$ (however, the group ${\rm Pic}(G)$ is trivial for every $n$).

I am grateful to Shuai Wang for bringing this example to my attention; this led to writing of this note. I am also indebted to S. O. Gorchinsky whose comments led to the above proof of Lemma and emphasis on the canonical nature of the construction (the original proof of Lemma in preprint [5] was more geometric).