
Brief communications
Picard group of a connected affine algebraic group
V. L. Popov^{ab} ^{a} Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
^{b} National Research University Higher School of Economics, Moscow
Received: 03.04.2023
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 $\varepsilon_{G, H}\colon \operatorname{Hom}_{\rm alg}(H,\mathbb G_m)\to \operatorname{Pic}(G/H)$ is the homomorphism that maps each character $\chi\colon H\to \mathbb G_m$ to the class of the onedimensional homogeneous vector bundle over $G/H$ determined by $\chi$ (see [2], Theorem 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 homomorphism of Picard groups associated with $\varphi$ (see [3], Chap. III, § 1.2). Recall that the derived subgroup of a connected reductive algebraic group is connected and semisimple (see [1], § § 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 ${\mathcal R}_u(G)$ be its unipotent radical, let $\varrho\colon G\to G/{\mathcal R}_u(G)$ be the canonical homomorphism, let $S$ be the derived subgroup of the connected reductive group $G/{\mathcal R}_u(G)$, and let $\iota\colon S\hookrightarrow G/{\mathscr R}_u(G)$ be the identity embedding. Then the following canonical homomorphisms are isomorphisms:
$$
\begin{equation*}
\operatorname{Pic}(G)\xleftarrow{\varrho^*} \operatorname{Pic}(G/{\mathcal R}_u(G))\xrightarrow{\iota^*} \operatorname{Pic}(S)\xleftarrow{\varepsilon_{\widehat{S},\pi(S)}} \operatorname{Hom}_{\rm alg}(\pi(S),\mathbb G_m).
\end{equation*}
\notag
$$
Corollary. The group $\operatorname{Pic}(G)$ is canonically isomorphic to the group $\operatorname{Hom}_{\rm alg}(\pi(S),\mathbb G_m)$ and is noncanonically isomorphic to the group $\pi(S)$.
Example. Let $G=\operatorname{GL}_n$. Then the group ${\mathcal R}_u(G)$ is trivial, and the derived group of the group $G$ is the semisimple group $\operatorname{SL}_n$. The latter is simply connected, so the group $\pi(\operatorname{SL}_n)$ is trivial. Therefore, by the theorem, the group $\operatorname{Pic}(G)$ is trivial. This agrees with the fact that the group variety of the group $\operatorname{GL}_n$ is isomorphic to an open subset of $\mathbb A^{n^2}$.
The following lemma is used in the proof of the 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
$$
\begin{equation}
\alpha\colon X\times U \to X,\ \ (x,u)\mapsto x,\quad\textit{and}\quad \beta\colon X \to X\times U,\ \ x\mapsto (x,u_0),
\end{equation}
\tag{1}
$$
the homomorphisms $\alpha^*$ and $\beta^*$ are mutually inverse isomorphisms.
Proof of the lemma. Consider the morphisms
$$
\begin{equation}
\gamma\colon X\times \mathbb A^d \to X,\ \ (x,a)\mapsto x,\quad\text{and}\quad \delta\colon X\times U \to X\times\mathbb A^d,\ \ (x, u)\mapsto (x, u).
\end{equation}
\tag{2}
$$
It follows from (1) and (2) that the following equalities hold:
$$
\begin{equation}
\begin{aligned} \, \gamma\circ\delta\circ\beta=\operatorname{id}_X\!\!\quad\text{and}\quad \alpha\circ\beta=\operatorname{id}_X\!. \end{aligned}
\end{equation}
\tag{3}
$$
As is known, $\gamma^*$ is an isomorphism (see [4], Chap. II, Proposition 6.6 and its proof) and $\delta^*$ is a surjection (see [4], Chap. II, Proposition 6.5, (a)). From this and the lefthand equality in (3) it follows that $\beta^*$ is an isomorphism. In view of the righthand equality in (3), this shows that $\alpha^*$ also is an isomorphism, and $\alpha^*$ and $\beta^*$ are mutually inverse. $\Box$
Proof of the theorem and corollary. Since the connected affine algebraic group ${\mathcal R}_u(G)$ is unipotent, it follows from [5], Propositions 1 and 2, that
(a) the group variety of ${\mathcal R}_u(G)$ is isomorphic to an affine space;
(b) the following commutative diagram exists: 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 the lemma it follows from (a) and (4) that $\varrho^*$ is an isomorphism.
According to [6], Theorem 1, the group $G/{\mathcal R}_u(G)$ contains a torus $Z$ such that the mapping
$$
\begin{equation*}
\mu\colon S\times Z\to G/{\mathcal R}_u(G),\quad (s,z)\mapsto sz
\end{equation*}
\notag
$$
is an isomorphism of group varieties (but not of groups in general). Consider the commutative diagram 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 the lemma imply that $\iota^*$ is an isomorphism.
In view of the semisimplicity of the group $\widehat S$ the group $\operatorname{Hom}_{\rm alg}(\widehat S,\mathbb G_m)$ is trivial, and since the group $\widehat S$ is simply connected, the group $\operatorname{Pic}(\widehat S)$ is trivial (see [2], Proposition 1). According to [2], Theorem 4, it follows from this that $\varepsilon_{\widehat{S}, \pi(S)}$ is an isomorphism. This completes the proof of the theorem.
The first part of the corollary follows directly from the theorem, while the second part follows from the fact that $\pi (S)$ is a finite abelian group. $\Box$
Remark. The above theorem corrects Theorem 6 in [2]. The latter asserts that the group $\operatorname{Pic}(G)$ is isomorphic to $\pi\bigl(G/{\mathcal R}(G)\bigr)$, where ${\mathcal R}(G)$ is the solvable radical of $G$. If the group extension $1\to {\mathcal R}(G)\to G\to G/{\mathcal R}(G)\to 1$ splits, then the group $G/{\mathcal R}(G)$ is isomorphic to the derived group of $G/{\mathcal R}_u(G)$, and so the formulated assertion is true in view of the above theorem. But in general this is not the case, as the example above shows: in it the group $G/{\mathcal R}(G)$ is isomorphic to $\operatorname{PGL}_n$, and $\pi(\operatorname{PGL}_n)$ 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 $\operatorname{Pic}(G)$ 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. Gorchinskiy, whose comments led to the above proof of the lemma and the emphasis on the canonical nature of the construction (the original proof of Lemma in the preprint [7] was more geometric).



Bibliography



1. 
A. Borel, Linear algebraic groups, Grad. Texts in Math., 126, 2nd ed., SpringerVerlag, New York, 1991, xii+288 с. 
2. 
V. L. Popov, Math. USSRIzv., 8:2 (1974), 301–327 
3. 
I. R. Shafarevich, Basic algebraic geometry, v. 1, Varieties in projective space, 3rd ed., Springer, Heidelberg, 2013, xviii+310 pp. ; v. 2, Schemes and complex manifolds, 3rd ed., 2013, xiv+262 pp. 
4. 
R. Hartshorne, Algebraic geometry, Grad. Texts in Math., 52, SpringerVerlag, New York–Heidelberg, 1977, xvi+496 pp. 
5. 
A. Grothendieck, Séminaire C. Chevalley, t. 3, Anneaux de Chow et applications, 2e année, Secrétariat mathématique, Paris, 1958, Exp. No. 5, , 29 pp. 
6. 
V. L. Popov, “Group varieties and group structures”, Izv. Math., 86:5 (2022), 903–924 
7. 
V. L. Popov, Picard group of connected affine algebraic group, 2023, 3 pp., arXiv: 2302.13374v1 
Citation:
V. L. Popov, “Picard group of a connected affine algebraic group”, Uspekhi Mat. Nauk, 78:4(472) (2023), 209–210; Russian Math. Surveys, 78:4 (2023), 794–796
Linking options:
https://www.mathnet.ru/eng/rm10107https://doi.org/10.4213/rm10107e https://www.mathnet.ru/eng/rm/v78/i4/p209

Statistics & downloads: 
Abstract page:  378  Russian version PDF:  12  English version PDF:  35  Russian version HTML:  102  English version HTML:  115  References:  80  First page:  19 
