The Stacks project

Lemma 39.8.7. Let $k$ be a field. Let $G$ be an algebraic group scheme over $k$. Then $G$ is quasi-projective over $k$.

Proof. By Varieties, Lemma 33.15.1 we may assume that $k$ is algebraically closed. Let $G^0 \subset G$ be the connected component of $G$ as in Proposition 39.7.11. Then every other connected component of $G$ has a $k$-rational point and hence is isomorphic to $G^0$ as a scheme. Since $G$ is quasi-compact and Noetherian, there are finitely many of these connected components. Thus we reduce to the case discussed in the next paragraph.

Let $G$ be a connected algebraic group scheme over an algebraically closed field $k$. If the characteristic of $k$ is zero, then $G$ is smooth over $k$ by Lemma 39.8.2. If the characteristic of $k$ is $p > 0$, then we let $H = G_{red}$ be the reduction of $G$. By Divisors, Proposition 31.17.9 it suffices to show that $H$ has an ample invertible sheaf. (For an algebraic scheme over $k$ having an ample invertible sheaf is equivalent to being quasi-projective over $k$, see for example the very general More on Morphisms, Lemma 37.45.1.) By Lemma 39.7.6 we see that $H$ is a group scheme over $k$. By Lemma 39.8.4 we see that $H$ is smooth over $k$. This reduces us to the situation discussed in the next paragraph.

Let $G$ be a quasi-compact irreducible smooth group scheme over an algebraically closed field $k$. Observe that the local rings of $G$ are regular and hence UFDs (Varieties, Lemma 33.25.3 and More on Algebra, Lemma 15.118.2). The complement of a nonempty affine open of $G$ is the support of an effective Cartier divisor $D$. This follows from Divisors, Lemma 31.16.6. (Observe that $G$ is separated by Lemma 39.7.3.) We conclude there exists an effective Cartier divisor $D \subset G$ such that $G \setminus D$ is affine. We will use below that for any $n \geq 1$ and $g_1, \ldots , g_ n \in G(k)$ the complement $G \setminus \bigcup D g_ i$ is affine. Namely, it is the intersection of the affine opens $G \setminus Dg_ i \cong G \setminus D$ in the separated scheme $G$.

We may choose the top row of the diagram

\[ \xymatrix{ G & U \ar[l]_ j \ar[r]^\pi & \mathbf{A}^ d_ k \\ & W \ar[r]^{\pi '} \ar[u] & V \ar[u] } \]

such that $U \not= \emptyset $, $j : U \to G$ is an open immersion, and $\pi $ is ├ętale, see Morphisms, Lemma 29.36.20. There is a nonempty affine open $V \subset \mathbf{A}^ d_ k$ such that with $W = \pi ^{-1}(V)$ the morphism $\pi ' = \pi |_ W : W \to V$ is finite ├ętale. In particular $\pi '$ is finite locally free, say of degree $n$. Consider the effective Cartier divisor

\[ \mathcal{D} = \{ (g, w) \mid m(g, j(w)) \in D\} \subset G \times W \]

(This is the restriction to $G \times W$ of the pullback of $D \subset G$ under the flat morphism $m : G \times G \to G$.) Consider the closed subset1 $T = (1 \times \pi ')(\mathcal{D}) \subset G \times V$. Since $\pi '$ is finite locally free, every irreducible component of $T$ has codimension $1$ in $G \times V$. Since $G \times V$ is smooth over $k$ we conclude these components are effective Cartier divisors (Divisors, Lemma 31.15.7 and lemmas cited above) and hence $T$ is the support of an effective Cartier divisor $E$ in $G \times V$. If $v \in V(k)$, then $(\pi ')^{-1}(v) = \{ w_1, \ldots , w_ n\} \subset W(k)$ and we see that

\[ E_ v = \bigcup \nolimits _{i = 1, \ldots , n} D j(w_ i)^{-1} \]

in $G$ set theoretically. In particular we see that $G \setminus E_ v$ is affine open (see above). Moreover, if $g \in G(k)$, then there exists a $v \in V$ such that $g \not\in E_ v$. Namely, the set $W'$ of $w \in W$ such that $g \not\in Dj(w)^{-1}$ is nonempty open and it suffices to pick $v$ such that the fibre of $W' \to V$ over $v$ has $n$ elements.

Consider the invertible sheaf $\mathcal{M} = \mathcal{O}_{G \times V}(E)$ on $G \times V$. By Varieties, Lemma 33.30.5 the isomorphism class $\mathcal{L}$ of the restriction $\mathcal{M}_ v = \mathcal{O}_ G(E_ v)$ is independent of $v \in V(k)$. On the other hand, for every $g \in G(k)$ we can find a $v$ such that $g \not\in E_ v$ and such that $G \setminus E_ v$ is affine. Thus the canonical section (Divisors, Definition 31.14.1) of $\mathcal{O}_ G(E_ v)$ corresponds to a section $s_ v$ of $\mathcal{L}$ which does not vanish at $g$ and such that $G_{s_ v}$ is affine. This means that $\mathcal{L}$ is ample by definition (Properties, Definition 28.26.1). $\square$

[1] Using the material in Divisors, Section 31.17 we could take as effective Cartier divisor $E$ the norm of the effective Cartier divisor $\mathcal{D}$ along the finite locally free morphism $1 \times \pi '$ bypassing some of the arguments.

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