The Stacks project

39.7 Properties of group schemes over a field

In this section we collect some properties of group schemes over a field. In the case of group schemes which are (locally) algebraic over a field we can say a lot more, see Section 39.8.

Lemma 39.7.1. If $(G, m)$ is a group scheme over a field $k$, then the multiplication map $m : G \times _ k G \to G$ is open.

Proof. The multiplication map is isomorphic to the projection map $\text{pr}_0 : G \times _ k G \to G$ because the diagram

\[ \xymatrix{ G \times _ k G \ar[d]^ m \ar[rrr]_{(g, g') \mapsto (m(g, g'), g')} & & & G \times _ k G \ar[d]^{(g, g') \mapsto g} \\ G \ar[rrr]^{\text{id}} & & & G } \]

is commutative with isomorphisms as horizontal arrows. The projection is open by Morphisms, Lemma 29.23.4. $\square$

Lemma 39.7.2. If $(G, m)$ is a group scheme over a field $k$. Let $U \subset G$ open and $T \to G$ a morphism of schemes. Then the image of the composition $T \times _ k U \to G \times _ k G \to G$ is open.

Proof. For any field extension $k \subset K$ the morphism $G_ K \to G$ is open (Morphisms, Lemma 29.23.4). Every point $\xi $ of $T \times _ k U$ is the image of a morphism $(t, u) : \mathop{\mathrm{Spec}}(K) \to T \times _ k U$ for some $K$. Then the image of $T_ K \times _ K U_ K = (T \times _ k U)_ K \to G_ K$ contains the translate $t \cdot U_ K$ which is open. Combining these facts we see that the image of $T \times _ k U \to G$ contains an open neighbourhood of the image of $\xi $. Since $\xi $ was arbitrary we win. $\square$

Lemma 39.7.3. Let $G$ be a group scheme over a field. Then $G$ is a separated scheme.

Proof. Say $S = \mathop{\mathrm{Spec}}(k)$ with $k$ a field, and let $G$ be a group scheme over $S$. By Lemma 39.6.1 we have to show that $e : S \to G$ is a closed immersion. By Morphisms, Lemma 29.20.2 the image of $e : S \to G$ is a closed point of $G$. It is clear that $\mathcal{O}_ G \to e_*\mathcal{O}_ S$ is surjective, since $e_*\mathcal{O}_ S$ is a skyscraper sheaf supported at the neutral element of $G$ with value $k$. We conclude that $e$ is a closed immersion by Schemes, Lemma 26.24.2. $\square$

Lemma 39.7.4. Let $G$ be a group scheme over a field $k$. Then

  1. every local ring $\mathcal{O}_{G, g}$ of $G$ has a unique minimal prime ideal,

  2. there is exactly one irreducible component $Z$ of $G$ passing through $e$, and

  3. $Z$ is geometrically irreducible over $k$.

Proof. For any point $g \in G$ there exists a field extension $k \subset K$ and a $K$-valued point $g' \in G(K)$ mapping to $g$. If we think of $g'$ as a $K$-rational point of the group scheme $G_ K$, then we see that $\mathcal{O}_{G, g} \to \mathcal{O}_{G_ K, g'}$ is a faithfully flat local ring map (as $G_ K \to G$ is flat, and a local flat ring map is faithfully flat, see Algebra, Lemma 10.38.17). The result for $\mathcal{O}_{G_ K, g'}$ implies the result for $\mathcal{O}_{G, g}$, see Algebra, Lemma 10.29.5. Hence in order to prove (1) it suffices to prove it for $k$-rational points $g$ of $G$. In this case translation by $g$ defines an automorphism $G \to G$ which maps $e$ to $g$. Hence $\mathcal{O}_{G, g} \cong \mathcal{O}_{G, e}$. In this way we see that (2) implies (1), since irreducible components passing through $e$ correspond one to one with minimal prime ideals of $\mathcal{O}_{G, e}$.

In order to prove (2) and (3) it suffices to prove (2) when $k$ is algebraically closed. In this case, let $Z_1$, $Z_2$ be two irreducible components of $G$ passing through $e$. Since $k$ is algebraically closed the closed subscheme $Z_1 \times _ k Z_2 \subset G \times _ k G$ is irreducible too, see Varieties, Lemma 33.8.4. Hence $m(Z_1 \times _ k Z_2)$ is contained in an irreducible component of $G$. On the other hand it contains $Z_1$ and $Z_2$ since $m|_{e \times G} = \text{id}_ G$ and $m|_{G \times e} = \text{id}_ G$. We conclude $Z_1 = Z_2$ as desired. $\square$

Remark 39.7.5. Warning: The result of Lemma 39.7.4 does not mean that every irreducible component of $G/k$ is geometrically irreducible. For example the group scheme $\mu _{3, \mathbf{Q}} = \mathop{\mathrm{Spec}}(\mathbf{Q}[x]/(x^3 - 1))$ over $\mathbf{Q}$ has two irreducible components corresponding to the factorization $x^3 - 1 = (x - 1)(x^2 + x + 1)$. The first factor corresponds to the irreducible component passing through the identity, and the second irreducible component is not geometrically irreducible over $\mathop{\mathrm{Spec}}(\mathbf{Q})$.

Lemma 39.7.6. Let $G$ be a group scheme over a perfect field $k$. Then the reduction $G_{red}$ of $G$ is a closed subgroup scheme of $G$.

Proof. Omitted. Hint: Use that $G_{red} \times _ k G_{red}$ is reduced by Varieties, Lemmas 33.6.3 and 33.6.7. $\square$

Lemma 39.7.7. Let $k$ be a field. Let $\psi : G' \to G$ be a morphism of group schemes over $k$. If $\psi (G')$ is open in $G$, then $\psi (G')$ is closed in $G$.

Proof. Let $U = \psi (G') \subset G$. Let $Z = G \setminus \psi (G') = G \setminus U$ with the reduced induced closed subscheme structure. By Lemma 39.7.2 the image of

\[ Z \times _ k G' \longrightarrow Z \times _ k U \longrightarrow G \]

is open (the first arrow is surjective). On the other hand, since $\psi $ is a homomorphism of group schemes, the image of $Z \times _ k G' \to G$ is contained in $Z$ (because translation by $\psi (g')$ preserves $U$ for all points $g'$ of $G'$; small detail omitted). Hence $Z \subset G$ is an open subset (although not necessarily an open subscheme). Thus $U = \psi (G')$ is closed. $\square$

Lemma 39.7.8. Let $i : G' \to G$ be an immersion of group schemes over a field $k$. Then $i$ is a closed immersion, i.e., $i(G')$ is a closed subgroup scheme of $G$.

Proof. To show that $i$ is a closed immersion it suffices to show that $i(G')$ is a closed subset of $G$. Let $k \subset k'$ be a perfect extension of $k$. If $i(G'_{k'}) \subset G_{k'}$ is closed, then $i(G') \subset G$ is closed by Morphisms, Lemma 29.25.12 (as $G_{k'} \to G$ is flat, quasi-compact and surjective). Hence we may and do assume $k$ is perfect. We will use without further mention that products of reduced schemes over $k$ are reduced. We may replace $G'$ and $G$ by their reductions, see Lemma 39.7.6. Let $\overline{G'} \subset G$ be the closure of $i(G')$ viewed as a reduced closed subscheme. By Varieties, Lemma 33.24.1 we conclude that $\overline{G'} \times _ k \overline{G'}$ is the closure of the image of $G' \times _ k G' \to G \times _ k G$. Hence

\[ m\Big(\overline{G'} \times _ k \overline{G'}\Big) \subset \overline{G'} \]

as $m$ is continuous. It follows that $\overline{G'} \subset G$ is a (reduced) closed subgroup scheme. By Lemma 39.7.7 we see that $i(G') \subset \overline{G'}$ is also closed which implies that $i(G') = \overline{G'}$ as desired. $\square$

Lemma 39.7.9. Let $G$ be a group scheme over a field $k$. If $G$ is irreducible, then $G$ is quasi-compact.

Proof. Suppose that $k \subset K$ is a field extension. If $G_ K$ is quasi-compact, then $G$ is too as $G_ K \to G$ is surjective. By Lemma 39.7.4 we see that $G_ K$ is irreducible. Hence it suffices to prove the lemma after replacing $k$ by some extension. Choose $K$ to be an algebraically closed field extension of very large cardinality. Then by Varieties, Lemma 33.14.2, we see that $G_ K$ is a Jacobson scheme all of whose closed points have residue field equal to $K$. In other words we may assume $G$ is a Jacobson scheme all of whose closed points have residue field $k$.

Let $U \subset G$ be a nonempty affine open. Let $g \in G(k)$. Then $gU \cap U \not= \emptyset $. Hence we see that $g$ is in the image of the morphism

\[ U \times _{\mathop{\mathrm{Spec}}(k)} U \longrightarrow G, \quad (u_1, u_2) \longmapsto u_1u_2^{-1} \]

Since the image of this morphism is open (Lemma 39.7.1) we see that the image is all of $G$ (because $G$ is Jacobson and closed points are $k$-rational). Since $U$ is affine, so is $U \times _{\mathop{\mathrm{Spec}}(k)} U$. Hence $G$ is the image of a quasi-compact scheme, hence quasi-compact. $\square$

Lemma 39.7.10. Let $G$ be a group scheme over a field $k$. If $G$ is connected, then $G$ is irreducible.

Proof. By Varieties, Lemma 33.7.13 we see that $G$ is geometrically connected. If we show that $G_ K$ is irreducible for some field extension $k \subset K$, then the lemma follows. Hence we may apply Varieties, Lemma 33.14.2 to reduce to the case where $k$ is algebraically closed, $G$ is a Jacobson scheme, and all the closed points are $k$-rational.

Let $Z \subset G$ be the unique irreducible component of $G$ passing through the neutral element, see Lemma 39.7.4. Endowing $Z$ with the reduced induced closed subscheme structure, we see that $Z \times _ k Z$ is reduced and irreducible (Varieties, Lemmas 33.6.7 and 33.8.4). We conclude that $m|_{Z \times _ k Z} : Z \times _ k Z \to G$ factors through $Z$. Hence $Z$ becomes a closed subgroup scheme of $G$.

To get a contradiction, assume there exists another irreducible component $Z' \subset G$. Then $Z \cap Z' = \emptyset $ by Lemma 39.7.4. By Lemma 39.7.9 we see that $Z$ is quasi-compact. Thus we may choose a quasi-compact open $U \subset G$ with $Z \subset U$ and $U \cap Z' = \emptyset $. The image $W$ of $Z \times _ k U \to G$ is open in $G$ by Lemma 39.7.2. On the other hand, $W$ is quasi-compact as the image of a quasi-compact space. We claim that $W$ is closed. If the claim is true, then $W \subset G \setminus Z'$ is a proper open and closed subset of $G$, which contradicts the assumption that $G$ is connected.

Proof of the claim. Since $W$ is quasi-compact, we see that points in the closure of $W$ are specializations of points of $W$ (Morphisms, Lemma 29.6.5). Thus we have to show that any irreducible component $Z'' \subset G$ of $G$ which meets $W$ is contained in $W$. As $G$ is Jacobson and closed points are rational, $Z'' \cap W$ has a rational point $g \in Z''(k) \cap W(k)$ and hence $Z'' = Zg$. But $W = m(Z \times _ k W)$ by construction, so $Z'' \cap W \not= \emptyset $ implies $Z'' \subset W$. $\square$

Proposition 39.7.11. Let $G$ be a group scheme over a field $k$. There exists a canonical closed subgroup scheme $G^0 \subset G$ with the following properties

  1. $G^0 \to G$ is a flat closed immersion,

  2. $G^0 \subset G$ is the connected component of the identity,

  3. $G^0$ is geometrically irreducible, and

  4. $G^0$ is quasi-compact.

Proof. Let $G^0$ be the connected component of the identity with its canonical scheme structure (Morphisms, Definition 29.26.3). By Varieties, Lemma 33.7.13 we see that $G^0$ is geometrically connected. Thus $G^0 \times _ k G^0$ is connected (Varieties, Lemma 33.7.4). Thus $m(G^0 \times _ k G^0) \subset G^0$ set theoretically. To see that this holds scheme theoretically, note that $G^0 \times _ k G^0 \to G \times _ k G$ is a flat closed immersion. By Morphisms, Lemma 29.26.1 it follows that $G^0 \times _ k G^0$ is a closed subscheme of $(G \times _ k G) \times _{m, G} G^0$. Thus we see that $m|_{G^0 \times _ k G^0} : G^0 \times _ k G^0 \to G$ factors through $G^0$. Hence $G^0$ becomes a closed subgroup scheme of $G$. By Lemma 39.7.10 we see that $G^0$ is irreducible. By Lemma 39.7.4 we see that $G^0$ is geometrically irreducible. By Lemma 39.7.9 we see that $G^0$ is quasi-compact. $\square$

Lemma 39.7.12. Let $k$ be a field. Let $T = \mathop{\mathrm{Spec}}(A)$ where $A$ is a directed colimit of algebras which are finite products of copies of $k$. For any scheme $X$ over $k$ we have $|T \times _ k X| = |T| \times |X|$ as topological spaces.

Proof. By taking an affine open covering we reduce to the case of an affine $X$. Say $X = \mathop{\mathrm{Spec}}(B)$. Write $A = \mathop{\mathrm{colim}}\nolimits A_ i$ with $A_ i = \prod _{t \in T_ i} k$ and $T_ i$ finite. Then $T_ i = |\mathop{\mathrm{Spec}}(A_ i)|$ with the discrete topology and the transition morphisms $A_ i \to A_{i'}$ are given by set maps $T_{i'} \to T_ i$. Thus $|T| = \mathop{\mathrm{lim}}\nolimits T_ i$ as a topological space, see Limits, Lemma 32.4.6. Similarly we have

\begin{align*} |T \times _ k X| & = |\mathop{\mathrm{Spec}}(A \otimes _ k B)| \\ & = |\mathop{\mathrm{Spec}}(\mathop{\mathrm{colim}}\nolimits A_ i \otimes _ k B)| \\ & = \mathop{\mathrm{lim}}\nolimits |\mathop{\mathrm{Spec}}(A_ i \otimes _ k B)| \\ & = \mathop{\mathrm{lim}}\nolimits |\mathop{\mathrm{Spec}}(\prod \nolimits _{t \in T_ i} B)| \\ & = \mathop{\mathrm{lim}}\nolimits T_ i \times |X| \\ & = (\mathop{\mathrm{lim}}\nolimits T_ i) \times |X| \\ & = |T| \times |X| \end{align*}

by the lemma above and the fact that limits commute with limits. $\square$

The following lemma says that in fact we can put a “algebraic profinite family of points” in an affine open. We urge the reader to read Lemma 39.8.6 first.

Lemma 39.7.13. Let $k$ be an algebraically closed field. Let $G$ be a group scheme over $k$. Assume that $G$ is Jacobson and that all closed points are $k$-rational. Let $T = \mathop{\mathrm{Spec}}(A)$ where $A$ is a directed colimit of algebras which are finite products of copies of $k$. For any morphism $f : T \to G$ there exists an affine open $U \subset G$ containing $f(T)$.

Proof. Let $G^0 \subset G$ be the closed subgroup scheme found in Proposition 39.7.11. The first two paragraphs serve to reduce to the case $G = G^0$.

Observe that $T$ is a directed inverse limit of finite topological spaces (Limits, Lemma 32.4.6), hence profinite as a topological space (Topology, Definition 5.22.1). Let $W \subset G$ be a quasi-compact open containing the image of $T \to G$. After replacing $W$ by the image of $G^0 \times W \to G \times G \to G$ we may assume that $W$ is invariant under the action of left translation by $G^0$, see Lemma 39.7.2. Consider the composition

\[ \psi = \pi \circ f : T \xrightarrow {f} W \xrightarrow {\pi } \pi _0(W) \]

The space $\pi _0(W)$ is profinite (Topology, Lemma 5.23.9 and Properties, Lemma 28.2.4). Let $F_\xi \subset T$ be the fibre of $T \to \pi _0(W)$ over $\xi \in \pi _0(W)$. Assume that for all $\xi $ we can find an affine open $U_\xi \subset W$ with $F \subset U$. Since $\psi : T \to \pi _0(W)$ is proper as a map of topological spaces (Topology, Lemma 5.17.7), we can find a quasi-compact open $V_\xi \subset \pi _0(W)$ such that $\psi ^{-1}(V_\xi ) \subset f^{-1}(U_\xi )$ (easy topological argument omitted). After replacing $U_\xi $ by $U_\xi \cap \pi ^{-1}(V_\xi )$, which is open and closed in $U_\xi $ hence affine, we see that $U_\xi \subset \pi ^{-1}(V_\xi )$ and $U_\xi \cap T = \psi ^{-1}(V_\xi )$. By Topology, Lemma 5.22.4 we can find a finite disjoint union decomposition $\pi _0(W) = \bigcup _{i = 1, \ldots , n} V_ i$ by quasi-compact opens such that $V_ i \subset V_{\xi _ i}$ for some $i$. Then we see that

\[ f(T) \subset \bigcup \nolimits _{i = 1, \ldots , n} U_{\xi _ i} \cap \pi ^{-1}(V_ i) \]

the right hand side of which is a finite disjoint union of affines, therefore affine.

Let $Z$ be a connected component of $G$ which meets $f(T)$. Then $Z$ has a $k$-rational point $z$ (because all residue fields of the scheme $T$ are isomorphic to $k$). Hence $Z = G^0 z$. By our choice of $W$, we see that $Z \subset W$. The argument in the preceding paragraph reduces us to the problem of finding an affine open neighbourhood of $f(T) \cap Z$ in $W$. After translation by a rational point we may assume that $Z = G^0$ (details omitted). Observe that the scheme theoretic inverse image $T' = f^{-1}(G^0) \subset T$ is a closed subscheme, which has the same type. After replacing $T$ by $T'$ we may assume that $f(T) \subset G^0$. Choose an affine open neighbourhood $U \subset G$ of $e \in G$, so that in particular $U \cap G^0$ is nonempty. We will show there exists a $g \in G^0(k)$ such that $f(T) \subset g^{-1}U$. This will finish the proof as $g^{-1}U \subset W$ by the left $G^0$-invariance of $W$.

The arguments in the preceding two paragraphs allow us to pass to $G^0$ and reduce the problem to the following: Assume $G$ is irreducible and $U \subset G$ an affine open neighbourhood of $e$. Show that $f(T) \subset g^{-1}U$ for some $g \in G(k)$. Consider the morphism

\[ U \times _ k T \longrightarrow G \times _ k T,\quad (t, u) \longrightarrow (uf(t)^{-1}, t) \]

which is an open immersion (because the extension of this morphism to $G \times _ k T \to G \times _ k T$ is an isomorphism). By our assumption on $T$ we see that we have $|U \times _ k T| = |U| \times |T|$ and similarly for $G \times _ k T$, see Lemma 39.7.12. Hence the image of the displayed open immersion is a finite union of boxes $\bigcup _{i = 1, \ldots , n} U_ i \times V_ i$ with $V_ i \subset T$ and $U_ i \subset G$ quasi-compact open. This means that the possible opens $Uf(t)^{-1}$, $t \in T$ are finite in number, say $Uf(t_1)^{-1}, \ldots , Uf(t_ r)^{-1}$. Since $G$ is irreducible the intersection

\[ Uf(t_1)^{-1} \cap \ldots \cap Uf(t_ r)^{-1} \]

is nonempty and since $G$ is Jacobson with closed points $k$-rational, we can choose a $k$-valued point $g \in G(k)$ of this intersection. Then we see that $g \in Uf(t)^{-1}$ for all $t \in T$ which means that $f(t) \in g^{-1}U$ as desired. $\square$

Remark 39.7.14. If $G$ is a group scheme over a field, is there always a quasi-compact open and closed subgroup scheme? By Proposition 39.7.11 this question is only interesting if $G$ has infinitely many connected components (geometrically).

Lemma 39.7.15. Let $G$ be a group scheme over a field. There exists an open and closed subscheme $G' \subset G$ which is a countable union of affines.

Proof. Let $e \in U(k)$ be a quasi-compact open neighbourhood of the identity element. By replacing $U$ by $U \cap i(U)$ we may assume that $U$ is invariant under the inverse map. As $G$ is separated this is still a quasi-compact set. Set

\[ G' = \bigcup \nolimits _{n \geq 1} m_ n(U \times _ k \ldots \times _ k U) \]

where $m_ n : G \times _ k \ldots \times _ k G \to G$ is the $n$-slot multiplication map $(g_1, \ldots , g_ n) \mapsto m(m(\ldots (m(g_1, g_2), g_3), \ldots ), g_ n)$. Each of these maps are open (see Lemma 39.7.1) hence $G'$ is an open subgroup scheme. By Lemma 39.7.7 it is also a closed subgroup scheme. $\square$


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