Section 39.6 (045W): Properties of group schemes

39.6 Properties of group schemes

In this section we collect some simple properties of group schemes which hold over any base.

Lemma 39.6.1. Let $S$ be a scheme. Let $G$ be a group scheme over $S$. Then $G \to S$ is separated (resp. quasi-separated) if and only if the identity morphism $e : S \to G$ is a closed immersion (resp. quasi-compact).

Proof. We recall that by Schemes, Lemma 26.21.11 we have that $e$ is an immersion which is a closed immersion (resp. quasi-compact) if $G \to S$ is separated (resp. quasi-separated). For the converse, consider the diagram

\[ \xymatrix{ G \ar[r]_-{\Delta _{G/S}} \ar[d] & G \times _ S G \ar[d]^{(g, g') \mapsto m(i(g), g')} \\ S \ar[r]^ e & G } \]

It is an exercise in the functorial point of view in algebraic geometry to show that this diagram is cartesian. In other words, we see that $\Delta _{G/S}$ is a base change of $e$. Hence if $e$ is a closed immersion (resp. quasi-compact) so is $\Delta _{G/S}$, see Schemes, Lemma 26.18.2 (resp. Schemes, Lemma 26.19.3). $\square$

Lemma 39.6.2. Let $S$ be a scheme. Let $G$ be a group scheme over $S$. Let $T$ be a scheme over $S$ and let $\psi : T \to G$ be a morphism over $S$. If $T$ is flat over $S$, then the morphism

\[ T \times _ S G \longrightarrow G, \quad (t, g) \longmapsto m(\psi (t), g) \]

is flat. In particular, if $G$ is flat over $S$, then $m : G \times _ S G \to G$ is flat.

Proof. Consider the diagram

\[ \xymatrix{ T \times _ S G \ar[rrr]_{(t, g) \mapsto (t, m(\psi (t), g))} & & & T \times _ S G \ar[r]_{\text{pr}} \ar[d] & G \ar[d] \\ & & & T \ar[r] & S } \]

The left top horizontal arrow is an isomorphism and the square is cartesian. Hence the lemma follows from Morphisms, Lemma 29.25.8. $\square$

reference

Lemma 39.6.3. Let $(G, m, e, i)$ be a group scheme over the scheme $S$. Denote $f : G \to S$ the structure morphism. Then there exist canonical isomorphisms

\[ \Omega _{G/S} \cong f^*\mathcal{C}_{S/G} \cong f^*e^*\Omega _{G/S} \]

where $\mathcal{C}_{S/G}$ denotes the conormal sheaf of the immersion $e$. In particular, if $S$ is the spectrum of a field, then $\Omega _{G/S}$ is a free $\mathcal{O}_ G$-module.

Proof. By Morphisms, Lemma 29.32.10 we have

\[ \Omega _{G \times _ S G/G} = \text{pr}_0^*\Omega _{G/S} \]

where on the left hand side we view $G \times _ S G$ as a scheme over $G$ using $\text{pr}_1$. Let $\tau : G \times _ S G \to G \times _ S G$ be the “shearing map” given by $(g, h) \mapsto (m(g, h), h)$ on points. This map is an automorphism of $G \times _ S G$ viewed as a scheme over $G$ via the projection $\text{pr}_1$. Combining these two remarks we obtain an isomorphism

\[ \tau ^*\text{pr}_0^*\Omega _{G/S} \to \text{pr}_0^*\Omega _{G/S} \]

Since $\text{pr}_0 \circ \tau = m$ this can be rewritten as an isomorphism

\[ m^*\Omega _{G/S} \to \text{pr}_0^*\Omega _{G/S} \]

Pulling back this isomorphism by $(e \circ f, \text{id}_ G) : G \to G \times _ S G$ and using that $m \circ (e \circ f, \text{id}_ G) = \text{id}_ G$ and $\text{pr}_0 \circ (e \circ f, \text{id}_ G) = e \circ f$ we obtain an isomorphism

\[ \Omega _{G/S} \to f^*e^*\Omega _{G/S} \]

as desired. By Morphisms, Lemma 29.32.16 we have $\mathcal{C}_{S/G} \cong e^*\Omega _{G/S}$. If $S$ is the spectrum of a field, then any $\mathcal{O}_ S$-module on $S$ is free and the final statement follows. $\square$

Lemma 39.6.4. Let $S$ be a scheme. Let $G$ be a group scheme over $S$. Let $s \in S$. Then the composition

\[ T_{G/S, e(s)} \oplus T_{G/S, e(s)} = T_{G \times _ S G/S, (e(s), e(s))} \rightarrow T_{G/S, e(s)} \]

is addition of tangent vectors. Here the $=$ comes from Varieties, Lemma 33.16.7 and the right arrow is induced from $m : G \times _ S G \to G$ via Varieties, Lemma 33.16.6.

Proof. We will use Varieties, Equation (33.16.3.1) and work with tangent vectors in fibres. An element $\theta $ in the first factor $T_{G_ s/s, e(s)}$ is the image of $\theta $ via the map $T_{G_ s/s, e(s)} \to T_{G_ s \times G_ s/s, (e(s), e(s))}$ coming from $(1, e) : G_ s \to G_ s \times G_ s$. Since $m \circ (1, e) = 1$ we see that $\theta $ maps to $\theta $ by functoriality. Since the map is linear we see that $(\theta _1, \theta _2)$ maps to $\theta _1 + \theta _2$. $\square$

Comments (2)

Comment #5354 by Zhenhua Wu on June 29, 2020 at 15:56

tag 047I. Sorry for spam, the input of asterisk keeps bugging. This is my last try. The assumption of flatness is not needed for the result. See Prop (3.15) of "abelian varieties" by Ben Moonen. In short, define $\tau :G\times_S G\to G\times_S G, (g,h)\mapsto (gh,h)$ which is an automorphism. Denote $p_1$ as the left projection and $p_2$ by the right one. So we have an isomorphism: $\begin{equation}\tau^{*} p_1^{*} \Omega^1_{G/S}\cong p_1^{*} \Omega^1_{G/S}\end{equation}$ Let $\psi :G\to G\times_S G,g\mapsto (e,g)$ , so we have $\begin{equation}\psi^{*}\tau^{*} p_1^{*} \Omega^1_{G/S}\cong \psi^{*}p_1^{*} \Omega^1_{G/S}\end{equation}$ $\begin{equation}(p_1\circ \tau\circ \psi)^{*} \Omega^1_{G/S}\cong (p_1\circ \psi)^{*} \Omega^1_{G/S}\end{equation}$ $\begin{equation}\text{id}\_G^{*} \Omega^1_{G/S}\cong (e\circ f)^{*} \Omega^1_{G/S}\end{equation}$ $\begin{equation}\Omega^1_{G/S}\cong f^{*} e^{*}\Omega^1_{G/S}\end{equation}$ Q.E.D.

Comment #5593 by Johan on November 12, 2020 at 19:14

Thanks very much. I added the new argument here.

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39.6 Properties of group schemes

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