The Stacks project

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. $\square$

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. Assume $f$ is flat. 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. In Morphisms, Lemma 29.32.7 we identified $\Omega _{G/S}$ with the conormal sheaf of the diagonal morphism $\Delta _{G/S}$. In the proof of Lemma 39.6.1 we showed that $\Delta _{G/S}$ is a base change of the immersion $e$ by the morphism $(g, g') \mapsto m(i(g), g')$. This morphism is isomorphic to the morphism $(g, g') \mapsto m(g, g')$ hence is flat by Lemma 39.6.2. Hence we get the first isomorphism by Morphisms, Lemma 29.31.4. 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 $G \to S$ is flat, and any $\mathcal{O}_ S$-module on $S$ is free. $\square$

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 (3)

Comment #5352 by Zhenhua Wu on

tag 047I. The assumption of flatness is not needed for the result. See Prop (3.15) of this link \ref{http://page.mi.fu-berlin.de/elenalavanda/BMoonen.pdf}Define. In short, define which is an automorphism. Denote as the left projection and by the right one. So we have an isomorphism: Let , so we have Q.E.D.

Comment #5353 by Zhenhua Wu on

tag 047I. The assumption of flatness is not needed for the result. See Prop (3.15) of this link \ref{http://page.mi.fu-berlin.de/elenalavanda/BMoonen.pdf} In short, define which is an automorphism. Denote as the left projection and by the right one. So we have an isomorphism: Let , so we have Q.E.D.

Comment #5354 by Zhenhua Wu on

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 which is an automorphism. Denote as the left projection and by the right one. So we have an isomorphism: Let , so we have Q.E.D.


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