Lemma 39.6.2 (047H)—The Stacks project

Processing math: 100%

Table of contents
Part 2: Schemes
Chapter 39: Groupoid Schemes
Section 39.6: Properties of group schemes
Lemma 39.6.2 (cite)

numberstags

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. $\square$

Comments (0)

There are also:

2 comment(s) on Section 39.6: Properties of group schemes

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment