Lemma 79.9.6 (06E2)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 79: More on Groupoids in Spaces
Section 79.9: Properties of groups over fields and groupoids on fields
Lemma 79.9.6 (cite)

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Lemma 79.9.6. In Situation 79.9.2. For any point $r \in |R|$ there exist

a field extension $k'/k$ with $k'$ algebraically closed,
a point $r' : \mathop{\mathrm{Spec}}(k') \to R'$ where $(U', R', s', t', c')$ is the restriction of $(U, R, s, t, c)$ via $\mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k)$

such that

the point $r'$ maps to $r$ under the morphism $R' \to R$ , and
the maps $s' \circ r', t' \circ r' : \mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k')$ are automorphisms.

Proof. Let's represent $r$ by a morphism $r : \mathop{\mathrm{Spec}}(K) \to R$ for some field $K$ . To prove the lemma we have to find an algebraically closed field $k'$ and a commutative diagram

$\xymatrix{ k' & k' \ar[l]^1 & \\ k' \ar[u]^\tau & K \ar[lu]^\sigma & k \ar[l]^-s \ar[lu]_ i \\ & k \ar[lu]^ i \ar[u]_ t }$

where $s, t : k \to K$ are the field maps coming from $s \circ r$ and $t \circ r$ . In the proof of More on Groupoids, Lemma 40.10.5 it is shown how to construct such a diagram. $\square$

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