Bourbaki

Proposition 2

Tag 00IG, permanent. Cite it as [Bourbaki, Tag 00IG].

Algebra, VIII, p. 290, § 16. Other Descriptions of the Brauer Group

In the notation above, the following diagram commutes:

F ι^{\iota} // Γ π^{\pi} // G

vφ(5) v\varphi \tag{5}

F′ι′{F'}^{\iota'} // Γ′π′{\Gamma'}^{\pi'} // G .

Let E1′=(Γ1′,ι1′,π1′)\mathscr{E}_1'= (\Gamma '_1, \iota '_1, \pi_1') be a τ′\tau '-extension of G by F′F', and let φ1:Γ→Γ1′\varphi_1: \Gamma \rightarrow \Gamma '_1 be a group homomorphism such that the following diagram commutes:

F ι^{\iota} // Γ π^{\pi} // G

vφ1v\varphi_1

F′ι1′{F'}^{\iota'_1} // Γ1′π1′{\Gamma'_1}^{\pi'_1} // G.

Then there exists a unique morphism ψ\psi of τ′\tau '-extensions from v∗(E)v_*(\mathscr{E}) to E1′\mathscr{E}_1' such that we have φ1=ψ∘φ\varphi_1=\psi \circ \varphi.

The commutativity of the first diagram follows by construction. The existence and uniqueness of ψ\psi follow from Lemma 3 below.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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Transcribed from the printed edition, 2023, Springer Nature. Not translated.