Bourbaki

Proposition 1

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

Algebra, VIII, p. 287, § 16. Other Descriptions of the Brauer Group

The following diagram commutes:

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

(2)

φu\varphi u

F ι^{\iota} // Γ π^{\pi} // G.

Moreover, if E1′=(Γ1′,ι1′,π1′)\mathscr{E}_1'= (\Gamma '_1, \iota '_1, \pi '_1) is a τ′\tau '-extension and φ1:Γ1′→Γ\varphi_1: \Gamma '_1\rightarrow \Gamma is a group homomorphism such that the diagram

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

φ1u\varphi_1u

F ι^{\iota} // Γ π^{\pi} // G

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

The commutativity of the first diagram follows from the definition of φ\varphi. The existence and uniqueness of ψ\psi follow from Lemma 2 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.