Proposition 18

Tag 05T9, permanent. Cite it as [Bourbaki, Tag 05T9].

Let $\varphi$ be a homomorphism of a field $K$ into a ring $A$ . For every exact sequence of vector $K$ -spaces and $K$ -linear mappings

E' \xrightarrow{u} E \xrightarrow{v} E''

the sequence

E'_{(A)} \xrightarrow{u_{(A)}} E_{(A)} \xrightarrow{v_{(A)}} E''_{(A)}

is exact.

This is a special case of no. 7, Proposition 14, taking account of § 1, no. 4, Remark 4.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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