Corollary 1

Tag 05V9, permanent. Cite it as [Bourbaki, Tag 05V9].

In the notation of Proposition 3, $\operatorname{Im}(f) = (\operatorname{Im}(f'))_{(\mathbf{K})}$ ,

\operatorname{Ker}(f') = (\operatorname{Ker}(f'))_{(\mathbf{K})}, \quad \operatorname{Coker}(f) = (\operatorname{Coker}(f'))_{(\mathbf{K})}.

In particular, for

f

to be injective (resp. surjective, zero), it is necessary and sufficient that

f'

be so. If

f

is bijective, its inverse mapping is rational over

\mathbf{K}'

This is a particular case of § 7, no. 9, Corollary to Proposition 18.

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

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