Proposition 1

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

Algebra, VIII, p. 184, § 11. Grothendieck Groups

Suppose that $\mathscr{C}$ is hereditary. Let E and F be A-modules and $u : E \to F$ a linear mapping. a) If E or F is of type $\mathscr{C}$ , then so is the image of u. b) If E is of type $\mathscr{C}$ , then so is the kernel of u, and we have

(2)

\varphi(E) = \varphi(\operatorname{Ker}\,u) + \varphi(\operatorname{Im}\,u).

c) If F is of type $\mathscr{C}$ , then so is the cokernel of u, and we have

(3)

\varphi(F) = \varphi(\operatorname{Im}\,u) + \varphi(\operatorname{Coker}\,u).

The proposition follows from the existence of exact sequences

0 \longrightarrow \operatorname{Ker}\,u \longrightarrow E \longrightarrow \operatorname{Im}\,u \longrightarrow 0,

0 \longrightarrow \operatorname{Im}\,u \longrightarrow F \longrightarrow \operatorname{Coker}\,u \longrightarrow 0.

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.

Cited by

Corollary, § 11

Cites

Nothing.