Remark 4

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

Algebra, VIII, p. 192, § 11. Grothendieck Groups

We keep the assumptions and notation of Proposition 8. Let E be an A-module of finite length, and let $a_1A, . . . , a_nA$ be its invariant factors (VII, §4, No. 5, p. 20). Since E is isomorphic to $\bigoplus_{i=1}^nA/a_iA$ , we have

\varphi ([E]) =\prod_{i=1}^n\varphi ([A/a_iA]) =a_1\cdots a_nA^* \tag{10}

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

Nothing.

Cites

VII, §4, No. 5, p. 20
Proposition 8