Proposition 20 ("Cayley–Hamilton theorem")

Tag 0651, permanent. Cite it as [Bourbaki, Tag 0651].

For every endomorphism $u$ of a finite-dimensional free $A$ -module, $\chi_u(u) = 0$ .

In the notation of Proposition 18 ( $§ 3$ , no. 10), for all $x \in M$ , $\chi_u(u)(x)$ is the image under $\phi$ of $\chi_u(X) \otimes x$ . But if $v$ is the endomorphism of $M[X]$ , the co-transpose of $X - \bar{u}$ (no. 6), then

\chi_u(X) \otimes x = \chi_u(X)(1 \otimes x) = (X - \bar{u})(v(1 \otimes x))

and the conclusion follows from Proposition 18 of no. 10.

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

Cited by

Proposition 4, § 9
Corollary 2, § 17

Cites

Proposition 18