Proposition 3

Let E be a projective A-module.

a) Let $u\in$ End$^f_A(E)$. Let F be a projective A-submodule of E containing Im $u$, and let $u_F$ be the endomorphism of F induced by $u$. We have

(5) det(1$_E+u$) $=$ det(1$_F+u_F$).

b) Let $u$ and $v$ be two elements of End$^f_A(E)$. We have

(6) det $(1_E+u)\circ (1_E+v)$ = det(1$_E+u$) det(1$_E+v$).

Let us prove a). For every integer $p\geqslant 0$, the projective A-module $\wedge^pF$ can be identified with a submodule of $\wedge^pE$ (III, §7, No. 9, p. 520, Corollary). The image of $\wedge^pu$ is contained in $\wedge^pF$, and the endomorphism of $\wedge^pF$ induced by $\wedge^pu$ is equal to $\wedge^pu_F$. We consequently have Tr($\wedge^pu$) $=$ Tr($\wedge^pu_F$) by the corollary of Proposition 2, and therefore a).

Let us prove b). Let G be an A-module such that the A-module $L = E\oplus G$ is free. Denote by $u'$ and $v'$ the endomorphisms $u\oplus 0_G$ and $v\oplus 0_G$ of L. By a), we have the relations det(1$_L+u'$) $=$ det(1$_E+u$), det(1$_L+v'$) $=$ det(1$_E+v$), and

det(1$_L+u'+v'+u'\circ v'$) $=$ det(1$_E+u+v+u\circ v$).

It therefore suffices to prove assertion b) when the A-module E is free. There then exists a finitely generated free submodule F of E that contains the image of $u$ and that of $v$. Set $w=u+v+u\circ v$. The image of $w$ is contained in F, and we have $w_F=u_F+v_F+u_F\circ v_F$. Therefore, by (5), we have det(1$_E+u$) $=$ det(1$_F+u_F$), det(1$_E+v$) $=$ det(1$_F+v_F$), and

det$((1_E+u)\circ (1_E+v))$

= det(1$_E+w$) $=$ det(1$_F+w_F$) $=$ det((1$_F+u_F$)$\circ (1_F+v_F)$).

Since F is a finitely generated free A-module, we have

det((1$_F+u_F$)$\circ (1_F+v_F)$) $=$ det(1$_F+u_F$) det(1$_F+v_F$) $=$ det(1$_E+u$) det(1$_E+v$).