Proposition 2

Let E, F be projective A-modules. Let $u\in$ Hom$^f_A(E,F)$ and $v\in$ Hom$_A(F,E)$. We have the relation

(3) Tr($v\circ u$) $=$ Tr($u\circ v$).

It suffices to prove the proposition when $u$ is of the form $\theta (x^*\otimes y)$ with $x^*\in E^*$ and $y\in F$. In this case, we have

$v\circ u=\theta_E(x^*\otimes v(y))$ and $u\circ v=\theta_F(^tv(x^*)\otimes y)$,

and therefore

Tr($v\circ u$) $=\langle v(y), x^*\rangle =\langle y,^tv(x^*)\rangle =$ Tr($u\circ v$).