Proposition 1

Let E be a free A-module. Let $(e_i)_{i\in I}$ be a basis of E, and let $(e^*_i)_{i\in I}$ be its dual basis. Let $u\in$ End$^f_A(E)$. The family $(\langle u(e_i), e^*_i\rangle )_{i\in I}$ has finite support, and its sum is equal to Tr($u$).

It suffices to treat the case when $u$ is of the form $\theta_E(x^*\otimes x)$ with $x\in E$ and $x^*\in E^*$. The family $(\langle x, e^*_i\rangle )_{i\in I}$ then has finite support, and we have $x=\sum_{i\in I}\langle x, e^*_i\rangle e_i$. Consequently, the family $(\langle x, e^*_i\rangle \langle e_i, x^*\rangle )_{i\in I}$ also has finite support, and we have $\langle x, x^*\rangle =\sum_{i\in I}\langle x, e^*_i\rangle \langle e_i, x^*\rangle$. Now, we have $\langle u(e_i), e^*_i\rangle =\langle x, e^*_i\rangle \langle e_i, x^*\rangle$ for every $i\in I$. This proves the proposition.