Exercise 110.22.6. Let $A$ be a ring.

1. Suppose that $M$ is a finite locally free $A$-module, and suppose that $\varphi : M \to M$ is an endomorphism. Define/construct the trace and determinant of $\varphi$ and prove that your construction is “functorial in the triple $(A, M, \varphi )$”.

2. Show that if $M, N$ are finite locally free $A$-modules, and if $\varphi : M \to N$ and $\psi : N \to M$ then $\text{Trace}(\varphi \circ \psi ) = \text{Trace}(\psi \circ \varphi )$ and $\det (\varphi \circ \psi ) = \det (\psi \circ \varphi )$.

3. In case $M$ is finite locally free show that $\text{Trace}$ defines an $A$-linear map $\text{End}_ A(M) \to A$ and $\det$ defines a multiplicative map $\text{End}_ A(M) \to A$.

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