Proof for isomorphism $\text{Hom}_A(V,W)\cong V^*\otimes_A W$

Question

Let $A$ be a f.d. semisimple associative algebra over $\mathbb{C}$ with identity element. Let $V,W$ be two f.d. representation (left modules) of $A$. The dual representation (right module) $V^*$ of $V$ is defined such that $\langle f\cdot a,v\rangle=\langle f,a\cdot v\rangle$ for any $f\in V^*,v\in V,a\in A$. Denote by $V^*\otimes_A W$ the quotient space $V^*\otimes W/\text{span}\{(f\cdot a)\otimes w-f\otimes (a\cdot v):f\in V^*,v\in V,a\in A\}$. Could you give a proof for $\text{Hom}_A(V,W)\cong V^*\otimes_A W$? Thanks.

Answer 1

It is not difficult to show that if $P$ is a finitely generated projective left module over a ring $A$ then for all left $A$ modules $Q$ there is an isomorphism $\phi:P^*\otimes_AQ\to\hom_A(P,Q)$ such that $\phi(f\otimes q)(p)=f(p)q$. This is proved in most textbooks that deal with all the terms involved (Here $P^*$ denotes $\hom_A(P,A)$, with its usual right $A$-module structure)

Now if $A$ is semisimple, then all its modules are projective. If it is moreover finite-dimensional, a module is finitely generated iff it is finite dimensional. What you want is therefore just a special case of the general statement above.