I have the following question and I don´t know how to answer it:
Let $k$ be an algebraically closed field (or just an arbitrary field), and $A$ and $B$ two $k$-algebras. Let $A$ be a $B$-module too. I want to know under what circumstances we have a bijection between the following sets: $$\operatorname{Hom}_{k-\rm{alg \hspace{0.1cm}grad}}(\operatorname{S}^{\bullet}_{B}A,k)=\operatorname{Hom}_{B-\rm{alg \hspace{0.1cm} grad}}(\operatorname{S}^{\bullet}_{B}A,B)\text{,}$$ where $\operatorname{S}^{\bullet}_{B}A$ is the symmetric algebra of $A$ as $B$-algebra. Here, $\operatorname{Hom}_{k-\rm{alg \hspace{0.1cm}grad}}$ stands for "set of homomorphisms of graded $k$-algebras".
Thank you for your time.