Definition of $\text{Hom}^{G}(V_{1} , V_{2})$

Question

Suppose that $V_{1} , V_{2}$ are $k[G]$ modules for a group $G$ and a field $k$. What does it mean for $\phi \in \text{Hom}^{G}(V_{1},V_{2})$? In the book I am reading, it says it is a $k[G]$ homomorphism of $V_{1}$ to $V_{2}$. So is it simply a $k[G]$ module homomorphism from $V_{1}$ to $V_{2}$?

Answer 1

Yes, $\phi$ is just a map of $k[G]$ modules. It's a very common abbreviation to call a morphism of $R$-modules an $R$ homomorphism, or an $R$ linear map.

In this specific case, a $k[G]$ morphism is equivalently just a $k$ linear map $V_1 \longrightarrow V_2$ such that $\phi(g v) = g \phi(v)$ for $g \in G$ and $v \in V_1$.