Let $k$ be a field and $R$ be a $k$-algebra. Let $M$ and $N$ be left $R$-modules. Finally, let $\ell$ be a field extension of $k$. We thus have an $\ell$-algebra $\ell \otimes R$, and both $\ell \otimes M$ and $\ell \otimes N$ are left modules over it.


**Question**: What is the relationship between the $k$-vector space $\text{Hom}\_R(M,N)$ and the $\ell$-vector space $\text{Hom}\_{\ell \otimes R}(\ell \otimes M,\ell \otimes N)$?


I am most interested in the case where $\text{char}(k)=0$ and $R = k[G]$ for a finite group $G$ and $\ell$ is a finite extension of $k$ and $M$ and $N$ are finitely generated.



Effect of extending scalars on maps of modules
We have that $\text{Hom}\_{\ell \otimes R}(\ell \otimes M,\ell \otimes N)$ is isomorphic to $\ell \otimes \text{Hom}\_R(M,N)$ as $\ell$-vector spaces for a general (possible infinite dimensional) $k$-algebra $R$ in case $M$ has finite $k$-dimension, see for example Lemma 7.4. in the book "A first course in noncommutative rings" by Lam.


