TITLE: Effect of extending scalars on maps of modules
QUESTION [4 upvotes]: 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.

REPLY [6 votes]: 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.
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