"$V$ is a finite dimensional vector space over number field $K$ and $L/K$ a finite extension. $M\subset V$ is a lattice. Then $M\otimes_{O_K} O_L\to V\otimes_K L$ is injection." $O_L,O_K$ are the number rings associated to number fields $L$ and $K$ respectively.
I could think $M=O_K^n$. Then $M\otimes_{O_K}O_L=(O_L)^n$ and this embeds well into $L^n=V\otimes_KL$ by $M$ lattice in $V$. Hence injectivity of the map follows.
$\textbf{Q:}$ Can I think $M\otimes_{O_K} O_L\to V\otimes_K L$ as $M\otimes_{O_K} O_L\to V\otimes_{O_K} O_L\to V\otimes_K L$ composition? Is $(-)\otimes_{O_K}O_L$ an exact functor here?
$\textbf{Q':} $ How should I think about the last step $V\otimes_{O_K} O_L\to V\otimes_K L$ formally as tensor product. This is surely induced from $V\to V$ and $O_L\to L$ map. However I have to extend the ring of coefficients here.
Ref: Algebraic Number Theory by Taylor Frolich