Let $A\to B$ be a morphism of (commutative) algebras and $M$ a $B$-module. The $A$-bilinear map $B\times M\to M$ given by $(b,m)\mapsto bm$ induces a surjective homomorphism $B\otimes\_{A}M\to M$. 


Give sufficient and necessary (or at least sufficient) conditions for this mapping to be an isomorphism of $B$-modules.



Criteria for Preservation of a Module Structure under Extension of Scalars.
[Frankild, Sather-Wagstaff, and Wiegand](http://www.math.unl.edu/~rwiegand1/ExtVanish/paper.pdf): As long as ${}\_BM$ is finitely generated, $A \longrightarrow B$ is flat local, $\mathfrak{m}\_A B = \mathfrak{m}\_B$, and the extension of residue fields is an isomorphism, this happens if and only if $\mathrm{Ext}\_A^i(B,M)$ is $A$-finitely generated for all $i \geq 1$.


