For $A$ a (commutative) ring, $f:A\to B$ an $A$-algebra, what conditions do we need on $A$ and $B$ (and $f$) for the functor $-\otimes\_A B:A-mod\to B-mod\quad$ to be faithful (i.e. injective on $Hom$-sets)?


I can't seem to come up with anything other than the rather obvious condition that tensoring with $B$ shouldn't kill anything (or at least not too much), but this is hardly satisfying. It *seems* like it would be faithful in general, but I fail to come up with an argument as to why this should be true, and precisely when it is (if at all). Dare I beg the aid of the MO?




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Note: this is basically the same as $f^\*$ being faithful for a morphism $f:X\to Y$ of schemes, which reduces to the above. Hence the algebraic geometry tag.



What conditions are needed for $-\otimes_A B$ to be faithful?
The functor you mention is faithful if and only if the functor $-\bigotimes\_A B :A-mod\to A-mod$ is faithful, ie iff $B$ is a faithful $A$-module.


For a concrete counterexample take $f:\mathbb{Z}\to \mathbb{Q}$ and like Graham says this kills the torsion stuff.


