TITLE: Projective modules and ring homomorphisms.
QUESTION [6 upvotes]: Let $\varphi:R\rightarrow S$ be a ring homomorphism between commutative rings with unity. If $P$ is a projective $R$-module, is its extension $P\otimes_RS$ a projective $S$-module?
I tried shows that the functor $Hom_S(P\otimes_RS,\_)$ is exact, but I didn't know as find a homomorphism which the induced homomorphism carries it on a fixed homomorphism.

REPLY [3 votes]: Extension of scalars is left-adjoint to restriction of scalars. Restriction of scalars preserves epimorphisms. It follows from abstract nonsense that extension of scalars sends projective objects to projective objects.