I have a brief passage I don't understand. Suppose $R$ is a commutative ring, $A$ is an $R$-algebra, which is projective and finitely generated as an $R$-module. Let $M$ be a progenerator for $A$, so $M$ is projective and finitely generated as an $A$-module.
Let $\varphi\colon A\to\operatorname{End}_R(M)$ be the canonical map, and $C$ the cokernel. The composition $$ M\xrightarrow{m\mapsto 1\otimes m}\operatorname{End}_R(M)\otimes_A M\xrightarrow{f\otimes m\mapsto f(m)} M $$ is the identity. So $\varphi\otimes_A 1_M$ is a split injection of $R$-modules.
Since $\operatorname{End}_R(M)$ is a projective $R$-module and $M$ is a projective $A$-module, $C\otimes_A M$ is a projective $R$-module.
I don't understand this last sentence. I know that the tensor product of projective modules is projective, but we're looking at $C=\operatorname{End}_R(M)/\varphi(A)$, not $\operatorname{End}_R(M)$, and it's not clear that $C$ is projective?