Inertial coefficient rings and the idempotent lifting property

Abstract:

A commutative ring $R$ with identity is called an inertial coefficient ring if every finitely generated $R$-algebra $A$ with $A/N$ separable over $R$ contains a separable $R$-subalgebra $S$ of $A$ such that $A = S + N$, where $N$ is the Jacobson radical of $A$. We say $A$ has the idempotent lifting property if every idempotent in $A/N$ is the image of an idempotent in $A$. Our main theorem is that any finitely generated algebra over an inertial coefficient ring has the idempotent lifting property.