F-splitting and F-purity from commutative algebra viewpoint

Question

First I define two terms: Let $R$ be a commutative ring with identity,let char$R$ = $p$, let $F:R\rightarrow R$ be the Frobenius ring homomorphism. This makes $R$ into an $R$-module with respect to the action $r.s := F(r)s=r^p s.$ We say that $R$ is F-split if there exists $G:R\rightarrow R$ such that $Go F = Id.$ R is called F-pure if $F\otimes Id_E:R\otimes E\rightarrow R\otimes E$ is injective $\forall$R-modules E.

I want to learn about F-Splitting and F-purity for my commutative algebra presentation. However, I have searched the internet and everywhere they start the discussion with schemes ,ideal sheaves etc. I want to avoid schemes and stuff and focus solely on commutative algebra aspect. Can somebody suggest good reference for these things, where the discussion is purely in terms of commutative algebra and not Algebraic Geometry? I need it urgently!

Thanks in advance.

Answer 1

This new wonderful note, F-singularities: a commutative algebra approach, written by Linquan Ma and Thomas Polstra, two card-carrying commutative algebraists, is perhaps what you need. From the Introduction: "Our approach is completely algebraic, and most of the results are stated and proved in the greatest general form."

$F$-split and $F$-pure rings are discussed starting from Section 2. (of course, there are other sources out there since the beginning of tight closure theory but this looks more convenient).