F-Purity, Frobenius Splitting, and Tight Closure

Abstract

Several applications of the technique of studying when the Frobenius endomorphism from a ring of positive prime characteristic to itself splits are discussed. These include some problems that, historically, motivated the development of the theory. One of these is the theorem that rings of invariants of linearly reductive groups acting on regular rings are Cohen-Macaulay, including normal rings generated by monomials. Another is the characterization of when Stanley-Reisner rings are Cohen-Macaulay. Another is the proof of the existence of finitely generated maximal Cohen-Macaulay modules for graded rings of positive prime characteristic when the normalization has an isolated singularity (or an isolated non-Cohen-Macaulay point), which includes all three-dimensional graded domains of positive characteristic. Applications of the closely related theory of tight closure are also discussed.