Applications of pseudocanonical covers to tight closure problems

It is well known that nice conditions on the canonical module of a ring have a strong impact in the study of strong F-regularity and F-purity. For example, in the Gorenstein (or even image-Gorenstein) case strong F-regularity is equivalent to weak F-regularity, and it is conjectured that this is true in general. In this note, we will show how to use the double cover of a ring to obtain sufficient conditions for strong F-regularity and F-purity. Our results involve a closure operation for a pair of ideals that could be of relevance for the conjecture mentioned above.