## Extension of weakly and strongly F-regular rings by flat maps

##### Abstract

Throughout this paper all rings will be Noetherian of positive characteristic p. Hence
tight closure theory [HH1-4] takes a prominent place (see §2 for tight closure definitions
and terminology). The purpose of this note is to help answer the following question: if
R is weakly (resp. strongly) F-regular and φ : R → S is a flat map then under what conditions on the fibers is S weakly (resp. strongly) F-regular. This question (among many others) is raised in [HH4] in section 7. It is shown there that if φ is a flat map of local rings, S is excellent and the generic and closed fibers are regular then weak Fregularity of R implies that of S (Theorem 7.24). One of our main results weakens the hypotheses considerably.