Strong F-regularity in Images of Regular Rings

We characterize strong F-regularity, a property associated with tight closure, in a large class of rings. A special case of our results is a workable criterion in complete intersection rings. Tight closure is a recently introduced operation linked to a variety of results in commutative algebra. Among the major results achieved with its use are a generalization of the Direct Summand Conjecture for rings containing a field [HH2] and a greatly simplified proof that invariant rings of regular rings by reductive groups are Cohen-Macaulay [HH1]. (Other applications are found in [Ho], [HH3], [HH4], and [S].) Hoping to discover additional results and to understand what makes tight closure so effective, one is led to examining tight closure itself. One way to do this is to study related concepts such as strong F-regularity. The definition of strong F-regularity is not intuitively clear. An F-finite reduced ring R of prime characteristic p is strongly F-regular if for every element c of R that is not in any minimal prime of R, the R-linear map R → R sending 1 7→ c has an R-linear retraction for all sufficiently large powers q of p [HH3]. (Here, R denotes the set of qth roots of the elements of R, regarded as an R-module in the natural way.) While its definition may seem largely technical, this concept turns out to fit nicely with a number of ring properties. Strongly F-regular rings are between regular rings and Cohen-Macaulay normal rings [HH3]. They are always F-pure and have a negative a-invariant in the graded case ([HH3],[HH2]). And, of course, they’re connected with tight closure: Every ideal in a strongly F-regular ring is tightly closed ([HH3]). What we contribute to an understanding of strong F-regularity is a characterization (Theorems 2.3 and 3.1) in a large class of rings: A homomorphic image S/I of an F-finite regular local or F-finite regular graded ring S of characteristic p with (homogeneous) maximal ideal m, assumed to have infinite residue field in the graded case, by a (homogeneous) radical ideal I is strongly F-regular if and only if s(I [p ] : I) 6⊆ m[pe] for some e ≥ 1, where s is a (homogeneous) element of S at Received by the editors March 2, 1994 and, in revised form, August 9, 1994; the contents of this paper were presented to the AMS Special Session on Commutative Noetherian Rings and Modules, San Francisco, CA, January 4, 1995. 1991 Mathematics Subject Classification. Primary 13A35.