Frobenius Test Exponents for Parameter Ideals in Generalized Cohen–macaulay Local Rings

This paper studies Frobenius powers of parameter ideals in a commutative Noetherian local ring R of prime characteristic p. For a given ideal a of R, there is a power Q of p, depending on a, such that the Q-th Frobenius power of the Frobenius closure of a is equal to the Q-th Frobenius power of a. The paper addresses the question as to whether there exists a uniform Q0 which ‘works’ in this context for all parameter ideals of R simultaneously. In a recent paper, Katzman and Sharp proved that there does exists such a uniform Q0 when R is Cohen–Macaulay. The purpose of this paper is to show that such a uniform Q0 exists when R is a generalized Cohen–Macaulay local ring. A variety of concepts and techniques from commutative algebra are used, including unconditioned strong d-sequences, cohomological annihilators, modules of generalized fractions, and the Hartshorne–Speiser–Lyubeznik Theorem employed by Katzman and Sharp in the Cohen–Macaulay case.