Resumen de Graded annihilators of modules over the Frobenius skew polynomial ring, and tight closure

Rodney Y. Sharp

• This paper is concerned with the tight closure of an ideal [a] in a commutative Noetherian local ring R of prime characteristic P. Several authors, including R. Fedder, K-i. Watanabe, K. E. Smith, N. Hara and F. Enescu, have used the natural Frobenius action on the top local cohomology module of such an R to good effect in the study of tight closure, and this paper uses that device. The main part of the paper develops a theory of what are here called `special annihilator submodules' of a left module over the Frobenius skew polynomial ring associated to R; this theory is then applied in the later sections of the paper to the top local cohomology module of R and used to show that, if R is Cohen-Macaulay, then it must have a weak parameter test element, even if it is not excellent.