A Computation of Tight Closure in Diagonal Hypersurfaces

Abstract

The aim of this paper is to settle a question about the tight closure of Ž 2 2 2 . w x Ž 3 3 3. the ideal x , y , z in the ring R s K X, Y, Z r X q Y q Z where Ž K is a field of prime characteristic p / 3. Lower case letters denote the . images of the corresponding variables. M. McDermott has studied the tight closure of various irreducible ideals in R and has established that Ž 2 2 2 . w x xyz g x , y , z * when p 200, see Mc . The general case however existed as a classic example of the difficulty involved in tight closure w x Ž 2 2 2 . computations, see also Hu, Example 1.2 . We show that xyz g x , y , z * in arbitrary prime characteristic p, and furthermore establish that xyz g Ž 2 2 2 .F x , y , z whenever R is not F-pure, i.e., when p ' 2 mod 3. We move on to generalize these results to the diagonal hypersurfaces R s w x Ž n n. K X , . . . , X r X q ??? qX . 1 n 1 n These issues relate to the question whether the tight closure I* of an ideal I agrees with its plus closure, Iqs IRql R, where R is a domain over a field of characteristic p and Rq is the integral closure of R in an algebraic closure of its fraction field. In this setting, we may think of the Frobenius closure of I as I F s IR l R where R is the extension of R obtained by adjoining pth roots of all nonzero elements of R for e g N. It is not difficult to see that Iq: I*, and equality in general is a formidable open question. It should be mentioned that in the case when I is an ideal generated by part of a system of parameters, the equality is a result