p-TORSION ELEMENTS IN LOCAL COHOMOLOGY MODULES

For every prime integer p, M. Hochster conjectured the existence of certain p-torsion elements in a local cohomology module over a regular ring of mixed characteristic. We show that Hochster’s conjecture is false. We next construct an example where a local cohomology module over a hypersurface has p-torsion elements for every prime integer p, and consequently has infinitely many associated prime ideals. For a commutative Noetherian ring R and an ideal a ⊂ R, the finiteness properties of the local cohomology modules H a (R) have been studied by various authors. In this paper we focus on the following question raised by C. Huneke [Hu, Problem 4]: if M is a finitely generated R-module, is the number of associated primes ideals of H a (M) always finite? In the case that the ring R is regular and contains a field of prime characteristic p > 0, Huneke and Sharp showed in [HS] that the set of associated prime ideals of H a (R) is finite. If R is a regular local ring containing a field of characteristic zero, G. Lyubeznik showed that H a (R) has only finitely many associated prime ideals, see [Ly1] and also [Ly2, Ly3]. Recently Lyubeznik has also proved this result for unramified regular local rings of mixed characteristic, [Ly4]. Our computations here support Lyubeznik’s conjecture [Ly1, Remark 3.7 (iii)] that local cohomology modules over all regular rings have only finitely many associated prime ideals. In Section 4 we construct an example of a hypersurface R such that the local cohomology module H a (R) has p-torsion elements for every prime integer p, and consequently has infinitely many associated prime ideals. For some of the other work related to this question, we refer the reader to the papers [BL, BRS, He]. 1. Hochster’s conjecture Consider the polynomial ring over the integers R = Z[u, v, w, x, y, z] where a is the ideal generated by the size two minors of the matrix M = (