p-TORSION ELEMENTS IN LOCAL COHOMOLOGY MODULES II

Gennady Lyubeznik conjectured that if R is a regular ring and a is an ideal of R, then the local cohomology modules H i a(R) have only finitely many associated prime ideals, [Ly1, Remark 3.7 (iii)]. While this conjecture remains open in this generality, several results are now available: if the regular ring R contains a field of prime characteristic p > 0, Huneke and Sharp showed in [HS] that the set of associated prime ideals of H i a(R) is finite. If R is a regular local ring containing a field of characteristic zero, Lyubeznik showed that H i 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]. In [Hu] Craig Huneke first raised the following question: for a Noetherian ring R, an ideal a ⊂ R, and a finitely generated R-moduleM , is the number of associated primes ideals of H i a(M) always finite? For some of the work on this problem, we refer the reader to the papers [BL, BRS, He] in addition to those mentioned above. In [Si] we constructed an example of a hypersurface R for which a local cohomology module H3 a (R) has p-torsion elements for every prime integer p, and consequently has infinitely many associated prime ideals. Since this is the only known source of infinitely many associated prime ideals so far, it is worthwhile to investigate whether similar techniques may yield an example of a regular ring R for which a local cohomology module H i a(R) has p-torsion elements for every prime integer p. This leads to some very intriguing questions as we shall see in this paper. Our results thus far support Lyubeznik’s conjecture that local cohomology modules of all regular rings have only finitely many associated prime ideals. Let R be a polynomial ring over the integers and Fi, Gi be elements of R for which F1G1 + F2G2 + · · ·+ FnGn = 0.