An example of an infinite set of associated primes of a local cohomology module

Abstract

Let (R, m) be a local Noetherian ring, let I ⊂ R be any ideal and let M be a finitely generated R-module. It has been long conjectured that the local cohomology modules H i I (M ) have finitely many associated primes for all i (see Conjecture 5. in [H] and [L].)If R is not required to be local these sets of associated primes may be infinite, as shown by Anurag Singh in [S], where he constructed an example of a local cohomology module of a finitely generated module over a finitely generated Z-algebra with infinitely many associated primes. This local cohomology module has p-torsion for all primes p ∈ Z.However, the question of the finiteness of the set of associated primes of local cohomology modules defined over local rings and over k-algebras (where k is a field) has remained open until now. In this paper I settle this question by constructing a local cohomology module of a local finitely generated k-algebra with an infinite set of associated primes, and I do this for any field k. The exampleLet k be any field, let R 0 = k[x, y, s, t] and let S = R 0 [u, v]. Define a grading on S by declaring deg(x) = deg(y) = deg(s) = deg(t) = 0 and deg(u) = deg(v) = 1. Let f = sx 2 v 2 −(t+s)xyuv+ty 2 u 2 and let R = S/f S. Notice that f is homogeneous and hence R is graded. Let S + be the ideal of S generated by u and v and let R + be the ideal of R generated by the images of u and v.

On the vanishing and the finiteness of supports of generalized local cohomology modules

“…But, there exist in general local cohomology modules of Noetherian local rings with infinitely many associated primes, see [11].…”

On the cofiniteness of local cohomology modules

“…A. Singh [18] and M. Katzman [11] have given counterexamples to this conjecture. However, it is known that this conjecture is true in many situations; see, [1], [2], [8], [10], [12], [14].…”

Local Cohomology Modules Supported on Monomial Ideals

“…R. Hartshorne [41] gave an example of a local cohomology module whose Bass numbers may be infinite if R is not regular. The finiteness of the associated primes set of H r I (R) for any Noetherian ring R and any ideal I was an open question until A. Singh [84] (non local case) and M. Katzman [51] (local case) have given examples of local cohomology modules having infinite associated primes.…”

“…R. Hartshorne [41] va donar un exemple en el qual els nombres de Bass d'un mòdul de cohomologia local poden ser infinits si R noés regular. La finitud dels primers associats de H r I (R) per a qualsevol anell Noetherià R i qualsevol ideal I era una qüestió oberta fins que A. Singh [84] (cas no local) i M. Katzman [51] (cas local) han donat exemples de mòduls de cohomologia local amb infinits primers associats.…”