On the associated primes of Matlis duals of top local cohomology modules

After motivating the question we prove various results about the set of associated primes of Matlis duals of local cohomology modules. In easy cases we can calculate this set. An easy application of this theory is the well-known fact that Krull dimension can be expressed by the vanishing of local cohomology modules. 0. Motivation and Notation Whenever (R,m) is a noetherian local ring, we denote by ER(R/m) an R-injective hull of the residue field R/m and by D(M) := HomR(M,ER(R/m)) the Matlis dual of M (here M is any R-module, for basics on injective modules and Matlis duals see [3], [4], [5], [9]). Let I be an ideal of a noetherian local ring (R,m) such that 0 = HI(R) = H 3 I(R) = . . .. For any f ∈ I we have √ I = √ fR ⇐⇒ f acts surjevtively on HI(R) (Proof: “⇒”: Clear, because f acts surjectively on HfR(R). “⇐”: For f ∈ p ∈ Spec(R) the element f ∈ R acts like zero on HI(R/p) and at the same time surjectively on H 1 I(R)⊗R (R/p) = HI(R/p); so we must have HI(R/p) = 0 which implies I ⊆ p.) f acts surjectively on HI(R) if and only if f acts injectively on D(H 1 I(R)), that is, if and only if f is not contained in any prime ideal associated to D(HI(R)). The above statement is easily generalized to the following one: Let I be an ideal in a noetherian local ring (R,m), n ∈ lN, 0 = H I (R) = H I (R) = . . . and f1, . . . fn ∈ I. We assume I 6= R is a proper ideal to avoid trivial cases. Then √ I = √ (f1, . . . , fn)R ⇐⇒ fi acts surjectively on H I (R/(f1, . . . , fi−1)R) for i = 1, . . . , n ⇐⇒ fi acts injectively on D(H I (R/(f1, . . . , fi−1)R)) for i = 1, . . . , n Under the additional assumption H l I(R) = 0 (l 6= n) (i. e. f1, . . . , fn form a regular sequence on R) we may formulate: √ I = √ (f1, . . . , fn)R ⇐⇒ f1, . . . , fn is a regular sequence on D(HI (R))