An Avoidance Principle with an Application to the Asymptotic Behaviour of Graded Local Cohomology

We present an Avoidance Principle for certain graded rings. As an application we fill a gap in the proof of a result by Brodmann, Rohrer and Sazeedeh about the antipolynomiality of the Hilbert-Samuel multiplicity of the graded components of the local cohomology modules of a finitely generated module over a Noetherian homogeneous ring with two-dimensional local base ring. Let R = ⊕ n∈N0 Rn be a Noetherian homogeneous ring with local base ring (R0,m0) and irrelevant ideal R+ := ⊕ n∈NRn, let q0 ⊆ R0 be an m0-primary ideal and let M = ⊕ n∈ZMn be a finitely generated graded R-module. For i ∈ N0 we denote by H i R+(M) the i-th local cohomology module of M with respect to R+, and for n ∈ Z we denote by H i R+(M)n its n-th graded component. By eq0(T ) we denote the Hilbert-Samuel multiplicity of a finitely generated R0-module T with respect to q0. If A is a ring, a ⊆ A is an ideal and T is an A-module, by Gr(a) := ⊕ n∈N0 a/a and Gr(a, T ) := ⊕ n∈N0 aT/aT we denote the graded ring associated with a and the graded module associated with T with respect to a respectively. If (A,m) is a local ring, by an unramified A-algebra we mean an A-algebra B which is a local ring with maximal ideal mB. The following Theorem is stated in [B-R-S, Theorem 5.7]: Theorem Let i ∈ N0, let dim(R0) = 2 and let dimR0(H i R+(M)n) ≥ 1 for all n ≪ 0. Then, there exists a polynomial Q ∈ Q[X] such that deg(Q) < i and eq0(H i R+(M)n) = Q(n) for all n ≪ 0. But Alas!, there is a gap in the proof of this Theorem in [B-R-S]. In this note we will fix this. We will not reproduce the entire proof and refer the reader (as we do for unexplained notations) to the article cited above. The argument lacks in the case where dimR0(H i R+ (M)n) = 1 for all n ≪ 0 and under the additional assumptions that Γm0R(M) = 0 and that R0 is complete. For n ∈ Z let Tn := H i R+(M)n/Γm0(H i R+(M)n). In order to continue the proof there must exist – possibly after a replacement of R0 by an appropriate ring – an element Date: Zürich, September 2006. 2000 Mathematics Subject Classification. Primary 13D45; Secondary 13D40, 13F30.