Tony J. Puthenpurakal

Hilbert polynomials and powers of ideals

The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal I in the polynomial ring S = K[x1, . . ., xn] and a finitely generated graded S-module M, the Hilbert coefficients ei(M/IkM) are polynomial functions. Given two families of graded ideals (Ik)k=0 and (Jk)k=0 with Jk Ik for all k with the property that JkJl Jk+l and IkIl Ik+l for all k and l, and such that the algebras and are finitely generated, we show the function k e0(Ik/Jk) is of quasi-polynomial type, say given by the polynomials P0,. . ., Pg-1. If Jk = Jk for all k, for a graded ideal J, then we show that all the Pi have the same degree and the same leading coefficient. As one of the applications it is shown that , if I is a monomial ideal. We also study analogous statements in the local case.

Hilbert coefficients of a Cohen–Macaulay module

Let (A,m) be a d-dimensional Noetherian local ring, M a finite Cohen–Macaulay A-module of dimension r, and let I be an ideal of definition for M. We define the notion of minimal multiplicity of Cohen–Macaulay modules with respect to I and show that if M has minimal multiplicity with respect to I then the associated graded module GI(M) is Cohen–Macaulay. When A is Cohen–Macaulay, M is maximal Cohen–Macaulay, and I is m-primary, we find a relation between the first Hilbert coefficient of M, A, and Syz1A(M).

DE RHAM COHOMOLOGY OF LOCAL COHOMOLOGY MODULES: THE GRADED CASE

Let K be a field of characteristic zero, and let R = K [ X 1 ,… ,X n ]. Let A n (K) = K⟨X 1 ,… ,X n ,∂ 1 ,… ,∂ n ⟩ be the n th Weyl algebra over K . We consider the case when R and A n ( K ) are graded by giving deg X i = ω i and deg ∂ i = – ω i for i = 1,…, n (here ω i are positive integers). Set . Let I be a graded ideal in R . By a result due to Lyubeznik the local cohomology modules are holonomic ( A n ( K ))-modules for each i≥ 0. In this article we prove that the de Rham cohomology modules are concentrated in degree — ω ; that is, for j ≠ –ω . As an application when A = R/ ( f ) is an isolated singularity, we relate to H n-1 ( ∂ ( f ); A ), the ( n – 1)th Koszul cohomology of A with respect to ∂ 1 ( f ),…, ∂ n ( f ).

Invariance of a Length Associated to a Reduction

ABSTRACT Let ( A , 𝔪) be a d -dimensional Cohen-Macaulay local ring with infinite residue field and let J be a minimal reduction of 𝔪. We show that λ(𝔪 3 / J 𝔪 2 ) is independent of J .

Hilbert-Samuel functions of modules over Cohen-Macaulay rings

For a finitely generated, non-free module M over a CM local ring (R,m,k), it is proved that for n » 0 the length of Tor R 1(M, R/m n+1 ) is given by a polynomial of degree dim R - 1. The vanishing of Tor R i (M, N/m n+1 N) is studied, with a view towards answering the question: If there exists a finitely generated R-module N with dim N ≥ 1 such that the projective dimension or, the injective dimension of N/m n+ 1 N is finite, then is R regular? Upper bounds are provided for n beyond which the question has an affirmative answer.

Local cohomology modules of invariant rings

Let

Asymptotic prime divisors over complete intersection rings

K

de Rham cohomology of local cohomology modules II

be a field and let

Associated primes of local cohomology modules over regular rings

R

On the upper bound of the multiplicity conjecture

be a regular domain containing