- Local cohomology under small perturbations(arXiv)
Author : Luís Duarte
Abstract : Let (R,m) be a Noetherian local ring and I an ideal of R. We study how local cohomology modules with support in m change for small perturbations J of I, that is, for ideals J such that I≡JmodmN for large N, under the hypothesis that I and J share the same Hilbert function. As one of our main results, we show that if R/I is generalized Cohen-Macaulay, then the local cohomology modules of R/J are isomorphic to the corresponding local cohomology modules of R/I, except possibly the top one. In particular, this answers a question raised by Quy and V. D. Trung. Our approach also allows us to prove that if R/I is Buchsbaum, then so is R/J. Finally, under some additional assumptions, we show that if R/I satisfies Serre’s property (Sn), then so does R/J
2.Annihilator of Top Local Cohomology and Lynch’s Conjecture (arXiv)
Author : Ali Fathi
Abstract : In this paper, we give a bound under inclusion for the annihilator of top local cohomology and we compute this annihilator in certain cases. Also, we give a counterexample to Lynch’s conjecture which extends the previous counterexamples given by Bahmanpour and by Singh and Walther.
3. Graded components of local cohomology modules supported on C-monomial ideals(arXiv)
Abstract : Let A be a Dedekind domain of characteristic zero such that its localization at every maximal ideal has mixed characteristic with finite residue field. Let R=A[X1,…,Xn] be a polynomial ring and I=(a1U1,…,acUc)⊆R an ideal, where aj∈A (not necessarily units) and Uj’s are monomials in X1,…,Xn. We call such an ideal I as a C-monomial ideal. Consider the standard multigrading on R. We produce a structure theorem for the multigraded components of the local cohomology modules HiI(R) for i≥0. We further analyze the torsion part and the torsion-free part of these components. We show that if A is a PID then each component can be written as a direct sum of its torsion part and torsion-free part. As a consequence, we obtain that their Bass numbers are finite.