Josep Àlvarez Montaner

Local cohomology, arrangements of subspaces and monomial ideals

Let Ank denote the affine space of dimension n over a field k; let XCA n k be an arrangement of linear subvarieties. Set R 1⁄4 k1⁄2x1;y; xn and let ICR denote an ideal which defines X : In this paper we study the local cohomology modules H IðRÞ :1⁄4 indlimj Ext i RðR=I ;RÞ; with special regard of the case where the ideal I is generated by monomials. If k is the field of complex numbers (or, more generally, a field of characteristic zero), the module H I ðRÞ is known to have a module structure over the Weyl algebra AnðkÞ; and one can therefore consider its characteristic cycle, denoted CCðH I ðRÞÞ in this paper (see e.g. [3, I.1.8.5]). On the other hand, the arrangement X defines a partially ordered set PðX Þ whose elements correspond to the intersections of irreducible components of X and where the order is given by inclusion. Our first result is the determination of the characteristic cycles CCðH I ðRÞÞ in terms of the cohomology of some simplicial complexes attached to the poset PðXÞ: It follows from the formulas obtained that, in either the complex or the real case, these characteristic cycles determine the Betti numbers of the complement of the arrangement in Ank: In fact, it was proved by Goresky and MacPherson that the

Characteristic cycles of local cohomology modules of monomial ideals

By using the theory of D-modules we express the characteristic cycle of a local cohomology module supported on a monomial ideal in terms of conormal bundles relative to a subvariety. As a consequence we can decide when a given local cohomology module vanishes and compute the cohomological dimension in terms of the minimal primary decomposition. We can also give a Cohen-Macualayness criterion for the quotient of a polynomial ring by a monomial ideal and compute its Lyubeznik numbers.

Lyubeznik numbers of monomial ideals

In this work we will study Bass numbers of local cohomology modules supported on a squarefree monomial ideal. Among them we are mainly interested in Lyubeznik numbers. We build a dictionary between these local cohomology modules and the minimal free resolution of the Alexander dual ideal that allow us to interpret Lyubeznik numbers as the obstruction to the acyclicity of the corresponding linear strands. The methods we develop also help us to give a bound for the injective dimension of the local cohomology modules in terms of the dimension of the small support

Some numerical invariants of local rings

Let

Multiplier ideals in two-dimensional local rings with rational singularities

R

Lyubeznik Table of Sequentially Cohen–Macaulay Rings

be a formal power series ring over a field of characteristic zero and

Operations with regular holonomic D-modules with support a normal crossing

I\subseteq R

Effective computation of base points of ideals in two-dimensional local rings

be any ideal. The aim of this work is to introduce some numerical invariants of the local rings

Local Cohomology Modules Supported on Monomial Ideals

R/I

Frobenius and Cartier algebras of Stanley–Reisner rings

by using theory of algebraic