A survey on the Lyubeznik numbers

Abstract

The Lyubeznik numbers are invariants of a local ring containing a field that cap- ture ring-theoretic properties, but also have numerous connections to geometry and topol- ogy. We discuss basic properties of these integer-valued invariants, as well as describe some significant results and recent developments (including certain generalizations) in the area. Since the introduction of the Lyubeznik numbers (Lyu93), the study of these invariants of local rings containing a field has grown in several compelling directions. One aim of this paper is to present definitions used in the study of the Lyubeznik numbers, along with examples and applications of these invariants, to those new to them. For those familiar with them, we also present recent results on, and generalizations of, the Lyubeznik numbers, as well as open problems in the area. Given a module M over a ring S, if Iis a minimal injective resolution of M, then each I i is isomorphic to a direct sum of indecomposable injective modules ES(S/p), p is a prime ideal of S; i.e., injective hulls over S of S/p. The number of copies of ES(S/p) in I i is the i-th Bass number of M with respect to p, denoted µi(p,M) and equals dimSp/pSp Ext i (Sp/pSp,Mp). Huneke and Sharp proved that if S is a regular ring of characteristic p > 0 and I is an ideal of S, then the Bass numbers of the local cohomology modules of the form H j I (S), j ∈ N, are finite, raising the analogous question in the characteristic zero case (HS93). Uti- lizing D-module theory, Lyubeznik proved the same statement for regular local rings of characteristic zero containing a field (Lyu93). Relying on the finiteness of the Bass numbers of local cohomology modules, Lyubeznik introduced a family of integer-valued invariants associated to a local ring containing a field, now called Lyubeznik numbers. They are defined as follows: Suppose that (R,m,K) is a local ring admitting a surjection from an n-dimensional regular local ring (S,n,K) containing a field, and let I denote the kernel of the surjection. Given i,j ∈ N, the Lyubeznik number of R with respect to i,j ∈ N, is defined as dimK Ext i