Lyubeznik Numbers of Local Rings and Linear Strands of Graded Ideals

Abstract

Abstract. In this work we introduce a new set of invariants associated to the linear strands of a minimal free resolution of a Z-graded ideal I ⊆ R = k[x 1 , . . . , x n ]. We also prove that these invariants satisfy some properties analogous to those of Lyubeznik numbers of local rings. In particular, they satisfy a consecutiveness property that we prove first for the Lyubeznik table. For the case of squarefree monomial ideals we get more insight on the relation between Lyubeznik numbers and the linear strands of their associated Alexander dual ideals. Finally, we prove that Lyubeznik numbers of StanleyReisner rings are not only an algebraic invariant but also a topological invariant, meaning that they depend on the homeomorphic class of the geometric realization of the associated simplicial complex and the characteristic of the base field.

Bass numbers of local cohomology of cover ideals of graphs

“…Another property that we are going to use in this work is the following Thom-Sebastiani type formula for the case of squarefree monomial ideals that was proved in [9]. ii) If both the height of I and the height of J are ≥ 2, then we have:…”

“…A way to interpret the Lyubeznik numbers for the case of squarefree monomial ideals is in terms of the linear strands of the free resolution of the Alexander dual of the ideal. This approach was given in [8] and further developed in [9], and we will briefly recall it here.…”