Lyubeznik numbers of local rings and linear strands of graded ideals

TL;DR

This paper introduces new invariants related to linear strands of graded ideals, explores their properties akin to Lyubeznik numbers, and reveals their topological nature in Stanley-Reisner rings.

Contribution

It defines new invariants for linear strands of graded ideals, establishes their properties similar to Lyubeznik numbers, and shows their topological significance in Stanley-Reisner rings.

Findings

Invariants satisfy properties analogous to Lyubeznik numbers.

For squarefree monomial ideals, links between Lyubeznik numbers and Alexander duals are clarified.

Lyubeznik numbers of Stanley-Reisner rings depend on the topology of the associated simplicial complex.

Abstract

In this work we introduce a new set of invariants associated to the linear strands of a minimal free resolution of a $\mathbb{Z}$-graded ideal $I\subseteq R=\Bbbk[x_1, \ldots, x_n]$. 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 Lyubeznik numbers. 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 Stanley-Reisner 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.