Subadditivity of shifts, Eilenberg-Zilber shuffle products and homology of lattices

TL;DR

This paper proves the subadditivity of maximal shifts in minimal free resolutions of monomial ideal quotients by introducing a new poset homology model and an Eilenberg-Zilber shuffle product, advancing understanding of lattice homology.

Contribution

It introduces the synor complex for poset homology and an Eilenberg-Zilber shuffle product, linking lattice homology to properties of free resolutions.

Findings

Maximal shifts are subadditive in minimal free resolutions.

Non-zero lattice homology classes imply lower interval homology classes.

Generalizes previous results on shifts in resolutions.

Abstract

We show that the maximal shifts in the minimal free resolution of the quotients of a polynomial ring by a monomial ideal are subadditive as a function of the homological degree. This answers a question that has received some attention in recent years. To do so, we define and study a new model for the homology of posets, given by the so called synor complex. We also introduce an Eilenberg-Zilber type shuffle product on the simplicial chain complex of lattices. Combining these concepts we prove that the existence of a non-zero homology class for a lattice forces certain non-zero homology classes in lower intervals. This result then translates into properties of the minimal free resolution. In particular, it implies a generalization of the original question.