The structure of DGA resolutions of monomial ideals

TL;DR

This paper investigates the conditions under which the minimal free resolution of squarefree monomial ideals admits a differential graded algebra structure, providing characterizations, examples, and modifications to achieve such structures.

Contribution

It characterizes the Betti number vectors for ideals with DGA structures, shows subadditivity of maximal shifts, and demonstrates how to modify resolutions to obtain DGA structures.

Findings

Characterization of Betti number vectors for DGA-structured resolutions

Existence of monomial ideals without DGA structures

Modification of the last map can induce a DGA structure

Abstract

Let $I \subset k[x_1, \dotsc, x_n]$ be a squarefree monomial ideal a polynomial ring. In this paper we study multiplications on the minimal free resolution $\mathbb{F}$ of $k[x_1, \dotsc, x_n]/I$. In particular, we characterize the possible vectors of total Betti numbers for such ideals which admit a differential graded algebra (DGA) structure on $\mathbb{F}$. We also show that under these assumptions the maximal shifts of the graded Betti numbers are subadditive. On the other hand, we present an example of a strongly generic monomial ideal which does not admit a DGA structure on its minimal free resolution. In particular, this demonstrates that the Hull resolution and the Lyubeznik resolution do not admit DGA structures in general. Finally, we show that it is enough to modify the last map of $\mathbb{F}$ to ensure that it admits the structure of a DG algebra.