On the ideal generated by all squarefree monomials of a given degree

TL;DR

This paper provides an explicit, characteristic-free minimal free resolution of ideals generated by all squarefree monomials of a fixed degree, utilizing symmetric group actions, with applications to algebraic structures like De Concini-Procesi rings and FI-modules.

Contribution

It introduces a natural, symmetric group-equivariant construction of the minimal free resolution for squarefree monomial ideals, applicable over any coefficient ring.

Findings

Resolution is characteristic free and explicit.

Provides equivariant resolution of De Concini-Procesi rings.

Offers a resolution framework for FI-modules.

Abstract

An explicit construction is given of a minimal free resolution of the ideal generated by all squarefree monomials of a given degree. The construction relies upon and exhibits the natural action of the symmetric group on the syzygy modules. The resolution is obtained over an arbitrary coefficient ring; in particular, it is characteristic free. Two applications are given: an equivariant resolution of De Concini-Procesi rings indexed by hook partitions, and a resolution of FI-modules.