Subcomplexes of Certain Free Resolutions

TL;DR

This paper investigates the structure of subcomplexes within certain well-known free resolutions, using the BGG correspondence to provide a complete characterization in the Koszul case and restrictions in the Eagon--Northcott case.

Contribution

It introduces a novel approach employing the BGG correspondence to analyze subcomplexes of Koszul and Eagon--Northcott resolutions, offering new characterizations and restrictions.

Findings

Complete characterization of subcomplex ranks in Koszul resolutions

Numerical restrictions on subcomplexes in Eagon--Northcott resolutions

Addresses computational challenges in analyzing free resolution subcomplexes

Abstract

What are the subcomplexes of a free resolution? This question is simple to state, but the naive approach leads to a computational quagmire that is infeasible even in small cases. In this paper, we invoke the Bernstein--Gelfand--Gelfand (BGG) correspondence to address this question for free resolutions given by two well-known complexes, the Koszul and the Eagon--Northcott. This novel approach provides a complete characterization of the ranks of free modules in a subcomplex in the Koszul case and imposes numerical restrictions in the Eagon--Northcott case.