Monomial Resolutions

Abstract

In this chapter we discuss free resolutions of monomial ideals; we call them monomial resolutions. The problem to describe the minimal free resolution of a monomial ideal (over a polynomial ring) was posed by Kaplansky in the early 1960’s. Despite the helpful combinatorial structure of monomial ideals, the problem turned out to be hard. The structure of a minimal free monomial resolution can be quite complex. There exists a minimal free monomial resolution which cannot be encoded in the structure of any CW-complex. In fact, even the minimal free resolutions of ideals generated by quadratic monomials are so complicated that it is beyond reach to obtain a description of them; we do not even know how to express the regularity of such ideals. In this situation, the guideline is to introduce new ideas and constructions which either have strong applications or/and are beautiful. Most proofs about monomial resolutions are easy. The key point is not to provide complicated proofs, but to introduce new beautiful ideas.