Free Resolutions

Abstract

In Chapter 5, we saw that to work with an R-module M, we needed not just the generators f ₁,..., f _t of M, but the relations they satisfy. Yet the set of relations Syz (f _l,..., f _t ) is an R-module in a natural way and, hence, to understand it, we need not just its generators g ₁,..., g _s , but the set of relations Syz (g _l,...,g _s ) on these generators, the so-called second syzygies. The second syzygies are again an R-module and to understand it, we again need a set of generators and relations, the third syzygies, and so on. We obtain a sequence, called a resolution, of generators and relations of successive syzygy modules of M. In this chapter, we will study resolutions and the information they encode about M. Throughout this chapter, R will denote the polynomial ring k[x _l,..., x _n ] or one of its localizations.