Using Algebraic Geometry pp 234-289 | Cite as

# 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.

## Keywords

Exact Sequence Free Module Degree Zero Hilbert Series Hilbert Function## Preview

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