Advertisement

Free Resolutions

  • David Cox
  • John Little
  • Donal O’Shea
Part of the Graduate Texts in Mathematics book series (GTM, volume 185)

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • David Cox
    • 1
  • John Little
    • 2
  • Donal O’Shea
    • 3
  1. 1.Department of Mathematics and Computer ScienceAmherst CollegeAmherstUSA
  2. 2.Department of MathematicsCollege of the Holy CrossWorcesterUSA
  3. 3.Department of Mathematics, Statistics and Computer ScienceMount Holyoke CollegeSouth HadleyUSA

Personalised recommendations