Free Resolutions

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


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.


Exact Sequence Free Module Degree Zero Hilbert Series Hilbert Function 
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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

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