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Graded Free Resolutions

  • Irena Peeva
Part of the Algebra and Applications book series (AA, volume 14)

Abstract

The study of free resolutions is a core and beautiful area in Commutative Algebra. The idea to associate a free resolution to a finitely generated module was introduced in two famous papers by Hilbert in 1890, 1893. Free resolutions provide a method for describing the structure of modules. There are several challenging and exciting conjectures involving resolutions. A number of open problems on graded syzygies and Hilbert functions are listed in a paper by Peeva and Stillman (2009).

We are using a grading on the polynomial ring S=k[x 1,…,x n ] and on the objects which we are interested to study: ideals, quotient rings, modules, complexes, and free resolutions. The grading is a powerful tool. The general principle using that tool is the following: in order to understand the properties of a graded object X, we consider X as a direct sum of vector spaces (its graded components) and we study the properties of each of these vector spaces.

Keywords

Simplicial Complex Short Exact Sequence Betti Number 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.

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References

  1. [Hilbert 1]
    D. Hilbert: Über the Theorie von algebraischen Formen, Math. Ann. 36 (1890), 473–534. An English translation is available in Hilbert’s Invariant Theory Papers, Math. Sci. Press, 1978. CrossRefMathSciNetGoogle Scholar
  2. [Hilbert 2]
    D. Hilbert: Über die vollen Invariantensysteme, Math. Ann. 42 (1893), 313–373. An English translation is available in Hilbert’s Invariant Theory Papers, Math. Sci. Press, 1978. CrossRefMathSciNetGoogle Scholar
  3. [Peeva-Stillman]
    I. Peeva and M. Stillman: Open problems on syzygies and Hilbert functions, J. Comm. Alg. 1 (2009), 159–195. MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  • Irena Peeva
    • 1
  1. 1.Department of MathematicsCornell UniversityIthacaUSA

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