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Derived Categories and Derived Functors

In: Methods of Homological Algebra

Author

Listed:
  • Sergei I. Gelfand

    (American Mathematical Society)

  • Yuri I. Manin

    (MPI für Mathematik)

Abstract

Homological algebra was founded by D. Hilbert. He considered, in particular, the following problem. Let $$ \sum\nolimits_{j = 1}^m {{a_{ij}}{x_j} = 0;\;i = 1, \ldots ,n;{a_{ij}}} \in k\left[ {{t_1}, \ldots {t_r}} \right] $$ , be a system of linear homogeneous equations with coefficients lying in the polynomial ring over a field. All polynomial solutions are linear combinations (with polynomial coefficients) of a finite subset of solutions. However, in general there exists no basis of solutions that are linearly independent over k[t 1..., t r]. Linear relations among elements of a generating system of solutions are, in turn, linear combinations of some finite set of relations, and again it might happen that there exists no free system of generators for relations.

Suggested Citation

  • Sergei I. Gelfand & Yuri I. Manin, 1996. "Derived Categories and Derived Functors," Springer Books, in: Methods of Homological Algebra, chapter 0, pages 139-238, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-03220-6_3
    DOI: 10.1007/978-3-662-03220-6_3
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