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The Derived Module Category

  • F. E. A. JohnsonEmail author
Chapter
  • 946 Downloads
Part of the Algebra and Applications book series (AA, volume 17)

Abstract

Linear algebra over a field is rendered tractable by the fact that every module over a field is free; that is, has a spanning set of linearly independent vectors. Over more general rings, when a module M is not free we make a first approximation to its being free by taking a surjective homomorphism φ:FM where F is free. The kernel Ker(φ) may then be regarded as the ‘first derivative’ module of M. These considerations may be made precise by working in the ‘derived module category’, the quotient of the category of Λ-modules obtained by quotienting out by morphisms which factorize through a projective. The invariants of this first derivative lead to a nonstandard definition of module cohomology.

Keywords

Exact Sequence Isomorphism Class Short Exact Sequence Natural Transformation Projective Module 
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.

References

  1. 5.
    Baer, R.: Erweiterung von Gruppen und ihren Isomorphismen. Math. Z. 38, 375–416 (1934) MathSciNetCrossRefGoogle Scholar
  2. 44.
    Humphreys, J.J.A.M.: Algebraic properties of semi-simple lattices and related groups. Ph.D. Thesis, University College London (2006) Google Scholar
  3. 68.
    MacLane, S.: Homology. Springer, Berlin (1963) zbMATHGoogle Scholar
  4. 74.
    Mitchell, B.: Theory of Categories. Academic Press, San Diego (1965) zbMATHGoogle Scholar
  5. 91.
    Swan, R.G.: Periodic resolutions for finite groups. Ann. Math. 72, 267–291 (1960) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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