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Applications of Homological Algebra to Stable Homotopy Theory

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Part of the Lecture Notes in Mathematics book series (LNM,volume 3)

Abstract

I ought to begin by running through the basic notions of homological algebra in the case where we have graded modules over a graded algebra A. Let M be such a module, i.e. \(M = \sum\limits_{o \leqslant t \leqslant \infty } {{M_t}.{A_s}} \cdot {M_t} \subseteq {M_{s + t}}.{M_t}\) is finitely generated. A resolution of M is a chain complex

$${C_o}\underleftarrow d{C_1} \leftarrow {C_2} \leftarrow \ldots \leftarrow {C_S} \leftarrow $$

in which

  1. i)

    each Cs is a free graded module over A

  2. ii)

    each d is an A-map preserving gradation

  3. iii)
    $${H_s}\left( C \right) = \left\{ {\begin{array}{*{20}{c}} {Mifs = 0} \\ {0ifs > 0} \end{array}} \right.$$

This amounts to the same thing as requiring a map є: Co → M so that

$$0 \leftarrow M\underleftarrow \varepsilon {C_o}\underleftarrow d{C_1} \leftarrow \ldots \leftarrow {C_s} \leftarrow \ldots $$

is exact at every stage. Such chain complexes always exist and they are unique up to chain equivalence.

Keywords

  • Exact Sequence
  • Spectral Sequence
  • Chain Complex
  • Homological Algebra
  • Exact Triangle

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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© 1964 Springer-Verlag Berlin Heidelberg

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Adams, J.F. (1964). Applications of Homological Algebra to Stable Homotopy Theory. In: Stable Homotopy Theory. Lecture Notes in Mathematics, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-15942-2_4

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  • DOI: https://doi.org/10.1007/978-3-662-15942-2_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-15944-6

  • Online ISBN: 978-3-662-15942-2

  • eBook Packages: Springer Book Archive