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
in which
-
i)
each Cs is a free graded module over A
-
ii)
each d is an A-map preserving gradation
-
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
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
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