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Published in:
2016 | OriginalPaper | Chapter
15. Appendix D: Homological Algebra
Author : Torsten Wedhorn
Published in: Manifolds, Sheaves, and Cohomology
Publisher: Springer Fachmedien Wiesbaden
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Abstract
In this appendix we recall some basic notions from homological algebra. We start by defining complexes of modules and the notion of homotopy between morphisms of complexes. This is in fact the cohomological shadow of the topological notion of homotopy (see the remark after Proposition 11.12). One obtains the homotopy category of modules, which is a fundamental notion in homological algebra. After a short interlude on diagram chases we introduce a central notion for the definition of cohomology: injective modules and K-injective complexes. Until then all notions were explained for modules over a ring, but in fact they make sense much more generally in arbitrary abelian categories. This is explained in the last section of this appendix.