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Results tagged with homological-algebra
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user 23571
(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
14
votes
Accepted
Geometric intuition behind this chain homotopy
When thinking about chain homotopies in a setting involving simplices it can be helpful to consider the product $\Delta^p\times I$ where $\Delta^p$ is a $p$-simplex and $I=[0,1]$. The formula $h\sigm …
- 20k
48
votes
Spectral sequences: opening the black box slowly with an example
Two simple examples with lots of interesting differentials are given by the Serre spectral sequences for integer homology (rather than cohomology) for the fibrations $$K({\mathbb Z}_2,1) \to K({\mathb …
- 20k
11
votes
Accepted
Convergence of spectral sequences of cohomological type
The lemma you refer to has two halves. The first half covers the case of homology and the second half covers the case of cohomology. Proofs are given for both halves (though the last sentence of the …
- 20k
4
votes
Accepted
Realizing complexes with bases as cellular complexes
Here is a sketch of an argument to show that all based chain complexes are realizable. (This might end up being pretty similar to Tyler's argument.)
First one gives an algebraic argument that by a ch …
- 20k