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(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.
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Examples of complexes of modules for wich homomorphisms "homological" implies "homotopic"
Let $V$ and $V^\prime$ - complexes of modules over ring $A$, and $f, g$ - homomorphisms $V\rightarrow V^\prime$.
I am interested in various conditions on $A, V, V^\prime$: ($f$ and $g$ are homolo …