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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.
7
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Algebra Counterexample Request: Linear Quotients
If you take square-free monomials ideals and consider the Alexander duality, then linear resolution is dual to Cohen-Macaulay and linear quotient is dual to shellable. Hence you are looking for a non- …