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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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estimate on degree of generators in cohomology of differential graded module

Let $R=\mathbb{C}[X_1,\ldots,X_r]$ be a polynomial ring and consider a finitely generated and free differential graded $R$-module $M$ with differential $d$. Lets say that the degree of the variables $ …