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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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How to compute the derived functor of bounded derived categories of hereditary algebra?

Let $\Lambda$ be a finite dimensional algebra given by the quiver $$\cdot\leftarrow\cdot\leftarrow\cdot\rightarrow\cdot.$$ It can be view as an triangulated matrix algebra. $$\Lambda={A\ \ M\choose0\ …