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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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on a characterisation of regular D-modules
I don't think so. Here's a possible counter example.
Take $X=\mathbb{A}^1_\mathbb{C}$, $M=(\mathcal{O}_X,\nabla)$ where $\nabla(f)=df-fdz$, $z$ being the co-ordinate on $\mathbb{A}^1_\mathbb{C}$. The …