I would like to know if it is possible to generalize the notion of elliptic chain complex of differential operators to different contexts, whether geometric or non geometric. I have in mind D-geometry. For example, in these notes about D-modules, it is given the definition of symbol of a differential operator, but I do not see any analogue of the concept of elliptic operator and elliptic chain complex. There is also the notion of Hilbert Complex, a more general theory that implies some results about elliptic chain complexes. A good answer might also comment on the existence or not of analogue results in these new contexts.
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$\begingroup$ I haven't looked at what these specific notes say, but D-modules are just a language to talk about linear differential equations, unless you are outside ordinary geometry and work in positive characteristic or something. It's not hard to complete the dictionary (D-module <-> PDE, D-module morphism <-> diff.op.) to define elliptic complexes. You can also consider complexes of diff.ops. that are not elliptic, etc. As such your question is a bit vague. $\endgroup$– Igor KhavkineCommented Jan 24, 2023 at 20:45
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$\begingroup$ It is true that the question is vague as I am looking for constructions that share properties with elliptic operators. I am indeed interested in what is outside ordinary geometry and positive characteristic although not exclusively. I am looking for existing generalizations of some of the conventional ideas to non-conventional geometry or things like graphs or combinatorics. I hope this make it less vague. $\endgroup$– ArturoCommented Jan 24, 2023 at 21:42
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