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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 25th 2015
    • (edited Sep 25th 2015)

    Jim Stasheff asks whether we have an entry for 𝒟\mathcal{D}-algebra, some kind of 𝒟\mathcal{D}-module.

    I see Paugam speaks of them in his contribution to Urs and Sati’s book.

    we define the category of D-algebras, that solves the mathematical problem of finding a natural setting for a coordinate free study of polynomial non-linear partial differential equations with smooth superfunction coefficients.

    one can extend the jet functor to the category of smooth D-algebras (and even to smooth super-algebras), to extend the forthcoming results to the study of nonpolynomial smooth partial differential equations.

    Surely this will have much to do with your current interest, Urs, in jet comonads.

    So a 𝒟\mathcal{D}-algebra is

    an algebra in (Mod(𝒟), 𝒪)(Mod(\mathcal{D}), \bigotimes_{\mathcal{O}})

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 25th 2015
    • (edited Sep 25th 2015)

    Yes, so these are just the function algebras of the 𝒟\mathcal{D}-schemes over the de Rham stack Σ\Im \Sigma of the given base scheme Σ\Sigma:

    𝒟\mathcal{D}-modules are just the quasicoherent sheaves over Σ\Im \Sigma. By the comonadic PDE ΣH /Σ\mathrm{PDE}_\Sigma \simeq \mathbf{H}_{/\Im \Sigma} a space over Σ\Im \Sigma is equivalently a differential equation, and in terms of algebraic geometry such a space, when affine, is an algebra in the modules over Σ\Im \Sigma, hence is a 𝒟\mathcal{D}-algebra.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 25th 2015

    I’ll sling these comments into a stub for D-algebra.