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

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

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

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

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

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

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeSep 25th 2015

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