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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 25th 2015
   • (edited Sep 25th 2015)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexJim 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](http://arxiv.org/abs/1106.4955) 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 $(Mod(\mathcal{D}), \bigotimes_{\mathcal{O}})$

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

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 25th 2015
   • (edited Sep 25th 2015)
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   Author: Urs
   Format: MarkdownItexYes, 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 25th 2015
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI'll sling these comments into a stub for [[D-algebra]].

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