𝒟\mathcal{D}-algebras provide a natural setting for a coordinate free study of polynomial non-linear partial differential equations with smooth superfunction coefficients.

A 𝒟\mathcal{D}-algebra is an algebra in (Mod(𝒟), 𝒪)(Mod(\mathcal{D}), \bigotimes_{\mathcal{O}}). These are 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.

Created on September 25, 2015 at 05:28:03. See the history of this page for a list of all contributions to it.