nLab D-algebra

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

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

Related concepts

• D-module