nLab differential bimodule

Idea

A differential bimodule over a ring $R$ is a bimodule that behaves like a collection of differential operators $R \to R$.

Definition

Given a commutative unital ring $R$, a filtration $M_n$ ($n\geq -1$) on a $R$-$R$-bimodule $M$ is a differential filtration if the commutator $[r,P]$ for any $P$ in $M_n$ is in $M_{n-1}$, and $M_{-1} = 0$. A bimodule is differential if it has an exhaustive ($\bigcup_n M_n = M$) differential filtration. Every $R$-$R$-bimodule has a differential part, i.e. the maximal differential subbimodule of $M$. The principal example of a differential bimodule is the ring of regular differential operators defined by Grothendieck.

There is a generalization in categorical setup developed by Lunts and Rosenberg in

• V. Lunts, A. L. Rosenberg, Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings

If a noncommutative scheme-like geometric object $X$ is represented by an abelian category, $C_X$ then morphism of schemes should correspond to pairs of adjoint functors, in fact bimodules (cf. the Eilenberg-Watts theorem). Then one considers thickenings of the diagonal …

to continue…

Examples

Let $X$ be a smooth manifold and let $R = C^\infty(X)$ be the ring of smooth functions. Consider inside $End(R)$ all differential operators, for instance

• multiplication operators by elements in $R$;

• operators that take the differential with respect to a vector field,

• etc.

Then we have a D-filtration by order of the differential operator. For instance the commutator of a vector field $v$ with a multiplication operator $a$ is $[v,a] = v(a)$, which is a multiplication operator. And since $R$ is commutative we have for $a,b$ two multiplication operators that $[a,b] = 0$.