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differential bimodule

Contents

Idea

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

Definition

Given a commutative unital ring RR, a filtration M nM_n (n1n\geq -1) on a RR-RR-bimodule MM is a differential filtration if the commutator [r,P][r,P] for any PP in M nM_n is in M n1M_{n-1}, and M 1=0M_{-1} = 0. A bimodule is differential if it has an exhaustive ( nM n=M\bigcup_n M_n = M) differential filtration. Every RR-RR-bimodule has a differential part, i.e. the maximal differential subbimodule of MM. 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 XX is represented by an abelian category, C XC_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 XX be a smooth manifold and let R=C (X)R = C^\infty(X) be the ring of smooth functions. Consider inside End(R)End(R) all differential operators, for instance

  • multiplication operators by elements in RR;

  • 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 vv with a multiplication operator aa is [v,a]=v(a)[v,a] = v(a), which is a multiplication operator. And since RR is commutative we have for a,ba,b two multiplication operators that [a,b]=0[a,b] = 0.

Revised on June 15, 2010 17:49:15 by Zoran Škoda (161.53.130.104)