nLab
regular differential operator

Grothendieck has developed a deep version of differential calculus, based on a linearization of O XO_X-bimodules. It is also related to the (de Rham) descent data for the stack of O XO_X-modules over the simplicial scheme resolving the diagonal of XX. As abstract descent data correspond to the flat connections for the corresponding monad, this was historically the first case in which this correspondence was noted; in positive characteristics Grothendieck called the corresponding descent data for the de Rham site “costratifications”, see

  • P. Berthelot, A. Ogus, Notes on crystalline cohomology, Princeton Univ.P. 1978. vi+243, ISBN0-691-08218-9

This corresponds to looking at a sequence of infinitesimal neighborhoods of the diagonal. This geometrical principle can be applied to other categories; it is the basis of the study of jet-schemes and close in spirit to some constructions in synthetic differential geometry.

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\cup_N M_n = M) differential filtration. Every RR-RR-bimodule has a differential part, i.e. the maximal differential submodule of MM.

Regular differential operators, as defined by Grothendieck, are the elements of the differential part Diff(R,R)Diff(R,R) of Hom(R,R)Hom(R,R) i.e. a maximal differential subbimodule in Hom(R,R)Hom(R,R). The operators in Diff(R,R) nDiff(R,R)_n are called the differential operators of degree n\leq n. If RBR\to B is a ring morphism, then the differential part of BB via its natural RR-RR-bimodule structure is also an object of R\RingR\backslash \mathrm{Ring}; in particular Diff(R,R)Diff(R,R) is a ring and RDiff(R,R)R\hookrightarrow Diff(R,R) is an embedding of rings.

More generally (and in the affine case equivalently), for a SS-scheme XX, let P X/S nP^n_{X/S} denote the sheaf (O X f 1(O S)O X)/I n+1(O_X\otimes_{f^{-1}(O_S)} O_X)/I^{n+1}, where II is the ideal of the diagonal (this makes sense since the diagonal morphism is an immersion), and f:XSf:X\rightarrow S the structure morphism. Consider P X/S nP^n_{X/S} as O XO_X-module via the morphism O XO X f 1(O S)O XO_X\rightarrow O_X\otimes_{f^{-1}(O_S)} O_X, aa1a\mapsto a\otimes 1.

Zoran: is here silent assumption of separatedness of XX over SS important ?

Lars: No, since the diagonal is an immersion (see EGAI, 5.3.9.), but what I wrote was nonsense nontheless. I fixed it, I hope.

For O XO_X-modules E,FE,F, Diff S(F,E) nDiff_S(F,E)_n is defined to be Hom O X(P X/S n O XF,E)Hom_{O_X}(P_{X/S}^n\otimes_{O_X} F, E). Note that P X/S nP^n_{X/S} has two canonical structures as O XO_X-module given by the projections p i:X× SXXp_i: X \times_S X\rightarrow X. The tensor product P X/S n O XFP_{X/S}^n \otimes_{O_X} F is understood to be constructed via p 1p_1 and considered as an O XO_X-module via p 0p_0.

In the affine case, and in characteristics zero, the sheaf of regular differential operators is locally isomorphic to the Weyl algebra. For that simple case, a good reference is

  • S. C. Coutinho, A primer of algebraic DD-modules, London Math. Soc. Stud. Texts, 33, Cambridge University Press, Cambridge, 1995. xii+207 pp.

Regular differential operators have been nontrivially generalized to noncommutative rings (and schemes) by V. Lunts and A. L. Rosenberg?, as well as to the setting of braided monoidal categories. Their motivation is an analogue of a Beilinson-Bernstein localization theorem for quantum groups. The category of differential bimodules is categorically characterized in their work as the minimal coreflective topologizing monoidal subcategory of the abelian monoidal category of RR-RR-bimodules which is containing RR. In the case of noncommutative rings, Lunts-Rosenberg definition of differential operators has been recovered from a different perspective in the setup of noncommutative algebraic geometry represented by monoidal categories; the emphasis is on the duality between infinitesimals and differential operators:

See also regular differential operator in noncommutative geometry.

Revised on March 6, 2013 19:25:56 by Zoran Škoda (161.53.130.104)