nLab crystalline differential operator


For XX a smooth scheme over a field the sheaf π’Ÿ=π’Ÿ X\mathcal{D} = \mathcal{D}_X of crystalline differential operators is the the β€œenveloping algebroid of the tangent Lie algebroid” of XX: to an affine Uβ†’XU \to X it assigns the algebra that is generated over π’ͺ(U)\mathcal{O}(U) from the π’ͺ(U)\mathcal{O}(U)-module Vect(U)Vect(U) of vector fields (derivations of π’ͺ(U)\mathcal{O}(U)), subject to the relations

v 1β‹…v 2βˆ’v 2β‹…v 1=[v 1,v 2] v_1 \cdot v_2 - v_2 \cdot v_1 = [v_1, v_2]


v 1β‹…fβˆ’fβ‹…v 1=v 1(f) v_1 \cdot f - f \cdot v_1 = v_1(f)

for all v 1,v 2∈Vect(U)v_1, v_2 \in Vect(U) and f∈π’ͺ(U)f \in \mathcal{O}(U).

If the field kk has characteristic 0 this is the ordinary sheaf of differential operators


for instance section 3.1 of

  • Roman Bezrukavnikov, Noncommutative Counterparts of the Springer Resolution (pdf)

Created on March 30, 2011 at 07:02:31. See the history of this page for a list of all contributions to it.