D-module - Holonomic Modules - Properties and Characterizations

Properties and Characterizations

Holonomic modules have a tendency to behave like finite-dimensional vector spaces. For example, their length is finite. Also, M is holonomic if and only if all cohomology groups of the complex Li∗(M) are finite-dimensional K-vector spaces, where i is the closed immersion of any point of X.

For any D-module M, the dual module is defined by

Holonomic modules can also be characterized by a homological condition: M is holonomic if and only if D(M) is concentrated (seen as an object in the derived category of D-modules) in degree 0. This fact is a first glimpse of Verdier duality and the Riemann–Hilbert correspondence. It is proven by extending the homological study of regular rings (especially what is related to global homological dimension) to the filtered ring D_X.

Another characterization of holonomic modules is via symplectic geometry. The characteristic variety Ch(M) of any D-module M is, seen as a subvariety of the cotangent bundle T∗X of X, an involutive variety. The module is holonomic if and only if Ch(M) is Lagrangian.