Holonomic Weyl algebra module ($d(M)=n$)

A nonzero finitely generated Weyl algebra module over $A_n(\mathbb{C})$ is holonomic when it attains the lower bound in the Bernstein inequality for Weyl algebra modules. The zero module is also conventionally called holonomic. Nonzero subquotients of a holonomic module remain holonomic. Additivity and integrality of the leading Hilbert-polynomial multiplicity then bound the number of nonzero factors in any submodule chain, giving finite composition length.

Every nonzero subquotient of a holonomic Weyl algebra module over a characteristic-zero field has dimension $n$, by Bernstein inequality for Weyl algebra modules and dimension and multiplicity in a filtered exact sequence. Each strict step in a submodule chain consumes positive multiplicity of a filtered module. With a standard graded algebra normalization these multiplicities are integral. There are therefore at most $m(M)$ strict steps. The module is Artinian and Noetherian, and its module length is bounded by that multiplicity.