[Paper Review] A generalization of the b-function lemma

Sam Raskin|arXiv (Cornell University)|Nov 15, 2016

Algebraic Geometry and Number Theory4 references3 citations

TL;DR

This paper generalizes the classical b-function lemma to non-holonomic D-modules by introducing the concept of holonomic defect, a numerical invariant measuring deviation from holonomicity. It proves that D-module operations—such as pushforward, pullback, and Verdier duality—preserve this defect, extending key results in D-module theory beyond the holonomic setting and enabling new applications in geometric representation theory and t-exactness for affine maps.

ABSTRACT

We establish some cohomological bounds in D-module theory that are known in the holonomic case and folklore in general. The method rests on a generalization of the b-function lemma for non-holonomic D-modules.

Motivation & Objective

• To extend the classical b-function lemma beyond holonomic D-modules to a broader class of coherent D-modules.
• To define and study a new numerical invariant, holonomic defect, measuring the failure of a D-module to be holonomic.
• To establish that D-module operations (pushforward, pullback, duality) preserve holonomic defect, generalizing known results in the holonomic case.
• To provide a conceptual foundation for the t-exactness of f! for affine morphisms in non-holonomic settings.
• To enable new applications in geometric representation theory, particularly for non-holonomic D-modules arising from transforms like Fourier-Deligne.

Proposed method

• Introduces the Gabber-Kashiwara-Sato (GKS) filtration as a tool to define holonomic defect via cohomological support conditions.
• Uses the GKS filtration to define holonomic defect δ as the smallest integer such that FδGKS(F) = F for a D-module F.
• Applies the GKS filtration to reduce questions to the smooth case, where singular support dimension controls the filtration.
• Proves that D-module operations (f*, f!, f*, dR, and Verdier duality) preserve the GKS filtration and thus holonomic defect.
• Employs pro-complexes to handle t-exactness of f! for affine maps f, even when f! is not classically defined on non-holonomic modules.
• Reduces the main preservation result to the case of affine open embeddings, using singular support analysis and dimension bounds.

Experimental results

Research questions

• RQ1 How can the b-function lemma be generalized to non-holonomic D-modules?
• RQ2 What numerical invariant captures the deviation of a D-module from holonomicity in a way that is preserved under D-module operations?
• RQ3 Does pushforward along an affine morphism preserve t-structure truncation (i.e., left t-exactness) for non-holonomic D-modules?
• RQ4 Can the preservation of holonomic defect under D-module operations be established without relying on finite-length arguments or holonomicity?
• RQ5 What is the role of the GKS filtration in controlling singular support dimensions and cohomological bounds in D-module theory?

Key findings

• The holonomic defect δ of a coherent D-module F is preserved under pushforward f*, pullback f!, and Verdier duality, generalizing the b-function lemma to non-holonomic settings.
• For a smooth variety X, a coherent D-module F has holonomic defect δ if and only if its singular support has dimension ≤ dim X + δ.
• The GKS filtration F•GKS(F) is compatible with open restrictions, closed embeddings, and filtered colimits, enabling reduction to the smooth case.
• The result implies that f! is left t-exact for affine morphisms f, even when f! is not defined on non-holonomic modules in the classical sense, by working in the pro-complex framework.
• The category of D-modules with holonomic defect δ is closed under submodules, quotients, extensions, and filtered colimits, forming a full subcategory of D(X).
• The proof of preservation under f! for affine maps relies on the GKS filtration and singular support dimension bounds, with the key lemma showing that j* preserves holonomic defect for affine open embeddings.