[Paper Review] Algorithms for D-modules --- restriction, tensor product, localization, and local cohomology groups

Toshinori Ôaku, Nobuki Takayama|ArXiv.org|May 2, 1998

Polynomial and algebraic computation4 references4 citations

TL;DR

This paper presents algorithmic methods for computing fundamental functors in D-module theory—restriction, tensor product, localization, and local cohomology groups—for systems of linear partial differential equations with polynomial coefficients. Using Gröbner basis techniques in the Weyl algebra, the authors provide constructive algorithms that enable effective computation of these invariants, offering a computational framework for algebraic D-modules with applications in algebraic geometry and mathematical physics.

ABSTRACT

We describe algorithms for computing various functors for algebraic D-modules, i.e. systems of linear partial differential equations with polynomial coefficients. We will give algorithms for restriction, tensor product, localization, and local cohomology groups for all degrees.

Motivation & Objective

• To develop effective computational algorithms for core functors in D-module theory.
• To address the lack of algorithmic tools for handling restriction, tensor product, localization, and local cohomology in systems of linear PDEs.
• To provide a constructive framework for algebraic D-modules using computational algebraic geometry.
• To enable practical computation of invariants such as local cohomology groups and derived functors in the context of polynomial coefficient differential operators.

Proposed method

• Utilization of Gröbner basis methods in the Weyl algebra to algorithmically compute D-module functors.
• Implementation of algorithms for restriction of D-modules along subvarieties via ideal-theoretic constructions.
• Computation of tensor products of D-modules using module structures over the Weyl algebra.
• Application of localization techniques via Ore-type extensions to invertive elements, particularly in the context of polynomial ideals.
• Computation of local cohomology groups for all degrees using derived functors and resolution techniques in the Weyl algebra.
• Integration of these algorithms into a coherent computational framework for algebraic D-modules.

Experimental results

Research questions

• RQ1 How can restriction functors for D-modules be effectively computed using algebraic algorithms?
• RQ2 What algorithmic approach enables the computation of tensor products of D-modules with polynomial coefficient systems?
• RQ3 Can localization of D-modules be systematically computed, especially in the context of singular support?
• RQ4 How can local cohomology groups be computed for all degrees using constructive algebraic methods?
• RQ5 What computational framework supports the effective manipulation of derived functors in D-module theory?

Key findings

• The paper successfully constructs algorithms for computing restriction functors of D-modules using Gröbner basis techniques in the Weyl algebra.
• Tensor product operations for D-modules are made algorithmically accessible through module-theoretic constructions over the Weyl algebra.
• Localization of D-modules is implemented via Ore-type extensions, enabling computation of D-modules localized at polynomial ideals.
• Local cohomology groups are computed for all degrees using derived functors and resolution methods, providing a complete computational framework.
• The algorithms are fully constructive and applicable to systems of linear partial differential equations with polynomial coefficients.
• The work establishes a foundation for effective computation in algebraic D-modules, supporting further applications in algebraic geometry and mathematical physics.