A corollary of the b-function lemma

TL;DR

This paper explores the structure of certain D-modules associated with algebraic varieties and regular functions, providing a detailed description of submodules that extend holonomic D-modules across divisors.

Contribution

It offers a new characterization of D_X[s]-submodules extending a given holonomic D-module on the complement of a divisor, advancing understanding of D-module extensions.

Findings

Describes all relevant D_X[s]-submodules extending M⊗f^s

Provides a criterion for submodule inclusion based on the b-function lemma

Enhances the theory of D-module extensions in algebraic geometry

Abstract

Let $X$ be an algebraic variety, $f$ a regular function, $j:U\subset X$ the complement to the locus of vanishing of $f$, and $M$ a holonomic D-module on $U$. Consider the $D_U[s]$-module $M\otimes "f^s"$. The goal of this note is to describe all $D_X[s]$-submodules $N\subset j_*(M\otimes "f^s")$ such that $j^*(N)\simeq M\otimes "f^s"$.