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Sandwich Bernstein-Sato Polynomials and Bernstein's Inequality

Jack Jeffries, David Lieberman

Abstract

Bernstein's inequality is a central result in the theory of $D$-modules on smooth varieties. While Bernstein's inequality fails for rings of differential operators on general singularities, recent work of Àlvarez Montaner, Hernández, Jeffries, Núñez-Betancourt, Teixeira, and Witt establishes Bernstein's inequality for invariants of finite groups in characteristic zero and certain other mild singularities in positive characteristic. Motivated by extending this result to new classes of singular rings, we introduce a ``two-sided'' analogue of the Bernstein-Sato polynomial which we call the sandwich Bernstein-Sato polynomial. We apply this notion to give an effective criterion to verify Bernstein's inequality, and apply this to show that Bernstein's inequality holds for the coordinate ring of $\mathbb{P}^a \times \mathbb{P}^b$ via the Segre embedding. We also establish a number of examples and basic results on sandwich Bernstein-Sato polynomials.

Sandwich Bernstein-Sato Polynomials and Bernstein's Inequality

Abstract

Bernstein's inequality is a central result in the theory of -modules on smooth varieties. While Bernstein's inequality fails for rings of differential operators on general singularities, recent work of Àlvarez Montaner, Hernández, Jeffries, Núñez-Betancourt, Teixeira, and Witt establishes Bernstein's inequality for invariants of finite groups in characteristic zero and certain other mild singularities in positive characteristic. Motivated by extending this result to new classes of singular rings, we introduce a ``two-sided'' analogue of the Bernstein-Sato polynomial which we call the sandwich Bernstein-Sato polynomial. We apply this notion to give an effective criterion to verify Bernstein's inequality, and apply this to show that Bernstein's inequality holds for the coordinate ring of via the Segre embedding. We also establish a number of examples and basic results on sandwich Bernstein-Sato polynomials.
Paper Structure (17 sections, 45 theorems, 165 equations)

This paper contains 17 sections, 45 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Rings of differential operators
  4. Filtered rings and modules
  5. Filtrations on rings of differential operators
  6. Generator filtrations
  7. Bernstein filtrations
  8. Localized filtrations
  9. Conditions for linear simplicity
  10. Bernstein's inequality and linear simplicity
  11. Sandwich Bernstein-Sato polynomials
  12. Definition
  13. The bimodule $D_{R_f}[s]\boldsymbol{f^s}$
  14. Bernstein inequality for bimodules and existence of sandwich Bernstein-Sato polynomials
  15. Linear simplicity from one sandwich Bernstein-Sato polynomial
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $R$ be a finitely generated $\mathbb{N}$-graded algebra over a field $K$ of characteristic zero. Suppose that there exists some nonzerodivisor $f$ in the singular locus of $R$ that admits a sandwich Bernstein-Sato polynomial with no nonnegative integer roots. Then Bernstein's inequality holds fo

Theorems & Definitions (114)

  • Theorem 1.1: Corollary \ref{['thm-main-BI']}, cf. Remark \ref{['rem:BIintro']}
  • Theorem 1.2: Theorem \ref{['thm:Segre']}
  • Theorem 1.3: Theorems \ref{['thm:LSimpliesSBS']} and \ref{['SBSimpliesLS']} and Corollary \ref{['cor:toric']}
  • Theorem 1.4: Propositions \ref{['prop-divides']} and \ref{['prop-equal']} and Example \ref{['ex-f=y']}
  • Theorem 1.5: Theorem \ref{['thm:diffsig']}
  • Definition 2.1
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Example 2.6
  • ...and 104 more