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Iarrobino's symmetric decomposition for self-dual modules

Maciej Wojtala

TL;DR

The paper extends Iarrobino's symmetric decomposition from Artinian Gorenstein algebras to finite-length self-dual modules over local rings, deriving a decomposition h_M(t) = ∑_{i=0}^d Δ_i(t) with Δ_i(t) symmetric about (d−i)/2 and showing h_M is built from symmetric blocks. It generalizes Macaulay bounds to modules and develops apolarity machinery, including a non-graded version of Kunte's self-duality criterion, to detect and construct self-dual modules. A central contribution is a complete classification of possible local Hilbert functions for self-dual modules of small degree (up to m ≤ 8), with explicit lists and constructive examples that reveal new phenomena not seen in the graded/algebra case. These results advance understanding of the structure of self-dual modules, providing tools for deformation theory, apolarity-based constructions, and potential applications to tensor structure and algebraic complexity theory.

Abstract

We generalize Iarrobino's symmetric decomposition for the associated graded algebra of an Artinian Gorenstein algebra to a symmetric decomposition of finite-length self-dual modules over a local algebra, and we deduce consequences for the Hilbert functions of such self-dual modules. We classify the local Hilbert functions for small degree modules. We generalize Kunte's criterion for self-duality in terms of Macaulay's inverse systems.

Iarrobino's symmetric decomposition for self-dual modules

TL;DR

The paper extends Iarrobino's symmetric decomposition from Artinian Gorenstein algebras to finite-length self-dual modules over local rings, deriving a decomposition h_M(t) = ∑_{i=0}^d Δ_i(t) with Δ_i(t) symmetric about (d−i)/2 and showing h_M is built from symmetric blocks. It generalizes Macaulay bounds to modules and develops apolarity machinery, including a non-graded version of Kunte's self-duality criterion, to detect and construct self-dual modules. A central contribution is a complete classification of possible local Hilbert functions for self-dual modules of small degree (up to m ≤ 8), with explicit lists and constructive examples that reveal new phenomena not seen in the graded/algebra case. These results advance understanding of the structure of self-dual modules, providing tools for deformation theory, apolarity-based constructions, and potential applications to tensor structure and algebraic complexity theory.

Abstract

We generalize Iarrobino's symmetric decomposition for the associated graded algebra of an Artinian Gorenstein algebra to a symmetric decomposition of finite-length self-dual modules over a local algebra, and we deduce consequences for the Hilbert functions of such self-dual modules. We classify the local Hilbert functions for small degree modules. We generalize Kunte's criterion for self-duality in terms of Macaulay's inverse systems.
Paper Structure (21 sections, 42 theorems, 49 equations, 1 table)

This paper contains 21 sections, 42 theorems, 49 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Self-dual algebras and modules
  4. Preliminaries
  5. Modules and submodules
  6. Local rings
  7. Modularity and filtrations
  8. Associated graded ring and module
  9. Macaulay's Bound
  10. Iarrobino's symmetric decomposition
  11. Apolarity
  12. Classification for small degrees
  13. m = 1:
  14. m = 2:
  15. m = 3:
  16. ...and 6 more sections

Key Result

Theorem 1.5

If $R$ is a local ring and $M$ is a self-dual module, then the following statements hold true:

Theorems & Definitions (101)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5: symmetric decomposition
  • Example 1.6
  • Theorem 1.7: Kunte's criterion
  • Example 1.8
  • Proposition 1.9
  • Proposition 1.10
  • ...and 91 more