Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cox-Gorenstein algebras

Ugo Bruzzo, Rodrigo Gondim, Rafael Holanda, William D. Montoya

TL;DR

This work develops the theory of Cox-Gorenstein algebras, i.e., Artinian $G$-graded $\\mathbb{K}$-algebras with Poincaré duality, and analyzes their Lefschetz properties in a toric context. It introduces a comprehensive $G$-graded Macaulay-Matlis duality, encodes Hilbert data via Hasse-Hilbert diagrams, and establishes a Hessian-based criterion for Lefschetz behavior in the toric/G-graded setting. A key contribution is the toric–algebra correspondence that connects Cox rings of toric varieties with $G$-gradings, including minimal reductions that preserve duality and allow embedding primitive cohomology into Cox-Gorenstein frameworks. The results yield a toric Codimension-One phenomenon for Picard-number-one cases and provide tools (Euler identities and Hessian criteria) to study Lefschetz properties in nonstandard gradings, with implications for primitive cohomology and principal embeddings in toric geometry.

Abstract

We study G-graded Artinian algebras having Poincaré duality, considering in particular their Lefschetz properties. We also prove a correspondence between the toric setup and the G-graded one, provide an application to toric geometry, and prove a Hessian criterion in the G-graded setup

Cox-Gorenstein algebras

TL;DR

This work develops the theory of Cox-Gorenstein algebras, i.e., Artinian -graded -algebras with Poincaré duality, and analyzes their Lefschetz properties in a toric context. It introduces a comprehensive -graded Macaulay-Matlis duality, encodes Hilbert data via Hasse-Hilbert diagrams, and establishes a Hessian-based criterion for Lefschetz behavior in the toric/G-graded setting. A key contribution is the toric–algebra correspondence that connects Cox rings of toric varieties with -gradings, including minimal reductions that preserve duality and allow embedding primitive cohomology into Cox-Gorenstein frameworks. The results yield a toric Codimension-One phenomenon for Picard-number-one cases and provide tools (Euler identities and Hessian criteria) to study Lefschetz properties in nonstandard gradings, with implications for primitive cohomology and principal embeddings in toric geometry.

Abstract

We study G-graded Artinian algebras having Poincaré duality, considering in particular their Lefschetz properties. We also prove a correspondence between the toric setup and the G-graded one, provide an application to toric geometry, and prove a Hessian criterion in the G-graded setup
Paper Structure (16 sections, 18 theorems, 67 equations)

This paper contains 16 sections, 18 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: $G$-graded algebras and toric varieties
  3. $G$-graded algebras
  4. Toric varieties
  5. $G$-graded polynomial rings and toric varieties
  6. Cox-Gorenstein algebras
  7. Hasse-Hilbert diagrams
  8. On the Codimension One conjecture
  9. Our special case.
  10. Toric Macaulay-Matlis duality
  11. Artinian and Gorenstein minimal reductions
  12. Toric Lefschetz properties and the Toric Hessian criteria
  13. Toric Lefschetz properties
  14. Toric differential Euler identity
  15. Toric Hessian Criterion
  16. ...and 1 more sections

Key Result

Proposition 2.5

Let $X$ be an irreducible normal variety with only constant invertible global functions and finitely generated divisor class group such that the Cox ring of $X$ is a polynomial ring. If $X$ is complete, then $X$ is a toric variety.

Theorems & Definitions (66)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Remark 3.1
  • Remark 3.2
  • Example 3.3
  • Example 3.4
  • ...and 56 more