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Rigidity of Graded Integral Domains and of their Veronese Subrings

Daniel Daigle

TL;DR

This work develops a comprehensive framework to study rigidity and locally nilpotent derivations on graded domains and their Veronese subrings. It introduces and interrelates key invariants ($\mathrm{LND}$, $\mathrm{HLND}$, $\mathrm{ML}$, $\mathbb{G}(B)$, $\bar{\mathbb{G}}(B)$) with Veronese constructions $B^{(H)}$, proving descent results for nonrigidity and providing an extension theory for derivations across Veronese subrings. The paper gives structural descriptions of the set $\mathscr{X}(B)$ of subgroups yielding nonrigidity, and a systematic treatment of the Z-graded and Pham–Brieskorn rings, including explicit formulas for saturation indices and rigidity behavior under localization and taking fibers. It connects algebraic derivations with geometric cylinders on $\operatorname{Proj} B$, yielding criteria that translate between cylindrical geometry and rigidity properties, and culminates in a detailed analysis of $\bar{\mathbb{G}}(B)$ and its implications for the Makar–Limanov invariant and the transcendence degree over its field of constants. The results unify descent, extension, and explicit computations across broad classes of graded rings, including Pham–Brieskorn rings, and offer tools to predict rigidity status of Veronese subrings from that of the ambient ring.

Abstract

A ring R is said to be rigid if the only locally nilpotent derivation of R is the zero derivation. Let G be an abelian group, and B = (direct sum of B_i for i in G) be a G-graded commutative integral domain of characteristic 0. For each subgroup H of G, consider the Veronese subring B(H) of B, defined by B(H) = (direct sum of the B_i for i in H). We study the following questions. If B is non-rigid, does it follow that B(H) is non-rigid? Can derivations of B(H) be extended to derivations of B? What are the properties of the set of subgroups H of G such that B(H) is non-rigid?

Rigidity of Graded Integral Domains and of their Veronese Subrings

TL;DR

This work develops a comprehensive framework to study rigidity and locally nilpotent derivations on graded domains and their Veronese subrings. It introduces and interrelates key invariants (, , , , ) with Veronese constructions , proving descent results for nonrigidity and providing an extension theory for derivations across Veronese subrings. The paper gives structural descriptions of the set of subgroups yielding nonrigidity, and a systematic treatment of the Z-graded and Pham–Brieskorn rings, including explicit formulas for saturation indices and rigidity behavior under localization and taking fibers. It connects algebraic derivations with geometric cylinders on , yielding criteria that translate between cylindrical geometry and rigidity properties, and culminates in a detailed analysis of and its implications for the Makar–Limanov invariant and the transcendence degree over its field of constants. The results unify descent, extension, and explicit computations across broad classes of graded rings, including Pham–Brieskorn rings, and offer tools to predict rigidity status of Veronese subrings from that of the ambient ring.

Abstract

A ring R is said to be rigid if the only locally nilpotent derivation of R is the zero derivation. Let G be an abelian group, and B = (direct sum of B_i for i in G) be a G-graded commutative integral domain of characteristic 0. For each subgroup H of G, consider the Veronese subring B(H) of B, defined by B(H) = (direct sum of the B_i for i in H). We study the following questions. If B is non-rigid, does it follow that B(H) is non-rigid? Can derivations of B(H) be extended to derivations of B? What are the properties of the set of subgroups H of G such that B(H) is non-rigid?
Paper Structure (11 sections, 69 theorems, 94 equations)

This paper contains 11 sections, 69 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. From derivations of $B$ to derivations of $B^{(H)}$
  4. The group $\operatorname{\bar{\mathbb{G}}}(B)$
  5. From derivations of $B^{(H)}$ to derivations of $B$
  6. Partial description of the set $\mathscr{X}(B)$
  7. $\mathbb{Z}$-gradings
  8. Pham-Brieskorn rings
  9. Rigidity of $B_{(x)}$
  10. Fibers of a homogeneous polynomial
  11. A result about $\operatorname{\bar{\mathbb{G}}}(B)$

Key Result

Theorem 1.2

Let ${\rm \bf k}$ be a field of characteristic $0$ and $B = \bigoplus_{i \in \mathbb{N}}B_i$ an $\mathbb{N}$-graded affine ${\rm \bf k}$-domain such that $\operatorname{\text{\rm trdeg}}(B:B_0) \ge 2$.

Theorems & Definitions (146)

  • Theorem 1.2
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.3: Thm 1.1 of GotoNoethGrRings83
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 136 more