Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Complete Classification of Ideal Chomp Games on Low-Rank Algebras

Leopold Karl

TL;DR

This work delivers a complete classification of the Ideal Chomp Game on $\\bar{K}$-algebras of rank at most 6, showing Alice wins except for five specific local algebras; the result combines the Structure Theorem for Artinian rings, Poonen’s local-algebra classification, and targeted reductions with computational verification. It situates the finite-rank classification within a broader context, referencing Henson’s classical results and outlining open questions for higher Krull-dimension algebras. The methodology couples rigorous game-theoretic analysis with algebraic structure theory to translate moves into ring quotients and ideals, producing a concrete, checkable map of outcomes across the rank-6 landscape. The work lays groundwork for systematic exploration of higher-dimensional cases and invites further refinement of strategies in the presence of more complex ring-theoretic invariants.

Abstract

We completely classify winning strategies in the Ideal Chomp Game played on $\bar{K}$-algebras R of rank at most 6. In this two-player combinatorial game, players alternately add generators to build an ideal inside a given ring R, with the player who builds an ideal equal to the entire ring losing. We prove that player A has a winning strategy on all $\bar{K}$-algebras R up to rank 6 except for five specific cases: $\bar{K}$ itself, $\bar{K}[x, y]/(x, y)^2$, and three other local algebras. Our methods combine game-theoretic analysis with the structure theory of Artinian rings and computational verification. We also discuss a classical result of Henson on winning strategies in the Ideal Chomp Game, as well as ideas and open questions about the Ideal Chomp Game on higher-dimensional $\bar{K}$-algebras.

A Complete Classification of Ideal Chomp Games on Low-Rank Algebras

TL;DR

This work delivers a complete classification of the Ideal Chomp Game on -algebras of rank at most 6, showing Alice wins except for five specific local algebras; the result combines the Structure Theorem for Artinian rings, Poonen’s local-algebra classification, and targeted reductions with computational verification. It situates the finite-rank classification within a broader context, referencing Henson’s classical results and outlining open questions for higher Krull-dimension algebras. The methodology couples rigorous game-theoretic analysis with algebraic structure theory to translate moves into ring quotients and ideals, producing a concrete, checkable map of outcomes across the rank-6 landscape. The work lays groundwork for systematic exploration of higher-dimensional cases and invites further refinement of strategies in the presence of more complex ring-theoretic invariants.

Abstract

We completely classify winning strategies in the Ideal Chomp Game played on -algebras R of rank at most 6. In this two-player combinatorial game, players alternately add generators to build an ideal inside a given ring R, with the player who builds an ideal equal to the entire ring losing. We prove that player A has a winning strategy on all -algebras R up to rank 6 except for five specific cases: itself, , and three other local algebras. Our methods combine game-theoretic analysis with the structure theory of Artinian rings and computational verification. We also discuss a classical result of Henson on winning strategies in the Ideal Chomp Game, as well as ideas and open questions about the Ideal Chomp Game on higher-dimensional -algebras.

Paper Structure

This paper contains 14 sections, 19 theorems, 28 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Game Instructions
  4. A Reformulation of the Game
  5. First Winning Conditions
  6. Poonen's Classification of local $\overline{K}$-Algebras up to Rank 6
  7. Classification of the Ideal Chomp Game on $\bar{K}$-algebras up to rank $6$
  8. Proof of the Classification
  9. Extensions and Outlook
  10. Henson's Theorem
  11. The Ideal Chomp Game on higher dimensional $\bar{K}$-Algebras
  12. $\bar{K}$-Algebras of Krull Dimension 1
  13. $\bar{K}$-Algebras of Krull Dimension 2
  14. $\bar{K}$-Algebras of Krull Dimension $\geqslant 3$

Key Result

Theorem 1.0.2

Alice has a winning strategy in the Ideal Chomp Game on any $\bar{K}$-algebra $R$ up to rank 6, except if $R$ is isomorphic to

Figures (1)

  • Figure 1: A game of Chomp where player A (red) loses.

Theorems & Definitions (29)

  • Example 1.0.1
  • Theorem 1.0.2: Classification of the Ideal Chomp Game on $\bar{K}$-algebras up to rank 6
  • Definition 2.1.1: Ideal Chomp Game
  • Example 2.1.2
  • Definition 2.2.1: Quotient Chomp Game
  • Proposition 2.2.2
  • Proposition 2.3.1
  • Proposition 2.3.2
  • Corollary 2.3.3
  • Proposition 2.3.4
  • ...and 19 more