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Determination of the stably free cancellation property for orders

Werner Bley, Tommy Hofmann, Henri Johnston

TL;DR

This work delivers practical algorithms to decide whether an $oldsymbol{ O}_K$-order in a finite-dimensional semisimple $K$-algebra has SFC, highlighting three complementary strategies: a naive maximal-order approach, a fail-fast sampling method, and a fiber-product reduction to smaller algebras. By combining local-global tools (idèles, reduced norms, and class groups) with the Eichler condition framework and Reiner--Ullom/Bley--Boltje machinery, the authors reduce complex SFC questions to explicit finite computations. They apply these methods to integral group rings $oldsymbol{ Z}[G]$, deriving new classifications for $|G|\, ext{up to}\, 383$ (and extending to $|G|\, ext{up to}\, 1023$ with gaps), showing, for instance, which groups force SFC or fail SFC and how quotients influence the outcome. The results advance the state-of-the-art by integrating deep algebraic tools with algorithmic techniques, enabling concrete verifications for groups previously out of reach and offering a framework for broader computational exploration of SFC in orders.

Abstract

Let $K$ be a number field, let $A$ be a finite-dimensional semisimple $K$-algebra, and let $Λ$ be an $\mathcal{O}_{K}$-order in $A$. We give practical algorithms that determine whether $Λ$ has stably free cancellation (SFC). As an application, we determine all finite groups $G$ of order at most $383$ such that the integral group ring $\mathbb{Z}[G]$ has SFC.

Determination of the stably free cancellation property for orders

TL;DR

This work delivers practical algorithms to decide whether an -order in a finite-dimensional semisimple -algebra has SFC, highlighting three complementary strategies: a naive maximal-order approach, a fail-fast sampling method, and a fiber-product reduction to smaller algebras. By combining local-global tools (idèles, reduced norms, and class groups) with the Eichler condition framework and Reiner--Ullom/Bley--Boltje machinery, the authors reduce complex SFC questions to explicit finite computations. They apply these methods to integral group rings , deriving new classifications for (and extending to with gaps), showing, for instance, which groups force SFC or fail SFC and how quotients influence the outcome. The results advance the state-of-the-art by integrating deep algebraic tools with algorithmic techniques, enabling concrete verifications for groups previously out of reach and offering a framework for broader computational exploration of SFC in orders.

Abstract

Let be a number field, let be a finite-dimensional semisimple -algebra, and let be an -order in . We give practical algorithms that determine whether has stably free cancellation (SFC). As an application, we determine all finite groups of order at most such that the integral group ring has SFC.
Paper Structure (58 sections, 72 theorems, 106 equations, 4 tables, 5 algorithms)

This paper contains 58 sections, 72 theorems, 106 equations, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries on lattices and orders
  3. Lattices and orders
  4. Localizations and completions
  5. Locally free lattices and the principal genus
  6. Stable isomorphism
  7. Locally free class groups
  8. Review of cancellation properties for orders
  9. Stably free and locally free cancellation
  10. Totally definite quaternion algebras and the Eichler condition
  11. Review of cancellation properties for integral group rings
  12. Fiber products and reduction criteria for SFC
  13. Fiber products
  14. Review of results of Reiner--Ullom
  15. A reduction criterion for SFC
  16. ...and 43 more sections

Key Result

Theorem 1.1

There exists an algorithm that, given an arbitrary $\mathcal{O}$-order $\Lambda$ in a finite-dimensional semisimple $K$-algebra, decides whether $\Lambda$ has SFC.

Theorems & Definitions (151)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 3.1: Fröhlich, Swan
  • proof
  • Corollary 3.2
  • Corollary 3.3
  • proof
  • ...and 141 more