Finiteness properties of congruence classes of infinite matrices

TL;DR

This paper investigates the structure of certain infinite matrix spaces, proving that their stable, closed subsets are characterized by finitely many equations, revealing finiteness properties in an infinite-dimensional setting.

Contribution

It establishes that all stable, closed subsets of infinite matrices are finitely defined, extending finiteness concepts to infinite-dimensional matrix spaces.

Findings

Any stable, closed subset of infinite matrices is finitely defined.

The result applies up to symmetry, indicating a universal finiteness property.

Provides a foundational understanding of the algebraic structure of infinite matrices.

Abstract

We look at spaces of infinite-by-infinite matrices, and consider closed subsets that are stable under simultaneous row and column operations. We prove that up to symmetry, any of these closed subsets is defined by finitely many equations.