On finite categories of algebraic varieties

TL;DR

This paper proves the decidability of whether a finitely generated category of irreducible algebraic varieties over characteristic zero fields is finite, and introduces a Burnside criterion with applications to algebraic dynamical systems.

Contribution

It establishes the decidability of finiteness for categories of algebraic varieties and provides a new Burnside criterion with practical applications.

Findings

Finiteness of such categories is decidable.

A Burnside finiteness criterion is developed.

Applications to algebraic dynamical systems are demonstrated.

Abstract

We prove that the finiteness of a finitely generated category of irreducible algebraic varieties over a field of characteristic zero is decidable. We also obtain a Burnside finiteness criterion for such a category, with applications to algebraic dynamical systems of several maps.