On finitude of the number of isomorphism classes in the category NT

TL;DR

This paper proves that for the category of commutative nil graded algebras encoded by matrices, the number of isomorphism classes is finite for each fixed size if and only if the ground field is finite, linking algebra classification to field finiteness.

Contribution

It establishes a precise criterion connecting the finiteness of isomorphism classes to the finiteness of the ground field for algebras of size at least four.

Findings

Number of isomorphism classes is finite iff the ground field is finite for n ≥ 4.

Finiteness of classification depends critically on the nature of the ground field.

Results extend understanding of algebra classification over different fields.

Abstract

The category $\bcalNT$ was defined in \cite{Lobos2}, it is a category whose objects are commutative nil graded algebras over a field, defined by presentation encoded by triangular matrices. A natural problem related to this category is to reach a complete classification up to isomorphism of its objects. Based in some results coming from \cite{Lobos2}, we can divide this problem by working with encoding matrices of a fixed size $n.$ In \cite{Lobos3} and \cite{Lobos4}, there are several advances for this search, in particular, in \cite{Lobos3} one can see that, for small matrices, the number of isomorphism classes seems to be finite and independent on the ground field. That fact, opened a series of questions related with the number of isomorphism classes and its relation with the ground field. At that point it was no clear, under which conditions of the ground field, this number could be finite. In this article, among other results, we prove that for each $n\geq4,$ the number of isomorphism classes is finite if and only if the ground field is finite.