On the generators of coordinate algebras of affine ind-varieties

TL;DR

This paper investigates the structure of coordinate rings of affine ind-varieties, revealing that non-algebraic affine ind-varieties have uncountably generated coordinate rings and dense subspaces of countable dimension.

Contribution

It demonstrates that coordinate rings of non-algebraic affine ind-varieties lack countable generators and possess dense subspaces of countable dimension, advancing understanding of their algebraic structure.

Findings

Non-isomorphic affine ind-varieties have uncountably generated coordinate rings.

Coordinate rings contain dense subspaces of countable dimension.

Results differentiate algebraic and non-algebraic affine ind-varieties.

Abstract

In this paper we study the structure of the coordinate ring of an affine ind-variety. We prove that any coordinate ring of an affine ind-variety which is not isomorphic to an affine algebraic variety does not have a countable set of generators. Also we prove that coordinate rings of affine ind-varieties have an everywhere dense subspace of countable dimension.