Varieties Associated to Linear operators

TL;DR

This paper introduces affine varieties linked to linear operators, establishing a Galois connection and a Nullstellensatz analogue, showing these varieties form a category equivalent to classical affine varieties.

Contribution

It generalizes affine varieties to those associated with linear operators and proves their categorical equivalence to classical affine varieties.

Findings

Established a Galois connection between subsets of $A^n_K$ and polynomial ideals.

Proved a Nullstellensatz version for varieties associated with linear operators.

Showed the category of these varieties is equivalent to the classical affine varieties category.

Abstract

We introduce and study the notion of affine varieties associated to ordered bases and establish Galois connection between the power set of $A^n_K$ and the power set of $K[x_1, . . ., x_n]$, and then induce a Galois correspondence. We generalize the idea by defining affine varieties associated to linear operators. We produce Hilbert's Nullstellensatz version for such varieties and show that there is a 1-1 correspondence between this kind of varieties in $A^n_K$ and the "usual" affine varieties in $A^n_K$. We prove that the \usual" affine varieties forms a skeleton for the category of all affine varieties associated to linear operators, and hence they are equivalent categories.