A Functorial Version of Chevalley's Theorem on Constructible Sets

TL;DR

This paper generalizes Chevalley's theorem to tensor spaces, showing that properties of high-dimensional tensors can be characterized by their lower-dimensional counterparts, simplifying the description of certain tensor subsets.

Contribution

It introduces a functorial approach to Chevalley's theorem, extending the classical result to a broad class of tensor space subsets.

Findings

Description of tensor subsets can be pulled back from lower-dimensional cases

Generalization of Chevalley's theorem to tensor spaces

Simplifies characterization of tensor properties

Abstract

To determine whether an $n\times n$-matrix has rank at most $r$ it suffices to check that the $(r+1)\times (r+1)$-minors have rank at most $r$. In other words, to describe the set of $n\times n$-matrices with the property of having rank at most $r$, we only need the description of the corresponding subset of $(r+1)\times (r+1)$-matrices. We will generalize this observation to a large class of subsets of tensor spaces. A description of certain subsets of a high-dimensional tensor space can always be pulled back from a description of the corresponding subset in a fixed lower-dimensional tensor space.