A Functorial Version of Chevalley's Theorem on Constructible Sets

To determine whether an n× n-matrix has rank at most r it suffices to check that the (r+1)× (r+1)-minors have rank at most r. In other words, to describe the set of n× n-matrices with the property of having rank at most r, we only need the description of the corresponding subset of (r+1)× (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.