Scheme structure of a finite-module rank stratum

First restrict to the open set $U$ where the residue-field dimension is at most $e$. This is open: from a finite presentation the condition is that its relation matrix have rank at least the number of generators minus $e$, an open minor condition. A rank-$e$ pullback necessarily maps into $U$. Around any point of $U$, Nakayama supplies a surjection ${\mathcal{O}}^e\twoheadrightarrow M$, permitting redundant zero generators when the dimension is smaller than $e$. Write a finite presentation ${\mathcal{O}}^a\stackrel{D}{\to }{\mathcal{O}}^e\to M\to 0$.

Let $J$ be the ideal generated by the entries of $D$. Under any map $T$ to this neighbourhood the pullback is free of rank $e$ exactly when $D_T=0$: if $D_T=0$ the presentation gives $M_T={\mathcal{O}}_T^e$; conversely the surjection from ${\mathcal{O}}_T^e$ to a rank-$e$ locally free module is an isomorphism, since its determinant is a unit in every local ring. Thus the vanishing ideal defines the required closed subscheme on the neighbourhood. These closed subschemes agree on overlaps by their identical functor, hence glue to a closed subscheme of $U$. Its underlying points are exactly the dimension-$e$ points. If all fibre dimensions on a subscheme are $e$, that subscheme is already contained in $U$, proving the last assertion, including nilpotents.