Tressl's Structure Theorem for Separable Algebras
Gabriel Ng
TL;DR
The paper extends Tressl's structure theorem from differential algebras over characteristic-0 rings to separable algebras over differential rings of any characteristic, enabling applications to differentially large fields in positive characteristic. It develops a robust differential-algebraic framework (characteristic sets, autoreduced/coherent sets, and initials/separants) and proves that, under separability, one can decompose a differentially finitely generated algebra into a finitely presented part over a polynomial base, achieving a finite presentation after localizing at a nonzero element. This generalization fills a gap for positive characteristic and supports uniform approaches to large differential fields, with explicit construction of $P$, $B$, and a nonzero $h$ such that $S_h = (B\cdot P)_h$. The results preserve a canonical isomorphism $B\otimes_R P \cong B\cdot P$ and extend the reach of differential-algebraic structure theorems beyond characteristic $0$.
Abstract
This short paper presents a generalisation of Tressl's structure theorem for differentially finitely generated algebras over differential rings of characteristic 0 to the case of separable algebras over differential rings of arbitrary characteristic.