Iterated power intersections of ideals in rings of iterated differential polynomials II

TL;DR

The paper addresses when iterated power intersections of ideals vanish in rings of iterated differential polynomials. It develops an elementary localization-and-factorization approach that preserves the iterated differential structure and reduces the problem to fewer variables, enabling an induction-based analysis. The main contribution is a general vanishing result: under conditions (a)–(c), every proper ideal $I$ of $S$ satisfies $I(n)=0$, with significant corollaries for universal enveloping algebras of completely solvable Lie algebras—showing that all projective modules are finitely generated or free—and extensions to finite-codimension ideals in $U(L)$ when $[L,L]$ is nilpotent. These results broaden the applicability to positive characteristic and connect ideal-theoretic vanishing to module-theoretic structure and AR-properties in differential-polynomial rings.

Abstract

This paper is a continuation of a previous work by the author and G. Puninski where iterated intersections of powers of ideals were studied in rings of iterated differential polynomials. We present a method which can be used to show that for every proper ideal $I$ of a suitable ring of iterated differential polynomials almost all iterated intersections of powers of $I$ have to be zero. The setback of the method is that it works only if the derivations used in the construction of the ring of iterated differential polynomials satisfy additional assumptions. On the other hand, it can be applied in cases which were not covered by the aforementioned work, for example for a universal enveloping algebra of a completely solvable Lie algebra over a field of positive characteristic.

Theorems & Definitions (13)

• Proposition 1.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Theorem 3.5
• Theorem 3.6
• Remark 3.7
• Corollary 3.8
• Lemma 3.9