Finiteness of Leaps in the sense of Hasse-Schmidt of reduced rings

TL;DR

This paper establishes conditions under which derivations of certain algebraic structures are infinitely integrable in the Hasse-Schmidt sense, and proves the finiteness of leaps in reduced rings containing a field.

Contribution

It provides new sufficient conditions for the infinite integrability of derivations in reduced rings and complete intersections, and demonstrates the finiteness of leaps in these contexts.

Findings

Derivations are $ $-integrable under specified conditions.

Finiteness of leaps is proven for reduced rings with a field.

Results apply to rings with regular base rings and complete intersections.

Abstract

We give sufficent conditions for a derivation of a $k$-algebra $A$ of finite type to be $\infty$-integrable in the sense of Hasse-Schmidt, when $A$ is a complete intersection, or when $A$ is reduced and $k$ is a regular ring. As a consequence, we prove that, if in addition $A$ contains a field, then the set of leaps of $A$ is finite along the minimal primes of certain Fitting ideal of $\Omega_{A/k}$.