Differential Projective Modules over Differential Rings, II

TL;DR

This paper explores the structure of differential projective modules over differential rings, focusing on their classification via a quotient monoid related to algebraic K-theory, revealing connections to constants of the ring.

Contribution

It introduces a new monoid structure to classify differential projective modules and relates it to the reduced K groups of the ring and its constants.

Findings

The quotient monoid contains the reduced K group of R as an image.

The subgroup of units in the monoid corresponds to the reduced K group of the constants.

Provides a classification framework for differential projective modules.

Abstract

Differential modules over a commutative differential ring R which are finitely generated projective as ring modules, with differential homomorphisms, form an additive category, so their isomorphism classes form a monoid. We study the quotient monoid of this monoid by the submonoid of isomorphism classes of free modules with component wise derivation. This quotient monoid has the reduced K group of R (ignoring the derivation) as an image and contains the reduced K group of the constants of R as its subgroup of units. This monoid provides a description of the isomorphism classes of differential projective R modules up to an equivalence.