Generalizing Serre's Splitting Theorem and Bass's Cancellation Theorem via free-basic elements

TL;DR

This paper introduces new proofs for generalized versions of Serre's and Bass's theorems on projective modules, utilizing basic element theory and extending their applicability to Cartier algebras.

Contribution

It provides novel proofs and generalizations of classical theorems on projective modules using basic element techniques, with applications to Cartier algebras.

Findings

Generalized Serre's Splitting Theorem with new splitting conditions

Extended Bass's Cancellation Theorem to broader contexts

Applied methods to Cartier algebra cases

Abstract

We give new proofs of two results of Stafford, which generalize two famous Theorems of Serre and Bass regarding projective modules. Our techniques are inspired by the theory of basic elements. Using these methods we further generalize Serre's Splitting Theorem by imposing a condition to the splitting maps, which has an application to the case of Cartier algebras.