Existence of unimodular element in a projective module over symbolic Rees algebras

TL;DR

This paper proves that finitely generated projective modules of rank at least the dimension over symbolic Rees algebras always contain a unimodular element, extending Serre's classical result to a broader class of algebras.

Contribution

It establishes the existence of unimodular elements in projective modules over symbolic Rees algebras, generalizing Serre's theorem beyond Noetherian cases.

Findings

Finitely generated projective modules of rank ≥ dimension have unimodular elements over symbolic Rees algebras.

Extends classical Serre's result to non-Noetherian symbolic Rees algebras.

Provides new insights into module structure over complex algebraic constructions.

Abstract

Let $A$ be a symbolic (or an extended symbolic) Rees algebra (need not be Noetherian) of dimension $d$. Let $P$ be a finitely generated projective $A$-module of rank $\geq$ $d$. Then P has a unimodular element. This improves the classical result of Serre for the mentioned class of algebras.