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

Chandan Bhaumik, Husney Parvez Sarwar

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.

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

Abstract

Let be a symbolic (or an extended symbolic) Rees algebra (need not be Noetherian) of dimension . Let be a finitely generated projective -module of rank . Then P has a unimodular element. This improves the classical result of Serre for the mentioned class of algebras.
Paper Structure (5 sections, 11 theorems, 14 equations)

This paper contains 5 sections, 11 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Recollection of basic definitions and results
  3. Notation.
  4. Recollection of Rees algebra and Symbolic Rees Algebras.
  5. Existence of a unimodular element

Key Result

Theorem 1.1

(Theorem uesra) Let $R$ be a commutative noetherian domain of dimension $d$ and $I$ an ideal of $R$. Let $A= \mathcal{R}_{s}(I)$ or $\mathcal{R}_{s}(I,x^{-1})$ (symbolic or extended symbolic Rees algebra)(see Definition defn:sym) and $P$ a finitely generated projective $A$-module of rank $\geq d+1$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Definition 2.1: Unimodular Element
  • Definition 2.2: Valuation Dimension
  • Theorem 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 10 more