The Picard group of algebras over an algebraically nonclosed field
type: Item
instance of: scholarly article
title: The Picard group of algebras over an algebraically nonclosed field
zbMATH Open document ID: 1227.13008
author: Wojciech Kucharz
author: Kamil Rusek
published in: Journal of Pure and Applied Algebra
publication date: 2011-08-19T00:00:00Z
review text: This short note calculates the order of elements in Picard groups of algebras over a non-algebraically closed field \(\mathbb K\) via certain numerical invariants associated to \(\mathbb K\). In particular, let \(d_n(\mathbb K)\) be the greatest common divisor of the degrees of homogeneous polynomials \(H\) in \(\mathbb K[T_0, \dots, T_n]\) satisfying \(H(x_0, \dots, x_n) = 0\) if and only if \((x_0, \dots, x_n) = (0, \dots, 0)\). \vskip.05in There are three main results. The first is that for any \(\mathbb K\)-algebra \(A\) with the property that every maximal ideal of \(A\) has residue field \(\mathbb K\), each element of Pic(\(A\)) has finite order. Moreover, if \(L\) is an invertible \(A\)-module generated by \(n+1\) elements, then the order of \([L]\) in Pic(\(A\)) divides \(d_n(\mathbb K)\). \vskip.05in The other results involve some special \(\mathbb K\)-algebras: let \(\Omega_n(d)\), for any integer \(d\), be the subset of \(\mathbb K(T_0, \dots, T_n)\) consisting of all rational functions \(F/G\), where \(F\) and \(G\) are homogeneous polynomials in \(\mathbb K[T_0, \dots, T_n]\) such that \(\deg F = d + \deg G\), and \(G(x_0, \dots, x_n) = 0\) if and only if \((x_0, \dots, x_n) = (0, \dots, 0)\). \vskip.05in The authors show that the \(\mathbb K\)-algebra \(\Omega_n: = \Omega_n(0)\) and the \(\Omega_n\)-modules \(\Omega_n(d)\) have the following properties: 1. Every maximal ideal of \(\Omega_n\) has residue field \(\mathbb K\); 2. \(\Omega_n\) is a Noetherian ring of dimension \(n\); 3. The localization of \(\Omega_n\) at each maximal ideal is a regular local ring of dimension \(n\); 4. \(\Omega_n(d)\) is an invertible \(\Omega_n\)-module and \([\Omega_n(d)]=[\Omega_n(1)]^d\); 5. Pic(\(\Omega_n\)) is finite cyclic of order \(d_n(\mathbb K)\), generated by \([\Omega_n(1)]\). \vskip.05in The last main result says that if \(A\) is a \(\mathbb K\)-algebra with the property that every maximal ideal of \(A\) has residue field \(\mathbb K\) and \(L\) is an invert…
reviewed by: Sandra Spiroff
Mathematics Subject Classification ID: 13C20
Mathematics Subject Classification ID: 14C22
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 5941003
zbMATH Keywords: Picard group
zbMATH Keywords: algebra over a non-algebraically closed field
MaRDI profile type: Publication
full work available at URL: j.jpaa.2011.02.001
OpenAlex ID: W2028798938
cites work: Q5562621
cites work: Q3328651
cites work: Q3676243
cites work: Serre's conjecture
cites work: Q4530238
DOI: 10.1016/J.JPAA.2011.02.001
description: scientific article; zbMATH DE number 5941003