The Picard group of algebras over an algebraically nonclosed field
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 invertible \(A\)-module generated by \(n+1\) elements, then there exists a \(\mathbb K\)-algebra homomorphism \(\psi: \Omega_n \to A\) such that the induced homomorphism Pic(\(\Omega_n) \to\) Pic(\(A\)) satisfies \([\Omega_n(1)] \mapsto [L]\). \vskip.05in The proofs follow from well-known results concerning algebras over an arbitrary field and localizations of modules over an arbitrary commutative ring.
algebra over a non-algebraically closed field
scientific article; zbMATH DE number 5941003