On Picard groups of \(\mathbb{Z}\)-graded domains
Let \(A = \sum_{n \in \mathbb{Z}} A_ n\) be a \(\mathbb{Z}\)-graded domain and let \(\Phi : \text{Pic} (A_ 0) \to \text{Pic} (A)\) be the canonical group homomorphism induced by the inclusion \(A_ 0 \subset A\) on the Picard groups. The main result on the kernel of \(\Phi\) is that it is generated by \([tA_ \alpha] \in \text{Pic} (A_ 0)\) over \(\mathbb{Z}\) where \(t \in A_{-d} \backslash \{0\}\) and \(\text{Ker} \Phi \simeq \mathbb{Z}/e(A) \mathbb{Z}\), where \(e(A) = m(A)/d(A)\) if the invertible index \(d(A)>0\) and \(e(A) = 1\) if \(d(A) = 0\). If \(A\) is quasinormal, then the canonical homomorphism \(\lambda : \text{H Pic} (A) \to \text{Pic} (A)\) is an isomorphism (theorem 2.7), where H Pic means the homogeneous part of Pic. Assuming that \(d=d(A)>0\) and \(\Phi^ H : \text{Pic} (A_ 0) \to \text{H Pic} (A)\) is the canonical homomorphism (the author shows that \(\text{Im} \Phi \subseteq \text{H Pic} A)\), \(\text{Coker} \Phi^ H\) is \(d\)- torsion, that is, \(d[I] \in \text{Im} \Phi^ H\) for any \([I] \in \text{H Pic} (A)\). If \(\text{Pic} (A_ 0)\) is a torsion group, then \(\text{H Pic} (A)\) is a torsion group (theorem 2.11). If \(A\) is a quasinormal domain, then \(\text{Coker} \Phi\) is \(d\)-torsion (corollary 2.12). If \(A_ 0\) is locally quasinormal, \(A_ 1\) and \(A_{-1}\) are invertible \(A_ 0\)-modules, \(d = d(A) > 0\), then \(A\) is almost quasinormal (theorem 4.8). If \(d(A) > 0\) and \(A_ 0\) is a Dedekind domain, then \(A\) is almost quasinormal and moreover quasinormal (corollaries 4.9, 4.12).
scientific article; zbMATH DE number 554126