On the class group of a pullback
Let \(R\) be an integral domain. The class group of \(R\), denoted \(\text{Cl}(R)\), is the group of \(t\)-invertible fractional \(t\)-ideals of \(R\) under \(t\)-multiplication modulo its subgroup of principal fractional ideals. It contains \(\text{Pic}(R)\), the Picard group of \(R\). The local class group of \(R\) is defined by \(G(R)=\text{Cl}(R)/\text{Pic}(R)\). This paper studies the class group, the Picard group, and the local class group of integral domains issued from pullback constructions. Here is the main theorem: Let \(T\) be an integral domain, \(I\) an ideal of \(T\), \(\varphi:T\rightarrow T/I\) the natural projection, \(D\) an integral domain which is a subring of \(T/I\). Let \(R:=\varphi^{-1}(D)\). Assume that qf\((D)\subseteq T/I\). Then the following statements are equivalent: (1) The map \(\phi:U(T)\rightarrow U(T/I)/U(D)\), \(u\mapsto \varphi(u)U(D)\) is surjective (here \(U(A)\) denotes the group of units of the ring \(A\)); (2) The sequence \(0 \rightarrow \text{Cl}(D) \rightarrow \text{Cl}(R) \rightarrow \text{Cl}(T)\) is an exact sequence of well-defined group homomorphisms; (3) The sequence \(0 \rightarrow \text{Pic}(D) \rightarrow \text{Pic}(R) \rightarrow \text{Pic}(T)\) is an exact sequence of well-defined group homomorphisms. A corollary of this result (theorem 1.13) recovers classic results by \textit{D. F. Anderson} and \textit{A. Ryckaert} [J. Pure Appl. Algebra 52, No.3, 199-212 (1988; Zbl 0668.13013)], \textit{M. Fontana} and \textit{S. Gabelli} [J. Algebra 181, No.3, 803-835 (1996; Zbl 0871.13006)] and \textit{M. Khalis} and \textit{D. Nour El Abidine} [in: Commutative ring theory. Proc. 2nd Int. Cong., Fés. 1995, Lect. Notes Pure Appl. Math. 185, 377-386 (1997; Zbl 0877.13011)]. Section 2 of the paper investigates the general case (i.e., qf\((D)\) not necessarily contained in \(T/I\)).
scientific article; zbMATH DE number 1758341
scientific article