On the class group and the local class group of a pullback
Let \(M\) be a non-zero maximal ideal of an integral domain \(T\), \(k=T/M\), \(\varphi: T\to k\) the natural homomorphism, and \(D\) a proper subring of \(k\). Let \(R\) be the pullback \[ \begin{tikzcd} R \ar[r] \ar[d] & D\ar[d] \\ T\ar[r,"\varphi"'] & k=T/M \quad . \end{tikzcd} \] The authors are interested in the Picard group, the class group \(\mathcal C(R)\) and the local class group \(\mathcal G(R)=\mathcal C(R)/\mathrm{Pic}(R)\) of \(R\) and their connections with the corresponding group of \(T\) and \(D\). After giving some general results on pullbacks in section 1, section 2 is devoted to the computation of \(\mathcal C(R)\), \(\mathrm{Pic}(R)\) and \(\mathcal G(R)\). In particular, under some hypothesis there is a canonical homomorphism \(\mathcal C(D)\to\mathcal C(R)\). Section 3 is devoted to the study of the map \(\mathcal C(D)\to\mathcal C(R)\) which in general is not surjective. Section 4 deals with particular properties which may or may be not inherited by \(R\) [ Bezout, \(\ldots\) ].
scientific article; zbMATH DE number 883787