On Nagata's theorem for the class group (Q753871)

Let \(A\) be a domain, \(S\) a multiplicative subset of \(A\) and \(f_ S: Cl(A)\to Cl(A_ S)\) the canonical homomorphism of class groups. Nagata's theorem states that \(f_ S\) is surjective if \(A\) (and therefore \(A_ S)\) are Krull domains. \(f_ S\) is always injective if \(S\) is generated by prime elements. The authors give several conditions which imply that \(f_ S\) is onto. For instance, this is the case if \(A\) is one-dimensional or if \(S\) is generated by primes and \(A\) satisfies the ascending chain condition on principal ideals. Some counterexamples illustrate that \(f_ S\) is not surjective in slightly more general situations.