The divisor class groups of some rings of holomorphic functions

Let (X, _x O) be a normal complex analytic space andAX a connected Stein compact set, i.e. a compact subset ofX which has a basis of open neighborhoods which are Stein spaces. We restrict attention to thoseA such thatR=H ⁰ O) is Noetherian. In Section I various exact sequences involving the divisor class group ofR, denotedC(R), are developed. (IfA is a point, one of these sequences is well-known 24], 39].)LetB be a connected compact Stein set on a normal varietyY such thatS=H ^o(B, _Y O) andT=H ^o(A×B, _X×Y O are Noetherian. In II we give a Künneth-type formula which relatesC(R), C(S) andC(T). In III we show certain analytic local rings are unique factorization domains, study the divisor class groups of local rings on the quotient of an analytic space by a finite group, and prove a simple result on the topology of germs of complex analytic sets. We give a function-theoretic proof that complete intersections of dimensions greater than three which have isolated singularities have local rings which are unique factorization domains.The author wishes to thank Columbia University and the Forschungsinstitut für Mathemathik der ETH (Zürich) for their hospitality and the NSF for financial support.