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The divisor class groups of some rings of holomorphic functions

Identifieur interne : 000E94 ( Istex/Curation ); précédent : 000E93; suivant : 000E95

The divisor class groups of some rings of holomorphic functions

Auteurs : David Prill [Suisse]

Source :

RBID : ISTEX:491014BA8FA92C7A077C5D8B45A1D28A907CC1F9

English descriptors

Abstract

Abstract: Let (X, x O) be a normal complex analytic space andA⋐X 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.

Url:
DOI: 10.1007/BF01110367

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ISTEX:491014BA8FA92C7A077C5D8B45A1D28A907CC1F9

Le document en format XML

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<div type="abstract" xml:lang="en">Abstract: Let (X, x O) be a normal complex analytic space andA⋐X 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.</div>
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