The divisor class groups of some rings of holomorphic functions
Identifieur interne : 000E94 ( Istex/Corpus ); précédent : 000E93; suivant : 000E95The divisor class groups of some rings of holomorphic functions
Auteurs : David PrillSource :
- Mathematische Zeitschrift [ 0025-5874 ] ; 1971-03-01.
English descriptors
- KwdEn :
- Abelian, Abelian group, Algebraic, Analytic space, Analytic spaces, Canonical mapping, Cech, Cohomology, Cohomology groups, Cohomology sequence, Commutative, Commutative diagram, Compact stein, Countable, Countable stein, Direct limit, Divisor, Divisor class, Divisor class group, Divisor class groups, Exact sequence, Factorization, Finite group, Finitely, First isomorphism, Germ, Holomorphic, Holomorphic function, Holomorphic functions, Homomorphism, Ideal sheaf, Injective, Intersection matrix, Irreducible, Irreducible subvariety, Isomorphism, Lemma, Lira, Local ring, Math, Meromorphic functions, Noetherian, Open neighborhood, Open neighborhoods, Open sets, Open stein neighborhood, Open subsets, Prill, Princeton university press, Quotient, Resp, Sheaf, Singular locus, Singular point, Singularity, Star neighborhood, Stein manifolds, Subset, Subvariety, Surjective, Topological, Topological vector spaces, Topology, Unique factorization domain, Unique factorization domains.
- Teeft :
- Abelian, Abelian group, Algebraic, Analytic space, Analytic spaces, Canonical mapping, Cech, Cohomology, Cohomology groups, Cohomology sequence, Commutative, Commutative diagram, Compact stein, Countable, Countable stein, Direct limit, Divisor, Divisor class, Divisor class group, Divisor class groups, Exact sequence, Factorization, Finite group, Finitely, First isomorphism, Germ, Holomorphic, Holomorphic function, Holomorphic functions, Homomorphism, Ideal sheaf, Injective, Intersection matrix, Irreducible, Irreducible subvariety, Isomorphism, Lemma, Lira, Local ring, Math, Meromorphic functions, Noetherian, Open neighborhood, Open neighborhoods, Open sets, Open stein neighborhood, Open subsets, Prill, Princeton university press, Quotient, Resp, Sheaf, Singular locus, Singular point, Singularity, Star neighborhood, Stein manifolds, Subset, Subvariety, Surjective, Topological, Topological vector spaces, Topology, Unique factorization domain, Unique factorization domains.
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
Links to Exploration step
ISTEX:491014BA8FA92C7A077C5D8B45A1D28A907CC1F9Le document en format XML
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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></front></TEI><istex><corpusName>springer-journals</corpusName><keywords><teeft><json:string>divisor</json:string><json:string>isomorphism</json:string><json:string>holomorphic</json:string><json:string>cohomology</json:string><json:string>divisor class groups</json:string><json:string>divisor class group</json:string><json:string>resp</json:string><json:string>subset</json:string><json:string>topology</json:string><json:string>holomorphic functions</json:string><json:string>noetherian</json:string><json:string>prill</json:string><json:string>finitely</json:string><json:string>open neighborhood</json:string><json:string>homomorphism</json:string><json:string>quotient</json:string><json:string>subvariety</json:string><json:string>countable</json:string><json:string>abelian</json:string><json:string>lira</json:string><json:string>local ring</json:string><json:string>topological</json:string><json:string>cech</json:string><json:string>irreducible</json:string><json:string>singularity</json:string><json:string>surjective</json:string><json:string>injective</json:string><json:string>commutative</json:string><json:string>exact sequence</json:string><json:string>factorization</json:string><json:string>open neighborhoods</json:string><json:string>holomorphic function</json:string><json:string>direct limit</json:string><json:string>divisor class</json:string><json:string>ideal sheaf</json:string><json:string>analytic space</json:string><json:string>commutative diagram</json:string><json:string>sheaf</json:string><json:string>algebraic</json:string><json:string>compact stein</json:string><json:string>finite group</json:string><json:string>unique factorization domain</json:string><json:string>irreducible subvariety</json:string><json:string>intersection matrix</json:string><json:string>singular point</json:string><json:string>abelian group</json:string><json:string>stein manifolds</json:string><json:string>germ</json:string><json:string>open stein neighborhood</json:string><json:string>open sets</json:string><json:string>unique factorization domains</json:string><json:string>cohomology sequence</json:string><json:string>analytic spaces</json:string><json:string>topological vector spaces</json:string><json:string>countable stein</json:string><json:string>open subsets</json:string><json:string>cohomology groups</json:string><json:string>first isomorphism</json:string><json:string>meromorphic functions</json:string><json:string>princeton university press</json:string><json:string>singular locus</json:string><json:string>canonical mapping</json:string><json:string>star neighborhood</json:string><json:string>math</json:string><json:string>lemma</json:string></teeft></keywords><author><json:item><name>Dr. David Prill</name><affiliations><json:string>Forschungsinstitut für Mathematik der ETH Zürich, CH-8006, Zürich, Schweiz</json:string></affiliations></json:item></author><articleId><json:string>BF01110367</json:string><json:string>Art5</json:string></articleId><arkIstex>ark:/67375/1BB-BM62N7H1-7</arkIstex><language><json:string>eng</json:string></language><originalGenre><json:string>OriginalPaper</json:string></originalGenre><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.</abstract><qualityIndicators><score>9.088</score><pdfWordCount>10441</pdfWordCount><pdfCharCount>44823</pdfCharCount><pdfVersion>1.3</pdfVersion><pdfPageCount>23</pdfPageCount><pdfPageSize>468 x 684 pts</pdfPageSize><refBibsNative>false</refBibsNative><abstractWordCount>174</abstractWordCount><abstractCharCount>1053</abstractCharCount><keywordCount>0</keywordCount></qualityIndicators><title>The divisor class groups of some rings of holomorphic functions</title><genre><json:string>research-article</json:string></genre><host><title>Mathematische Zeitschrift</title><language><json:string>unknown</json:string></language><publicationDate>1971</publicationDate><copyrightDate>1971</copyrightDate><issn><json:string>0025-5874</json:string></issn><eissn><json:string>1432-1823</json:string></eissn><journalId><json:string>209</json:string></journalId><volume>121</volume><issue>1</issue><pages><first>58</first><last>80</last></pages><genre><json:string>journal</json:string></genre><subject><json:item><value>Mathematics, general</value></json:item></subject></host><namedEntities><unitex><date></date><geogName></geogName><orgName></orgName><orgName_funder></orgName_funder><orgName_provider></orgName_provider><persName></persName><placeName></placeName><ref_url></ref_url><ref_bibl></ref_bibl><bibl></bibl></unitex></namedEntities><ark><json:string>ark:/67375/1BB-BM62N7H1-7</json:string></ark><categories><wos><json:string>1 - 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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.</p></abstract><textClass><keywords scheme="Journal Subject"><list><head>Mathematics</head><item><term>Mathematics, general</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="1970-12-16">Created</change><change when="1971-03-01">Published</change><change xml:id="refBibs-istex" who="#ISTEX-API" when="2017-11-30">References added</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/491014BA8FA92C7A077C5D8B45A1D28A907CC1F9/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="corpus springer-journals not found" wicri:toSee="no header"><istex:xmlDeclaration>version="1.0" encoding="UTF-8"</istex:xmlDeclaration><istex:docType PUBLIC="-//Springer-Verlag//DTD A++ V2.4//EN" URI="http://devel.springer.de/A++/V2.4/DTD/A++V2.4.dtd" name="istex:docType"></istex:docType><istex:document><Publisher><PublisherInfo><PublisherName>Springer-Verlag</PublisherName><PublisherLocation>Berlin/Heidelberg</PublisherLocation></PublisherInfo><Journal><JournalInfo JournalProductType="ArchiveJournal" NumberingStyle="Unnumbered"><JournalID>209</JournalID><JournalPrintISSN>0025-5874</JournalPrintISSN><JournalElectronicISSN>1432-1823</JournalElectronicISSN><JournalTitle>Mathematische Zeitschrift</JournalTitle><JournalAbbreviatedTitle>Math Z</JournalAbbreviatedTitle><JournalSubjectGroup><JournalSubject Type="Primary">Mathematics</JournalSubject><JournalSubject Type="Secondary">Mathematics, general</JournalSubject></JournalSubjectGroup></JournalInfo><Volume><VolumeInfo VolumeType="Regular" TocLevels="0"><VolumeIDStart>121</VolumeIDStart><VolumeIDEnd>121</VolumeIDEnd><VolumeIssueCount>4</VolumeIssueCount></VolumeInfo><Issue IssueType="Regular"><IssueInfo TocLevels="0"><IssueIDStart>1</IssueIDStart><IssueIDEnd>1</IssueIDEnd><IssueArticleCount>7</IssueArticleCount><IssueHistory><CoverDate><DateString>1971</DateString><Year>1971</Year><Month>3</Month></CoverDate></IssueHistory><IssueCopyright><CopyrightHolderName>Springer-Verlag</CopyrightHolderName><CopyrightYear>1971</CopyrightYear></IssueCopyright></IssueInfo><Article ID="Art5"><ArticleInfo Language="En" ArticleType="OriginalPaper" NumberingStyle="Unnumbered" TocLevels="0" ContainsESM="No"><ArticleID>BF01110367</ArticleID><ArticleDOI>10.1007/BF01110367</ArticleDOI><ArticleSequenceNumber>5</ArticleSequenceNumber><ArticleTitle Language="En">The divisor class groups of some rings of holomorphic functions</ArticleTitle><ArticleFirstPage>58</ArticleFirstPage><ArticleLastPage>80</ArticleLastPage><ArticleHistory><RegistrationDate><Year>2005</Year><Month>1</Month><Day>22</Day></RegistrationDate><Received><Year>1970</Year><Month>12</Month><Day>16</Day></Received></ArticleHistory><ArticleCopyright><CopyrightHolderName>Springer-Verlag</CopyrightHolderName><CopyrightYear>1971</CopyrightYear></ArticleCopyright><ArticleGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ArticleGrants><ArticleContext><JournalID>209</JournalID><VolumeIDStart>121</VolumeIDStart><VolumeIDEnd>121</VolumeIDEnd><IssueIDStart>1</IssueIDStart><IssueIDEnd>1</IssueIDEnd></ArticleContext></ArticleInfo><ArticleHeader><AuthorGroup><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><Prefix>Dr.</Prefix><GivenName>David</GivenName><FamilyName>Prill</FamilyName></AuthorName></Author><Affiliation ID="Aff1"><OrgName>Forschungsinstitut für Mathematik der ETH Zürich</OrgName><OrgAddress><Postcode>CH-8006</Postcode><City>Zürich</City><Country>Schweiz</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>Let (<Emphasis Type="Italic">X</Emphasis>,<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript><Emphasis FontCategory="NonProportional">O</Emphasis>) be a normal complex analytic space and<Emphasis Type="Italic">A</Emphasis>⋐<Emphasis Type="Italic">X</Emphasis> a connected Stein compact set, i.e. a compact subset of<Emphasis Type="Italic">X</Emphasis> which has a basis of open neighborhoods which are Stein spaces. We restrict attention to those<Emphasis Type="Italic">A</Emphasis> such that<Emphasis Type="Italic">R</Emphasis>=<Emphasis Type="Italic">H</Emphasis><Superscript>0</Superscript><Emphasis FontCategory="NonProportional">O</Emphasis>) is Noetherian. In Section I various exact sequences involving the divisor class group of<Emphasis Type="Italic">R</Emphasis>, denoted<Emphasis Type="Italic">C(R)</Emphasis>, are developed. (If<Emphasis Type="Italic">A</Emphasis> is a point, one of these sequences is well-known [24], [39].)</Para><Para>Let<Emphasis Type="Italic">B</Emphasis> be a connected compact Stein set on a normal variety<Emphasis Type="Italic">Y</Emphasis> such that<Emphasis Type="Italic">S</Emphasis>=<Emphasis Type="Italic">H</Emphasis><Superscript>o</Superscript>(B,<Subscript><Emphasis Type="Italic">Y</Emphasis></Subscript><Emphasis FontCategory="NonProportional">O</Emphasis>) and<Emphasis Type="Italic">T</Emphasis>=<Emphasis Type="Italic">H</Emphasis><Superscript>o</Superscript>(<Emphasis Type="Italic">A</Emphasis>×<Emphasis Type="Italic">B</Emphasis>,<Subscript><Emphasis Type="Italic">X</Emphasis>×<Emphasis Type="Italic">Y</Emphasis></Subscript><Emphasis FontCategory="NonProportional">O</Emphasis> are Noetherian. In II we give a Künneth-type formula which relates<Emphasis Type="Italic">C(R), C(S)</Emphasis> and<Emphasis Type="Italic">C(T)</Emphasis>. 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.</Para></Abstract><ArticleNote Type="Misc"><SimplePara>The author wishes to thank Columbia University and the Forschungsinstitut für Mathemathik der ETH (Zürich) for their hospitality and the NSF for financial support.</SimplePara></ArticleNote></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>The divisor class groups of some rings of holomorphic functions</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>The divisor class groups of some rings of holomorphic functions</title></titleInfo><name type="personal"><namePart type="termsOfAddress">Dr.</namePart><namePart type="given">David</namePart><namePart type="family">Prill</namePart><affiliation>Forschungsinstitut für Mathematik der ETH Zürich, CH-8006, Zürich, Schweiz</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Springer-Verlag</publisher><place><placeTerm type="text">Berlin/Heidelberg</placeTerm></place><dateCreated encoding="w3cdtf">1970-12-16</dateCreated><dateIssued encoding="w3cdtf">1971-03-01</dateIssued><dateIssued encoding="w3cdtf">1971</dateIssued><copyrightDate encoding="w3cdtf">1971</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract 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.</abstract><relatedItem type="host"><titleInfo><title>Mathematische Zeitschrift</title></titleInfo><titleInfo type="abbreviated"><title>Math Z</title></titleInfo><genre type="journal" displayLabel="Archive Journal" authority="ISTEX" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</genre><originInfo><publisher>Springer</publisher><dateIssued encoding="w3cdtf">1971-03-01</dateIssued><copyrightDate encoding="w3cdtf">1971</copyrightDate></originInfo><subject><genre>Mathematics</genre><topic>Mathematics, general</topic></subject><identifier type="ISSN">0025-5874</identifier><identifier type="eISSN">1432-1823</identifier><identifier type="JournalID">209</identifier><identifier type="IssueArticleCount">7</identifier><identifier type="VolumeIssueCount">4</identifier><part><date>1971</date><detail type="volume"><number>121</number><caption>vol.</caption></detail><detail type="issue"><number>1</number><caption>no.</caption></detail><extent unit="pages"><start>58</start><end>80</end></extent></part><recordInfo><recordOrigin>Springer-Verlag, 1971</recordOrigin></recordInfo></relatedItem><identifier type="istex">491014BA8FA92C7A077C5D8B45A1D28A907CC1F9</identifier><identifier type="ark">ark:/67375/1BB-BM62N7H1-7</identifier><identifier type="DOI">10.1007/BF01110367</identifier><identifier type="ArticleID">BF01110367</identifier><identifier type="ArticleID">Art5</identifier><accessCondition type="use and reproduction" contentType="copyright">Springer-Verlag, 1971</accessCondition><recordInfo><recordContentSource authority="ISTEX" authorityURI="https://loaded-corpus.data.istex.fr" valueURI="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-3XSW68JL-F">springer</recordContentSource><recordOrigin>Springer-Verlag, 1971</recordOrigin></recordInfo></mods><json:item><extension>json</extension><original>false</original><mimetype>application/json</mimetype><uri>https://api.istex.fr/document/491014BA8FA92C7A077C5D8B45A1D28A907CC1F9/metadata/json</uri></json:item></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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