On the \(\Gamma\)-cohomology of rings of numerical polynomials and \(E_\infty\) structures on \(K\)-theory
Saved in:
| Title: | On the \(\Gamma\)-cohomology of rings of numerical polynomials and \(E_\infty\) structures on \(K\)-theory |
|---|---|
| Authors: | Baker, Andrew, Richter, Birgit |
| Publisher Information: | EMS Press, Berlin |
| Subject Terms: | (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Johnson-Wilson spectra, Topological \(K\)-theory, \(\Gamma\)-cohomology, \(K\)-theory, Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.), Structured ring spectra |
| Description: | The aim of the paper is to prove the existence and uniqueness of \(E_\infty\)-ring structures on spectra related to \(K\)-theory. If \(E\) is a homotopy commutative ring spectrum satisfying the Künneth condition \[ E^\ast(E^{\wedge n})\cong \text{Hom}_{E_\ast}((E_\ast E)^{\oplus n}, E_\ast) \] there is an obstruction theory for the extension of the given structure to an \(E_\infty\)-structure developed by A. Robinson. He constructed a cohomology theory \(H\Gamma^\ast\) for commutative algebras called \(\Gamma\)-cohomology. The obstructions lie in groups \(H\Gamma^{n,2-n}(E_\ast E| E_\ast E)\) while the extensions are determined by elements in \(H\Gamma^{n,1-n}\), \(n\geq 3\). The bigrading \((s,t)\) refers to the cohomological degree \(s\) and the internal degree \(t\). The authors show that the obstruction groups for the spectra \(KU\), \(KO\), their localizations \(KU_{(p)}\), \(KO_{(p)}\), the Adams summand \(E(1)\), their completions \(KU^{\wedge}_p\), \(KO^{\wedge}_p\), \(E(1)^{\wedge}_p\), and the \(I_n\)-adic completion of \(\widehat{E(n)}\) of the Johnson-Wilson spectrum \(E(n)\) vanish in degrees \(\geq 2\). Hence these spectra have unique \(E_\infty\)-structures. Some of these results have been known. The present paper provides a unified proof. The results about \(KU^{\wedge}_p\) and \(E(1)^{\wedge}_p\) are not proven explicitly but follow directly by passage to continuous \(\Gamma\)-cohomology, the obstruction groups relevant for completed spectra. By standard techniques the existence results lift to the connective covers of the ring spectra considered. The main techniques involve continuous \(\Gamma\)-cohomology, the \(\Gamma\)-cohomology of numerical polynomials, and local-to-global arguments. |
| Document Type: | Article |
| File Description: | application/xml |
| DOI: | 10.4171/cmh/31 |
| Access URL: | https://zbmath.org/2242659 |
| Accession Number: | edsair.c2b0b933574d..a771296cc1b82f906826e4ebe2affd1b |
| Database: | OpenAIRE |
Nájsť tento článok vo Web of Science | FullText | Text: Availability: 0 CustomLinks: – Url: https://explore.openaire.eu/search/publication?articleId=c2b0b933574d%3A%3Aa771296cc1b82f906826e4ebe2affd1b Name: EDS - OpenAIRE (s4221598) Category: fullText Text: View record at OpenAIRE – Url: https://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=EBSCO&SrcAuth=EBSCO&DestApp=WOS&ServiceName=TransferToWoS&DestLinkType=GeneralSearchSummary&Func=Links&author=Baker%20A Name: ISI Category: fullText Text: Nájsť tento článok vo Web of Science Icon: https://imagesrvr.epnet.com/ls/20docs.gif MouseOverText: Nájsť tento článok vo Web of Science |
|---|---|
| Header | DbId: edsair DbLabel: OpenAIRE An: edsair.c2b0b933574d..a771296cc1b82f906826e4ebe2affd1b RelevancyScore: 685 AccessLevel: 3 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 685 |
| IllustrationInfo | |
| Items | – Name: Title Label: Title Group: Ti Data: On the \(\Gamma\)-cohomology of rings of numerical polynomials and \(E_\infty\) structures on \(K\)-theory – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Baker%2C+Andrew%22">Baker, Andrew</searchLink><br /><searchLink fieldCode="AR" term="%22Richter%2C+Birgit%22">Richter, Birgit</searchLink> – Name: Publisher Label: Publisher Information Group: PubInfo Data: EMS Press, Berlin – Name: Subject Label: Subject Terms Group: Su Data: <searchLink fieldCode="DE" term="%22%28Co%29homology+of+commutative+rings+and+algebras+%28e%2Eg%2E%2C+Hochschild%2C+André-Quillen%2C+cyclic%2C+dihedral%2C+etc%2E%29%22">(Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)</searchLink><br /><searchLink fieldCode="DE" term="%22Johnson-Wilson+spectra%22">Johnson-Wilson spectra</searchLink><br /><searchLink fieldCode="DE" term="%22Topological+%5C%28K%5C%29-theory%22">Topological \(K\)-theory</searchLink><br /><searchLink fieldCode="DE" term="%22%5C%28%5CGamma%5C%29-cohomology%22">\(\Gamma\)-cohomology</searchLink><br /><searchLink fieldCode="DE" term="%22%5C%28K%5C%29-theory%22">\(K\)-theory</searchLink><br /><searchLink fieldCode="DE" term="%22Spectra+with+additional+structure+%28%5C%28E%5F%5Cinfty%5C%29%2C+%5C%28A%5F%5Cinfty%5C%29%2C+ring+spectra%2C+etc%2E%29%22">Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)</searchLink><br /><searchLink fieldCode="DE" term="%22Structured+ring+spectra%22">Structured ring spectra</searchLink> – Name: Abstract Label: Description Group: Ab Data: The aim of the paper is to prove the existence and uniqueness of \(E_\infty\)-ring structures on spectra related to \(K\)-theory. If \(E\) is a homotopy commutative ring spectrum satisfying the Künneth condition \[ E^\ast(E^{\wedge n})\cong \text{Hom}_{E_\ast}((E_\ast E)^{\oplus n}, E_\ast) \] there is an obstruction theory for the extension of the given structure to an \(E_\infty\)-structure developed by A. Robinson. He constructed a cohomology theory \(H\Gamma^\ast\) for commutative algebras called \(\Gamma\)-cohomology. The obstructions lie in groups \(H\Gamma^{n,2-n}(E_\ast E| E_\ast E)\) while the extensions are determined by elements in \(H\Gamma^{n,1-n}\), \(n\geq 3\). The bigrading \((s,t)\) refers to the cohomological degree \(s\) and the internal degree \(t\). The authors show that the obstruction groups for the spectra \(KU\), \(KO\), their localizations \(KU_{(p)}\), \(KO_{(p)}\), the Adams summand \(E(1)\), their completions \(KU^{\wedge}_p\), \(KO^{\wedge}_p\), \(E(1)^{\wedge}_p\), and the \(I_n\)-adic completion of \(\widehat{E(n)}\) of the Johnson-Wilson spectrum \(E(n)\) vanish in degrees \(\geq 2\). Hence these spectra have unique \(E_\infty\)-structures. Some of these results have been known. The present paper provides a unified proof. The results about \(KU^{\wedge}_p\) and \(E(1)^{\wedge}_p\) are not proven explicitly but follow directly by passage to continuous \(\Gamma\)-cohomology, the obstruction groups relevant for completed spectra. By standard techniques the existence results lift to the connective covers of the ring spectra considered. The main techniques involve continuous \(\Gamma\)-cohomology, the \(\Gamma\)-cohomology of numerical polynomials, and local-to-global arguments. – Name: TypeDocument Label: Document Type Group: TypDoc Data: Article – Name: Format Label: File Description Group: SrcInfo Data: application/xml – Name: DOI Label: DOI Group: ID Data: 10.4171/cmh/31 – Name: URL Label: Access URL Group: URL Data: <link linkTarget="URL" linkTerm="https://zbmath.org/2242659" linkWindow="_blank">https://zbmath.org/2242659</link> – Name: AN Label: Accession Number Group: ID Data: edsair.c2b0b933574d..a771296cc1b82f906826e4ebe2affd1b |
| PLink | https://erproxy.cvtisr.sk/sfx/access?url=https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsair&AN=edsair.c2b0b933574d..a771296cc1b82f906826e4ebe2affd1b |
| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.4171/cmh/31 Languages: – Text: Undetermined Subjects: – SubjectFull: (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) Type: general – SubjectFull: Johnson-Wilson spectra Type: general – SubjectFull: Topological \(K\)-theory Type: general – SubjectFull: \(\Gamma\)-cohomology Type: general – SubjectFull: \(K\)-theory Type: general – SubjectFull: Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) Type: general – SubjectFull: Structured ring spectra Type: general Titles: – TitleFull: On the \(\Gamma\)-cohomology of rings of numerical polynomials and \(E_\infty\) structures on \(K\)-theory Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Baker, Andrew – PersonEntity: Name: NameFull: Richter, Birgit IsPartOfRelationships: – BibEntity: Identifiers: – Type: issn-locals Value: edsair |
| ResultId | 1 |