On the \(\Gamma\)-cohomology of rings of numerical polynomials and \(E_\infty\) structures on \(K\)-theory
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| Názov: | On the \(\Gamma\)-cohomology of rings of numerical polynomials and \(E_\infty\) structures on \(K\)-theory |
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| Autori: | Baker, Andrew, Richter, Birgit |
| Informácie o vydavateľovi: | EMS Press, Berlin |
| Predmety: | (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 |
| Popis: | 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. |
| Druh dokumentu: | Article |
| Popis súboru: | application/xml |
| DOI: | 10.4171/cmh/31 |
| Prístupová URL adresa: | https://zbmath.org/2242659 |
| Prístupové číslo: | edsair.c2b0b933574d..a771296cc1b82f906826e4ebe2affd1b |
| Databáza: | OpenAIRE |
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