{"title":"The tangent complex of K-theory","authors":"Benjamin Hennion","doi":"10.5802/JEP.161","DOIUrl":null,"url":null,"abstract":"We prove that the tangent complex of K-theory, in terms of (abelian) deformation problems over a characteristic 0 field k, is cyclic homology (over k). This equivalence is compatible with the $\\lambda$-operations. In particular, the relative algebraic K-theory functor fully determines the absolute cyclic homology over any field k of characteristic 0. \nWe also show that the Loday-Quillen-Tsygan generalized trace comes as the tangent morphism of the canonical map $BGL_\\infty \\to K$. \nThe proof builds on results of Goodwillie, using Wodzicki's excision for cyclic homology and formal deformation theory a la Lurie-Pridham.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":"2 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: K-Theory and Homology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5802/JEP.161","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that the tangent complex of K-theory, in terms of (abelian) deformation problems over a characteristic 0 field k, is cyclic homology (over k). This equivalence is compatible with the $\lambda$-operations. In particular, the relative algebraic K-theory functor fully determines the absolute cyclic homology over any field k of characteristic 0.
We also show that the Loday-Quillen-Tsygan generalized trace comes as the tangent morphism of the canonical map $BGL_\infty \to K$.
The proof builds on results of Goodwillie, using Wodzicki's excision for cyclic homology and formal deformation theory a la Lurie-Pridham.