{"title":"拓扑循环同调","authors":"L. Hesselholt, T. Nikolaus","doi":"10.1201/9781351251624-15","DOIUrl":null,"url":null,"abstract":"This survey of topological cyclic homology is a chapter in the Handbook on Homotopy Theory. We give a brief introduction to topological cyclic homology and the cyclotomic trace map following Nikolaus-Scholze, followed by a proof of B\\\"okstedt periodicity that closely resembles B\\\"okstedt's original unpublished proof. We explain the extension of B\\\"{o}kstedt periodicity by Bhatt-Morrow-Scholze from perfect fields to perfectoid rings and use this to give a purely p-adic proof of Bott periodicity. Finally, we evaluate the cofiber of the assembly map in p-adic topological cyclic homology for the cyclic group of order p and a perfectoid ring of coefficients.","PeriodicalId":378948,"journal":{"name":"Handbook of Homotopy Theory","volume":"7 5","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"237","resultStr":"{\"title\":\"Topological cyclic homology\",\"authors\":\"L. Hesselholt, T. Nikolaus\",\"doi\":\"10.1201/9781351251624-15\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This survey of topological cyclic homology is a chapter in the Handbook on Homotopy Theory. We give a brief introduction to topological cyclic homology and the cyclotomic trace map following Nikolaus-Scholze, followed by a proof of B\\\\\\\"okstedt periodicity that closely resembles B\\\\\\\"okstedt's original unpublished proof. We explain the extension of B\\\\\\\"{o}kstedt periodicity by Bhatt-Morrow-Scholze from perfect fields to perfectoid rings and use this to give a purely p-adic proof of Bott periodicity. Finally, we evaluate the cofiber of the assembly map in p-adic topological cyclic homology for the cyclic group of order p and a perfectoid ring of coefficients.\",\"PeriodicalId\":378948,\"journal\":{\"name\":\"Handbook of Homotopy Theory\",\"volume\":\"7 5\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-05-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"237\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Handbook of Homotopy Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1201/9781351251624-15\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Handbook of Homotopy Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1201/9781351251624-15","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
This survey of topological cyclic homology is a chapter in the Handbook on Homotopy Theory. We give a brief introduction to topological cyclic homology and the cyclotomic trace map following Nikolaus-Scholze, followed by a proof of B\"okstedt periodicity that closely resembles B\"okstedt's original unpublished proof. We explain the extension of B\"{o}kstedt periodicity by Bhatt-Morrow-Scholze from perfect fields to perfectoid rings and use this to give a purely p-adic proof of Bott periodicity. Finally, we evaluate the cofiber of the assembly map in p-adic topological cyclic homology for the cyclic group of order p and a perfectoid ring of coefficients.