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Diamonds 钻石
Berkeley Lectures on p-adic Geometry Pub Date : 2020-05-26 DOI: 10.2307/j.ctvwh8dr6.49
P. Scholze, Jared Weinstein
{"title":"Diamonds","authors":"P. Scholze, Jared Weinstein","doi":"10.2307/j.ctvwh8dr6.49","DOIUrl":"https://doi.org/10.2307/j.ctvwh8dr6.49","url":null,"abstract":"This chapter investigates the notion of a diamond. The idea is that there should be a functor which “forgets the structure morphism to Zp.” The desired quotient in the example provided in the chapter exists in a category of sheaves on the site of perfectoid spaces with pro-étale covers. The chapter then defines pro-étale morphisms between perfectoid spaces. A morphism of perfectoid spaces is pro-étale if it is locally (on the source and target) affinoid pro-étale. The intuitive definition of diamonds involved the tilting functor in case of perfectoid spaces of characteristic 0. For this reason, diamonds are defined as certain pro-étale sheaves on the category of perfectoid spaces of characteristic p.","PeriodicalId":270009,"journal":{"name":"Berkeley Lectures on p-adic Geometry","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132360670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Perfectoid rings Perfectoid环
Berkeley Lectures on p-adic Geometry Pub Date : 2020-05-26 DOI: 10.23943/princeton/9780691202082.003.0006
P. Scholze, Jared Weinstein
{"title":"Perfectoid rings","authors":"P. Scholze, Jared Weinstein","doi":"10.23943/princeton/9780691202082.003.0006","DOIUrl":"https://doi.org/10.23943/princeton/9780691202082.003.0006","url":null,"abstract":"This chapter examines perfectoid spaces. A Huber ring R is Tate if it contains a topologically nilpotent unit; such elements are called pseudo-uniformizers. One can more generally define when an analytic Huber ring is perfectoid. There are also notions of integral perfectoid rings which are not analytic. In this course, the perfectoid rings are all Tate. It would have been possible to proceed with the more general definition of perfectoid ring as a kind of analytic Huber ring. However, being analytic is critical for the purposes of the course. The chapter then looks at tilting and sousperfectoid rings. The class of sousperfectoid rings has good stability properties.","PeriodicalId":270009,"journal":{"name":"Berkeley Lectures on p-adic Geometry","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131328246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Affine flag varieties 仿射标志品种
Berkeley Lectures on p-adic Geometry Pub Date : 2020-05-26 DOI: 10.2307/j.ctvs32rc9.24
P. Scholze, Jared Weinstein
{"title":"Affine flag varieties","authors":"P. Scholze, Jared Weinstein","doi":"10.2307/j.ctvs32rc9.24","DOIUrl":"https://doi.org/10.2307/j.ctvs32rc9.24","url":null,"abstract":"This chapter reviews affine flag varieties. It generalizes some of the previous results to the case where G over Zp is a parahoric group scheme. In fact, slightly more generally, it allows the case that the special fiber is not connected, with connected component of the identity G? being a parahoric group scheme. This case comes up naturally in the classical definition of Rapoport-Zink spaces. The chapter first discusses the Witt vector affine flag variety over Fp. This is an increasing union of perfections of quasiprojective varieties along closed immersions. In the case that G° is parahoric, one gets ind-properness.","PeriodicalId":270009,"journal":{"name":"Berkeley Lectures on p-adic Geometry","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124886239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Local Shimura varieties 志村本地品种
Berkeley Lectures on p-adic Geometry Pub Date : 2020-05-26 DOI: 10.23943/PRINCETON/9780691202082.003.0024
P. Scholze, Jared Weinstein
{"title":"Local Shimura varieties","authors":"P. Scholze, Jared Weinstein","doi":"10.23943/PRINCETON/9780691202082.003.0024","DOIUrl":"https://doi.org/10.23943/PRINCETON/9780691202082.003.0024","url":null,"abstract":"This chapter specializes the theory back to the case of local Shimura varieties, and explains the relation with Rapoport-Zink spaces. It begins with a local Shimura datum. A local Shimura datum is a triple (G, b, µ) consisting of a reductive group G over Qp, a conjugacy class µ of minuscule cocharacters. Rapoport-Zink spaces are moduli of deformations of a fixed p-divisible group. After reviewing these, the chapter shows that the diamond associated with the generic fiber of a Rapoport-Zink space is isomorphic to a moduli space of shtukas of the form with µ minuscule. It then extends the results to general EL and PEL data.","PeriodicalId":270009,"journal":{"name":"Berkeley Lectures on p-adic Geometry","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114260922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Moduli spaces of shtukas
Berkeley Lectures on p-adic Geometry Pub Date : 2020-05-26 DOI: 10.2307/j.ctvs32rc9.26
P. Scholze, Jared Weinstein
{"title":"Moduli spaces of shtukas","authors":"P. Scholze, Jared Weinstein","doi":"10.2307/j.ctvs32rc9.26","DOIUrl":"https://doi.org/10.2307/j.ctvs32rc9.26","url":null,"abstract":"This chapter examines the moduli spaces of mixed-characteristic local G-shtukas and shows that they are representable by locally spatial diamonds. These will be the mixed-characteristic local analogues of the moduli spaces of global equal-characteristic shtukas introduced by Varshavsky. It may be helpful to briefly review the construction in the latter setting. The ingredients are a smooth projective geometrically connected curve X defined over a finite field Fq and a reductive group G/Fq. Each connected component is a quotient of a quasi-projective scheme by a finite group. From there, it is possible to add level structures to the spaces of shtukas, to obtain a tower of moduli spaces admitting an action of the adelic group. The cohomology of these towers of moduli spaces is the primary means by which V. Lafforgue constructs the “automorphic to Galois” direction of the Langlands correspondence for G over F.","PeriodicalId":270009,"journal":{"name":"Berkeley Lectures on p-adic Geometry","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124849771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Perfectoid rings Perfectoid环
Berkeley Lectures on p-adic Geometry Pub Date : 2020-05-26 DOI: 10.2307/j.ctvs32rc9.9
Noah Riggenbach
{"title":"Perfectoid rings","authors":"Noah Riggenbach","doi":"10.2307/j.ctvs32rc9.9","DOIUrl":"https://doi.org/10.2307/j.ctvs32rc9.9","url":null,"abstract":"In this talk I will discuss my recent computation of the NTC groups of perfectoid rings which have a system of pth power roots of unity and thus the Kgroups of the p-completed affine lineR〈x〉 over these rings relative to the ideal (x). This includes all perfect fields of positive characteristic, for which these groups vanish in non-negative degrees. This class of rings also contains many mixed characteristic rings, and perhaps surprisingly while the even nonnegative groups will still vanish, the odd groups will not.","PeriodicalId":270009,"journal":{"name":"Berkeley Lectures on p-adic Geometry","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128059700","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Adic spaces II
Berkeley Lectures on p-adic Geometry Pub Date : 2020-05-26 DOI: 10.2307/j.ctvs32rc9.6
P. Scholze, Jared Weinstein
{"title":"Adic spaces II","authors":"P. Scholze, Jared Weinstein","doi":"10.2307/j.ctvs32rc9.6","DOIUrl":"https://doi.org/10.2307/j.ctvs32rc9.6","url":null,"abstract":"This chapter defines adic spaces. A scheme is a ringed space which locally looks like the spectrum of a ring. An adic space will be something similar. The chapter then identifies the adic version of “locally ringed space.” Briefly, it is a topologically ringed topological space equipped with valuations. The chapter also reflects on the role of A+ in the definition of adic spaces. The subring A+ in a Huber pair may seem unnecessary at first: why not just consider all continuous valuations on A? Specifying A+ keeps track of which inequalities have been enforced among the continuous valuations. Finally, the chapter differentiates between sheafy and non-sheafy Huber pairs.","PeriodicalId":270009,"journal":{"name":"Berkeley Lectures on p-adic Geometry","volume":"114 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133497359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Complements on adic spaces 进进空间上的补码
Berkeley Lectures on p-adic Geometry Pub Date : 2020-05-26 DOI: 10.2307/j.ctvs32rc9.8
P. Scholze, Jared Weinstein
{"title":"Complements on adic spaces","authors":"P. Scholze, Jared Weinstein","doi":"10.2307/j.ctvs32rc9.8","DOIUrl":"https://doi.org/10.2307/j.ctvs32rc9.8","url":null,"abstract":"This chapter analyzes a collection of complements in the theory of adic spaces. These complements include adic morphisms, analytic adic spaces, and Cartier divisors. It turns out that there is a very general criterion for sheafyness. In general, uniformity does not guarantee sheafyness, but a strengthening of the uniformity condition does. Moreover, sheafyness, without any extra assumptions, implies other good properties. Ultimately, it is not immediately clear how to get a good theory of coherent sheaves on adic spaces. The chapter then considers Cartier divisors on adic spaces. The term closed Cartier divisor is meant to evoke a closed immersion of adic spaces.","PeriodicalId":270009,"journal":{"name":"Berkeley Lectures on p-adic Geometry","volume":"85 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114350143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Diamonds II
Berkeley Lectures on p-adic Geometry Pub Date : 2020-05-26 DOI: 10.2307/j.ctvs32rc9.12
P. Scholze, Jared Weinstein
{"title":"Diamonds II","authors":"P. Scholze, Jared Weinstein","doi":"10.2307/j.ctvs32rc9.12","DOIUrl":"https://doi.org/10.2307/j.ctvs32rc9.12","url":null,"abstract":"This chapter evaluates the complements on the pro-étale topology. It addresses two issues raised in the previous lecture on the pro-étale topology. The first issue concerned descent, or more specifically pro-étale descent for perfectoid spaces. The other issue was that the property of being a pro-étale morphism is not local for the pro-étale topology on the target. The chapter then looks at quasi-pro-étale morphisms, as well as G-torsors. A morphism of perfectoid spaces is quasi-pro-étale if for any strictly totally disconnected perfectoid space with a map, the pullback is pro-étale. Using this definition, one can give an equivalent characterization of diamonds.","PeriodicalId":270009,"journal":{"name":"Berkeley Lectures on p-adic Geometry","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131982088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Index 指数
Berkeley Lectures on p-adic Geometry Pub Date : 2020-05-26 DOI: 10.2307/j.ctvs32rc9.30
{"title":"Index","authors":"","doi":"10.2307/j.ctvs32rc9.30","DOIUrl":"https://doi.org/10.2307/j.ctvs32rc9.30","url":null,"abstract":"","PeriodicalId":270009,"journal":{"name":"Berkeley Lectures on p-adic Geometry","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128416774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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