{"title":"Shtukas with one leg II","authors":"P. Scholze, Jared Weinstein","doi":"10.2307/j.ctvs32rc9.16","DOIUrl":"https://doi.org/10.2307/j.ctvs32rc9.16","url":null,"abstract":"This chapter offers a second lecture on one-legged shtukas. It shows that a shtuka over Spa Cb, a priori defined over Y\u0000 [0,INFINITY) = Spa Ainf REVERSE SOLIDUS {xk, xL}, actually extends to Y = Spa Ainf REVERSE SOLIDUS {xk}. In doing so, the chapter considers the theory of φ-modules over the Robba ring, due to Kedlaya. These are in correspondence with vector bundles over the Fargues-Fontaine curve. The chapter then looks at the proposition that the space Y is an adic space. It also sketches a proof that the functor described in Theorem 13.2.1 is fully faithful. This is more general, and works if C is any perfectoid field (not necessarily algebraically closed).","PeriodicalId":270009,"journal":{"name":"Berkeley Lectures on p-adic Geometry","volume":"128 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":"134561783","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}
{"title":"Examples of diamonds","authors":"P. Scholze, Jared Weinstein","doi":"10.2307/j.ctvs32rc9.18","DOIUrl":"https://doi.org/10.2307/j.ctvs32rc9.18","url":null,"abstract":"This chapter assesses some interesting examples of diamonds. So far, the only example encountered is the self-product of copies of Spd Qp. The chapter first studies this self-product. It is useful to keep in mind that a diamond can have multiple “incarnations.” Another important class of diamonds, which in fact were one of the primary motivations for their definition, is the category of Banach-Colmez spaces. Recently, le Bras has reworked their theory in terms of perfectoid spaces. The category of Banach-Colmez spaces over C is the thick abelian subcategory of the category of pro-étale sheaves of Qp-modules. This is similar to a category considered by Milne in characteristic p.","PeriodicalId":270009,"journal":{"name":"Berkeley Lectures on p-adic Geometry","volume":"25 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":"131402299","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}
{"title":"Mixed-characteristic shtukas","authors":"P. Scholze, Jared Weinstein","doi":"10.2307/j.ctvs32rc9.14","DOIUrl":"https://doi.org/10.2307/j.ctvs32rc9.14","url":null,"abstract":"This chapter looks at mixed-characteristic shtukas. Much of the theory of mixed-characteristic shtukas is motivated by the structures appearing in (integral) p-adic Hodge theory. The chapter assesses Drinfeld's shtukas and local shtukas. In the mixed characteristic setting, X will be replaced with Spa Zp. The test objects S will be drawn from Perf, the category of perfectoid spaces in characteristic p. For an object, a shtuka over S should be a vector bundle over an adic space, together with a Frobenius structure. The product is not meant to be taken literally (if so, one would just recover S), but rather it is to be interpreted as a fiber product over a deeper base. Motivated by this, the chapter then defines an analytic adic space and shows that its associated diamond is the appropriate product of sheaves on Perf.","PeriodicalId":270009,"journal":{"name":"Berkeley Lectures on p-adic Geometry","volume":"603 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":"116452181","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}
{"title":"Drinfeld’s lemma for diamonds","authors":"P. Scholze, Jared Weinstein","doi":"10.2307/j.ctvs32rc9.19","DOIUrl":"https://doi.org/10.2307/j.ctvs32rc9.19","url":null,"abstract":"This chapter addresses Drinfeld's lemma for diamonds. It proves a local analogue of Drinfeld's lemma, thereby giving a first nontrivial argument involving diamonds. This lecture is entirely about fundamental groups. A diamond is defined to be connected if it is not the disjoint union of two open subsheaves. For a connected diamond, finite étale covers form a Galois category. As such, for a geometric point, one can define a profinite group, such that finite sets are equivalent to finite étale covers. In this proof, the chapter uses the formalism of diamonds rather heavily to transport finite étale maps between different presentations of a diamond as the diamond of an analytic adic space.","PeriodicalId":270009,"journal":{"name":"Berkeley Lectures on p-adic Geometry","volume":"4 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":"130835228","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}
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