棱镜晶体上同调的有限性和对偶性

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2021-09-02 DOI:10.1515/crelle-2023-0032

Yichao Tian

{"title":"棱镜晶体上同调的有限性和对偶性","authors":"Yichao Tian","doi":"10.1515/crelle-2023-0032","DOIUrl":null,"url":null,"abstract":"Abstract Let ( A , I ) {(A,I)} be a bounded prism, and let X be a smooth p-adic formal scheme over Spf ⁡ ( A / I ) {\\operatorname{Spf}(A/I)} . We consider the notion of crystals on Bhatt–Scholze’s prismatic site ( X / A ) Δ ⁢ Δ {(X/A)_{{\\kern-0.284528pt{\\Delta}\\kern-5.975079pt{\\Delta}}}} of X relative to A. We prove that if X is proper over Spf ⁡ ( A / I ) {\\operatorname{Spf}(A/I)} of relative dimension n, then the cohomology of a prismatic crystal is a perfect complex of A-modules with tor-amplitude in degrees [ 0 , 2 ⁢ n ] {[0,2n]} . We also establish a Poincaré duality for the reduced prismatic crystals, i.e. the crystals over the reduced structural sheaf of ( X / A ) Δ ⁢ Δ {(X/A)_{{\\kern-0.284528pt{\\Delta}\\kern-5.975079pt{\\Delta}}}} . The key ingredient is an explicit local description of reduced prismatic crystals in terms of Higgs modules.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2021-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"13","resultStr":"{\"title\":\"Finiteness and duality for the cohomology of prismatic crystals\",\"authors\":\"Yichao Tian\",\"doi\":\"10.1515/crelle-2023-0032\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Let ( A , I ) {(A,I)} be a bounded prism, and let X be a smooth p-adic formal scheme over Spf ⁡ ( A / I ) {\\\\operatorname{Spf}(A/I)} . We consider the notion of crystals on Bhatt–Scholze’s prismatic site ( X / A ) Δ ⁢ Δ {(X/A)_{{\\\\kern-0.284528pt{\\\\Delta}\\\\kern-5.975079pt{\\\\Delta}}}} of X relative to A. We prove that if X is proper over Spf ⁡ ( A / I ) {\\\\operatorname{Spf}(A/I)} of relative dimension n, then the cohomology of a prismatic crystal is a perfect complex of A-modules with tor-amplitude in degrees [ 0 , 2 ⁢ n ] {[0,2n]} . We also establish a Poincaré duality for the reduced prismatic crystals, i.e. the crystals over the reduced structural sheaf of ( X / A ) Δ ⁢ Δ {(X/A)_{{\\\\kern-0.284528pt{\\\\Delta}\\\\kern-5.975079pt{\\\\Delta}}}} . The key ingredient is an explicit local description of reduced prismatic crystals in terms of Higgs modules.\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2021-09-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"13\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/crelle-2023-0032\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/crelle-2023-0032","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}

引用次数: 13

摘要

摘要设(A,I) {(A,I)}是有界棱镜，设X是Spf (A/I) {\operatorname{Spf}(A/I)}上的光滑p进格式。我们考虑晶体的概念Bhatt-Scholze移动网站(X / A)Δ⁢Δ{(X / A) _ {{\ kern - 0.284528 - ptδ}{\ \ kern - 5.975079 - pt{\三角洲}}}}(X)相对于我们证明,如果X是适当的在Spf⁡(A / I) {\ operatorname {Spf} (A / I)}的相对尺寸n,那么一个移动的上同调水晶是一个完美的复杂的一个模块与tor-amplitude度[0,2⁢n] {[0, 2 n]}。我们还建立了简化棱柱形晶体的poincar对偶性，即(X/ a) Δ¹Δ {(X/ a)_{\kern-0.284528pt{\Delta}\kern-5.975079pt{\Delta}}}}的简化结构束上的晶体。关键因素是用希格斯模对还原棱柱晶体进行明确的局部描述。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Finiteness and duality for the cohomology of prismatic crystals

Abstract Let ( A , I ) {(A,I)} be a bounded prism, and let X be a smooth p-adic formal scheme over Spf ⁡ ( A / I ) {\operatorname{Spf}(A/I)} . We consider the notion of crystals on Bhatt–Scholze’s prismatic site ( X / A ) Δ ⁢ Δ {(X/A)_{{\kern-0.284528pt{\Delta}\kern-5.975079pt{\Delta}}}} of X relative to A. We prove that if X is proper over Spf ⁡ ( A / I ) {\operatorname{Spf}(A/I)} of relative dimension n, then the cohomology of a prismatic crystal is a perfect complex of A-modules with tor-amplitude in degrees [ 0 , 2 ⁢ n ] {[0,2n]} . We also establish a Poincaré duality for the reduced prismatic crystals, i.e. the crystals over the reduced structural sheaf of ( X / A ) Δ ⁢ Δ {(X/A)_{{\kern-0.284528pt{\Delta}\kern-5.975079pt{\Delta}}}} . The key ingredient is an explicit local description of reduced prismatic crystals in terms of Higgs modules.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.