{"title":"对估值域派生类中的粉碎理想进行分类","authors":"Scott Balchin, Florian Tecklenburg","doi":"arxiv-2407.11791","DOIUrl":null,"url":null,"abstract":"Building on results of Bazzoni-\\v{S}\\v{t}ov\\'{\\i}\\v{c}ek, we give a complete\nclassification of the frame of smashing ideals for the derived category of a\nfinite dimensional valuation domain. In particular, we give an explicit\nconstruction of an infinite family of commutative rings such that the telescope\nconjecture fails and which generalise an example of Keller. As a consequence,\nwe deduce that the Krull dimension of the Balmer spectrum and the Krull\ndimension of the smashing spectrum can differ arbitrarily for rigidly-compactly\ngenerated tensor-triangulated categories.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"72 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Classifying smashing ideals in derived categories of valuation domains\",\"authors\":\"Scott Balchin, Florian Tecklenburg\",\"doi\":\"arxiv-2407.11791\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Building on results of Bazzoni-\\\\v{S}\\\\v{t}ov\\\\'{\\\\i}\\\\v{c}ek, we give a complete\\nclassification of the frame of smashing ideals for the derived category of a\\nfinite dimensional valuation domain. In particular, we give an explicit\\nconstruction of an infinite family of commutative rings such that the telescope\\nconjecture fails and which generalise an example of Keller. As a consequence,\\nwe deduce that the Krull dimension of the Balmer spectrum and the Krull\\ndimension of the smashing spectrum can differ arbitrarily for rigidly-compactly\\ngenerated tensor-triangulated categories.\",\"PeriodicalId\":501475,\"journal\":{\"name\":\"arXiv - MATH - Commutative Algebra\",\"volume\":\"72 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-07-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Commutative Algebra\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2407.11791\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Commutative Algebra","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.11791","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Classifying smashing ideals in derived categories of valuation domains
Building on results of Bazzoni-\v{S}\v{t}ov\'{\i}\v{c}ek, we give a complete
classification of the frame of smashing ideals for the derived category of a
finite dimensional valuation domain. In particular, we give an explicit
construction of an infinite family of commutative rings such that the telescope
conjecture fails and which generalise an example of Keller. As a consequence,
we deduce that the Krull dimension of the Balmer spectrum and the Krull
dimension of the smashing spectrum can differ arbitrarily for rigidly-compactly
generated tensor-triangulated categories.