{"title":"零势商和三角范畴中的等价物","authors":"Teimuraz Pirashvili","doi":"10.1016/j.jpaa.2024.107755","DOIUrl":null,"url":null,"abstract":"<div><p>We will prove that if <span><math><mi>I</mi></math></span> is a nilpotent ideal of an additive category <span><math><mi>A</mi></math></span>, then an idempotent <em>e</em> of the category <span><math><mi>A</mi></math></span> splits iff its image splits in <span><math><mi>A</mi><mo>/</mo><mi>I</mi></math></span>. Based on this fact, we give a short proof of a crucial proposition of Le and Chen.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Idempotents in nilpotent quotients and triangulated categories\",\"authors\":\"Teimuraz Pirashvili\",\"doi\":\"10.1016/j.jpaa.2024.107755\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We will prove that if <span><math><mi>I</mi></math></span> is a nilpotent ideal of an additive category <span><math><mi>A</mi></math></span>, then an idempotent <em>e</em> of the category <span><math><mi>A</mi></math></span> splits iff its image splits in <span><math><mi>A</mi><mo>/</mo><mi>I</mi></math></span>. Based on this fact, we give a short proof of a crucial proposition of Le and Chen.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-06-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S002240492400152X\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S002240492400152X","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
我们将证明，如果 I 是一个可加范畴 A 的零幂理想，那么范畴 A 的幂等子 e 分裂，如果它的映像分裂于 A/I 中。基于这一事实，我们将对 Le 和 Chen 的一个关键命题给出简短证明。
Idempotents in nilpotent quotients and triangulated categories
We will prove that if is a nilpotent ideal of an additive category , then an idempotent e of the category splits iff its image splits in . Based on this fact, we give a short proof of a crucial proposition of Le and Chen.
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