{"title":"加法范畴与高等差分加法范畴之间的同构转移","authors":"Xi Tang, Zhao Yong Huang","doi":"10.1007/s10114-023-2193-8","DOIUrl":null,"url":null,"abstract":"<div><p>Given an additive category <span>\\({\\cal C}\\)</span> and an integer <i>n</i> ≥ 2. The higher differential additive category consists of objects <i>X</i> in <span>\\({\\cal C}\\)</span> equipped with an endomorphism <i>ϵ</i><sub><i>X</i></sub> satisfying <span>\\(\\epsilon_X^n = 0\\)</span>. Let <i>R</i> be a finite-dimensional basic algebra over an algebraically closed field and <i>T</i> the augmenting functor from the category of finitely generated left <i>R</i>-modules to that of finitely generated left <i>R</i>/(<i>t</i><sup><i>n</i></sup>)-modules. It is proved that a finitely generated left <i>R</i>-module <i>M</i> is <i>τ</i>-rigid (respectively, (support) <i>τ</i>-tilting, almost complete <i>τ</i>-tilting) if and only if so is <i>T</i>(<i>M</i>)as a left <i>R</i>[<i>t</i>]/(<i>t</i><sup><i>n</i></sup>)-module. Moreover, <i>R</i> is <i>τ</i><sub><i>m</i></sub>-selfinjective if and only if so is <i>R</i>[t]/(<i>t</i><sup><i>n</i></sup>).</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 5","pages":"1325 - 1344"},"PeriodicalIF":0.9000,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Homological Transfer between Additive Categories and Higher Differential Additive Categories\",\"authors\":\"Xi Tang, Zhao Yong Huang\",\"doi\":\"10.1007/s10114-023-2193-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Given an additive category <span>\\\\({\\\\cal C}\\\\)</span> and an integer <i>n</i> ≥ 2. The higher differential additive category consists of objects <i>X</i> in <span>\\\\({\\\\cal C}\\\\)</span> equipped with an endomorphism <i>ϵ</i><sub><i>X</i></sub> satisfying <span>\\\\(\\\\epsilon_X^n = 0\\\\)</span>. Let <i>R</i> be a finite-dimensional basic algebra over an algebraically closed field and <i>T</i> the augmenting functor from the category of finitely generated left <i>R</i>-modules to that of finitely generated left <i>R</i>/(<i>t</i><sup><i>n</i></sup>)-modules. It is proved that a finitely generated left <i>R</i>-module <i>M</i> is <i>τ</i>-rigid (respectively, (support) <i>τ</i>-tilting, almost complete <i>τ</i>-tilting) if and only if so is <i>T</i>(<i>M</i>)as a left <i>R</i>[<i>t</i>]/(<i>t</i><sup><i>n</i></sup>)-module. Moreover, <i>R</i> is <i>τ</i><sub><i>m</i></sub>-selfinjective if and only if so is <i>R</i>[t]/(<i>t</i><sup><i>n</i></sup>).</p></div>\",\"PeriodicalId\":50893,\"journal\":{\"name\":\"Acta Mathematica Sinica-English Series\",\"volume\":\"40 5\",\"pages\":\"1325 - 1344\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-11-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Sinica-English Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10114-023-2193-8\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Sinica-English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10114-023-2193-8","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
给定一个加法范畴 \({\cal C}\) 和一个整数 n ≥ 2。高微分可加范畴由 \({\cal C}\) 中的对象 X 组成，这些对象都有一个满足 \(\epsilon_X^n = 0\) 的内同态ϵX。让 R 是一个代数闭域上的有限维基代数，T 是有限生成的左 R 模块类别到有限生成的左 R/(tn)模块类别的增函数。本文证明，当且仅当 T(M)作为左 R[t]/(tn)- 模块时，有限生成的左 R 模块 M 是 τ- 刚性（分别是（支持）τ-倾斜、几乎完全 τ-倾斜）。此外，当且仅当 R[t]/(tn) 如此时，R 是 τm 自投影的。
Homological Transfer between Additive Categories and Higher Differential Additive Categories
Given an additive category \({\cal C}\) and an integer n ≥ 2. The higher differential additive category consists of objects X in \({\cal C}\) equipped with an endomorphism ϵX satisfying \(\epsilon_X^n = 0\). Let R be a finite-dimensional basic algebra over an algebraically closed field and T the augmenting functor from the category of finitely generated left R-modules to that of finitely generated left R/(tn)-modules. It is proved that a finitely generated left R-module M is τ-rigid (respectively, (support) τ-tilting, almost complete τ-tilting) if and only if so is T(M)as a left R[t]/(tn)-module. Moreover, R is τm-selfinjective if and only if so is R[t]/(tn).
期刊介绍：
Acta Mathematica Sinica, established by the Chinese Mathematical Society in 1936, is the first and the best mathematical journal in China. In 1985, Acta Mathematica Sinica is divided into English Series and Chinese Series. The English Series is a monthly journal, publishing significant research papers from all branches of pure and applied mathematics. It provides authoritative reviews of current developments in mathematical research. Contributions are invited from researchers from all over the world.
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