{"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}
引用次数: 0
Abstract
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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