{"title":"Relative torsionfreeness and Frobenius extensions","authors":"Yanhong Bao, Jiafeng Lü, Zhibing Zhao","doi":"arxiv-2409.11892","DOIUrl":null,"url":null,"abstract":"Let $S/R$ be a Frobenius extension with $_RS_R$ centrally projective over\n$R$. We show that if $_R\\omega$ is a Wakamatsu tilting module then so is\n$_SS\\otimes_R\\omega$, and the natural ring homomorphism from the endomorphism\nring of $_R\\omega$ to the endomorphism ring of $_SS\\otimes_R\\omega$ is a\nFrobenius extension in addition that pd$(\\omega_T)$ is finite, where $T$ is the\nendomorphism ring of $_R\\omega$. We also obtain that the relative\n$n$-torsionfreeness of modules is preserved under Frobenius extensions.\nFurthermore, we give an application, which shows that the generalized\nG-dimension with respect to a Wakamatsu module is invariant under Frobenius\nextensions.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"13 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11892","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Let $S/R$ be a Frobenius extension with $_RS_R$ centrally projective over
$R$. We show that if $_R\omega$ is a Wakamatsu tilting module then so is
$_SS\otimes_R\omega$, and the natural ring homomorphism from the endomorphism
ring of $_R\omega$ to the endomorphism ring of $_SS\otimes_R\omega$ is a
Frobenius extension in addition that pd$(\omega_T)$ is finite, where $T$ is the
endomorphism ring of $_R\omega$. We also obtain that the relative
$n$-torsionfreeness of modules is preserved under Frobenius extensions.
Furthermore, we give an application, which shows that the generalized
G-dimension with respect to a Wakamatsu module is invariant under Frobenius
extensions.