Gorenstein平坦模与Frobenius扩张

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英文篇名：Gorenstein Flat Modules and Frobenius Extensions 作者：任伟 英文作者：Wei REN;School of Mathematical Sciences, Chongqing Normal University; 关键词：Gorenstein平坦模 ; Gorenstein平坦维数 ; Frobenius扩张 ; 可分扩张 英文关键词：Gorenstein flat module;;Gorenstein flat dimension;;Frobenius extension;;separable extension 中文刊名：SXXB 英文刊名：Acta Mathematica Sinica(Chinese Series) 机构：重庆师范大学数学科学学院; 出版日期：2019-07-15 出版单位：数学学报(中文版) 年：2019 期：v.62 基金：国家自然科学基金资助项目(11871125);; 重庆市自然科学基金(cstc2018jcyjAX0541);; 市教委科学技术研究项目(KJQN201800509) 语种：中文; 页：SXXB201904011 页数：6 CN：04 ISSN：11-2038/O1 分类号：121-126

摘要

设R■A是环的Frobenius扩张,其中A是右凝聚环,M是任意左A-模.首先证明了_AM是Gorenstein平坦模当且仅当M作为左R-模也是Gorenstein平坦模.其次,证明了Nakayama和Tsuzuku关于平坦维数沿着Frobenius扩张的传递性定理的"Gorenstein版本":若_AM具有有限Gorenstein平坦维数,则Gfd_A(M)=Gfd_R(M).此外,证明了若R■S是可分Frobenius扩张,则任意A-模(不一定具有有限Gorenstein平坦维数),其Gorenstein平坦维数沿着该环扩张是不变的.
        Let R ■ A be a,Frobenius extension of rings,where A is right coherent.Let M be any left A-module.We first show that AM is Gorenstein flat if and only if the underlying R-module RM is Gorenstein flat.Then we prove a"Gorenstein version"of Nakayama and Tsuzuku's theorem on transfer of flat dimensions along Frobenius extensions:if AM has finite Gorenstein flat dimension,then Gfd_A(M)= Gfd_R(M).Moreover,it is proved that if R ■ S is a separable Frobenius extension,then for any A-module(not necessarily of finite Gorenstein flat dimension),its Gorenstein flat dimension is invariant along such ring extension.

引文

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