形式三角矩阵环上相对于对偶对的Gorenstein平坦模和维数

doi:10.6040/j.issn.1671-9352.0.2023.479

《山东大学学报(理学版)》 ›› 2025, Vol. 60 ›› Issue (11): 79-86.doi: 10.6040/j.issn.1671-9352.0.2023.479

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形式三角矩阵环上相对于对偶对的Gorenstein平坦模和维数

刘铃,陈文静^*

西北师范大学数学与统计学院, 甘肃兰州 730070

发布日期:2025-11-11
通讯作者: 陈文静(1989— ),女,副教授,博士,研究方向为环的同调理论. E-mail:chenwj@nwnu.edu.cn
作者简介:刘铃(1999— ),女,硕士研究生,研究方向为环的同调理论. E-mail:15282545860@163.com
基金资助:
国家自然科学基金资助项目(11901463,12361007);甘肃省青年科技基金计划项目(20JR5RA517)

Gorenstein flat modules and dimensions with respect to duality pairs over formal triangular matrix rings

LIU Ling, CHEN Wenjing^*

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China

Published:2025-11-11

摘要/Abstract

摘要： 设T=(A 0U B)是形式三角矩阵环,其中A和B是环,U是(B,A)-双模。G F_{(F(R),Y )}表示相对于完备对偶对(X,Y )的Gorenstein平坦左R-模类。假设(C₁,C₂)和(D₁,D₂)分别为A环和B环上的完备对偶对,(U^C₁_D₁,U_C₂,D₂)是由(C₁,C₂)和(D₁,D₂)诱导的T环上的完备对偶对。证明若U_A有有限平坦维数,G F_{(F(T),U_C₂,D₂)}关于扩张封闭,则左T-模(M₁M₂)_φ^M∈G F_{(F(T), U_C₂,D₂)}当且仅当 M₁∈G F_(F(A),C₂), M₂/Im(φ^M)∈G F_(F(B),D₂), φ^M:U⊗_AM₁→M₂是单同态。此外还给出左T-模的相对于完备对偶对(U^C₁_D₁,U_C₂,D₂)的Gorenstein平坦维数的估计。

关键词: 形式三角矩阵环, 相对于完备对偶对的Gorenstein平坦模, 相对于完备对偶对的Gorenstein平坦维数

Abstract: Let T=(A 0U B) be a formal triangular matrix ring, where A and B are rings and U is a (B,A)-bimodule. G F_{(F(R),Y )} denotes the class of all Gorenstein flat left R-modules with respect to a complete duality pair(X,Y ). Assume that(C₁,C₂)and(D₁,D₂)are complete duality pairs over the ring A and the ring B respectively, and(U^C₁_D₁,U_C₂,D₂)is a complete duality pair over the ring T induced by(C₁,C₂)and(D₁,D₂). It is proven that, if U_A has finite flat dimension and G F_{(F(T),U_C₂,D₂)} is closed under extensions, then a left T-module (M1M2)_φ^M∈G F_{(F(T),U_C₂,D₂)} if and only if M1∈G F_(F(A),C₂), M₂/Im(φ^M)∈G F_(F(B),D₂), and φ^M:U⊗_AM1→M2 is a monomorphism. Furthermore, an estimate of Gorenstein flat dimension with respect to the complete duality pair(U^C₁_D₁,U_C₂,D₂)of a left T-module is given.

Key words: formal triangular matrix ring, Gorenstein flat module with respect to a complete duality pair, Gorenstein flat dimension with respect to a complete duality pair

中图分类号:

O153.3

刘铃,陈文静. 形式三角矩阵环上相对于对偶对的Gorenstein平坦模和维数[J]. 《山东大学学报(理学版)》, 2025, 60(11): 79-86.

LIU Ling, CHEN Wenjing. Gorenstein flat modules and dimensions with respect to duality pairs over formal triangular matrix rings[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2025, 60(11): 79-86.

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形式三角矩阵环上相对于对偶对的Gorenstein平坦模和维数

Gorenstein flat modules and dimensions with respect to duality pairs over formal triangular matrix rings

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