形式三角矩阵环上强Gorenstein平坦模

设Γ是由环R、S和双模SMR组成的形式三角矩阵环.主要讨论环Γ上的模、模同态、模正合列以及模复形.研究了强Gorenstein平坦Γ-模的若干性质及等价刻画,并证明了由模RX和SY以及左-S同态φ:M⊗RX→Y组成的Γ-模是强Gorenstein平坦模,当且仅当RX和SCokerφ均是强Gorenstein平坦模且φ为单同态.     

相对Gorenstein投射模（英文）

设环A是环B的扩张环,即B是与A有相同单位的A的子环.记P(A,B)是由所有相对投射模构成的范畴.对于扩张B→A,本文介绍相对Gorenstein投射模的概念.由于Gorenstein投射模与投射模具有紧密的联系,并且关于Gorenstein维数有较好的性质,本文想给出相对Gorenstein投射模和相对投射模之间类似的关系.本文主要结果是:(1)设B→A是具有相同单位的环的扩张,则由所有相对Gorenstein投射模构成的范畴是相对可解的.(2)设B→A是具有相同单位的环的扩张,若gl.dim(A,B)≤n,则每一个相对Gorenstein投射模都是相对投射的,其中gl.dim(A,B)表示所有A-模的相对投射维数的上确界.     

关于半对偶化模的Gorenstein模的稳定性（英文）

本文研究了相对于半对偶化模C的Gorenstein模(即Gorenstein C-投射模,Gorenstein C-内射模和Gorenstein C-平坦模)的稳定性的问题.利用同调的方法,获得了Gorenstein C-投射(C-内射,C-平坦)模具有很好的稳定性的结果,推广了Gorenstein投射(内射,平坦)模具有很好的稳定性的结果.     

关于E_ω-投射模和E_ω-内射模(英文)

左R-模M称为Eω-内射模,如果对环R中任意的ω阶Euclid理想I来说,任何R-模同态能够拓展为R-模同态。左R-模M称为Eω-投射模,若对环R中任意的ω阶Euclid理想I和任何R-模同态f∈HomR(M,R/I),存在R-模同态g∈HomR(M,R)使得f=πg,其中π是自然同态。本文证明P和Q均是Eω-投射模当且仅当PQ是Eω-投射模。进而,又证明了每一个左R-模是Eω-投射的当且仅当每一个左R-模是Eω-内射。     

左wGF-封闭环上的弱Gorenstein平坦模

利用同调代数的工具,主要证明了弱Gorenstein平坦模类为投射预解的当且仅当它是扩张封闭的,进一步的刻画了左wGF-封闭环上弱Gorenstein平坦模的一些性质,这些内容丰富了D.Bennis等人的研究结果.     

Gorenstein平坦模与Frobenius扩张

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.     

奇点模型范畴的Quillen等价

令R是左Gorenstein环.我们构造了奇点反导出模型范畴和奇点余导出模型范畴(见文[Models for singularity categories,Adv Math.,2014,254:187-232])之间的Quillen等价.作为应用,给出了投射,内射模的正合复形的同伦范畴之间的一个具体的等价■.     

正合范畴的整体Gorenstein维数

设A是有足够多投射对象和足够多内射对象的正合范畴.本文研究了A的整体Gorenstein维数和A中的Gorenstein导出函子.利用同调的方法,证明了:如果A有可数直和与可数直积,那么sup{GpdM|M∈A}=sup{GidM|M∈A};对A中的对象M, N,若Gp M ∞, GidN ∞,则对任意的i≥0, Ext_(GP)~i(M, N)≌Ext_(GI)~i(M, N).     

正合范畴中的复形、余挠对及粘合

作者在弱幂等完备的正合范畴(A,ε)中引入了复形的新的定义,并且证明了ε-正合复形的同伦范畴κ_ex(ε)是同伦范畴κ_ε(A)的厚子范畴.给定(A,ε)中的余挠对(x,y),定义了正合范畴(c_ε(A),C(ε))中的两个余挠对(x^~_ε,dgy^~_ε)和(dgx^~_ε,y^~_ε),并且证明了当A是可数完备时,c_ε(A)中任意无界复形的dgx^~_ε,y^~_ε-分解存在.作为应用,建立了相对于范畴κ_ex(ε)和D_ε(A)的范畴κ_ε(A)的左粘合,给出了R-模范畴的粘合的例子.     

Gorenstein Prüfer整环的一些新刻画

设R是整环.众所周知,R是Prüfer整环当且仅当每个可除模是FP-内射模当且仅当每个h-可除模是FP-内射模.本文引进了一种新的Gorenstein FP-内射模,并且证明了R是Gorenstein Prüfer整环当且仅当每个可除模是Gorenstein FP-内射模,当且仅当每个h-可除模是Gorenstein FP-内射模.     

Complexes of Gorenstein Flat Modules and Gorenstein Cotorsion Modules

A complex (C, δ) is called strongly Gorenstein flat if C is exact and Ker δ ⁿ is Gorenstein flat in R-Mod for all n ∈ ?. Let 𝒮𝒢 stand for the class of strongly Gorenstein flat complexes. We show that a complex C of left R-modules over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C ⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ? and Hom^.(G, C) is exact for any strongly Gorenstein flat complex G. Furthermore, a bounded below complex C over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C ⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ?. Finally, strongly Gorenstein flat covers and 𝒮𝒢^⊥-envelopes of complexes are considered. For a right coherent ring R, we show that every bounded below complex has a 𝒮𝒢^⊥-envelope.     

弱半素子模的一些刻划

引入n′-系的概念,且给出弱半素子模的两个等价条件:(i)设K是左R-模M的子模,则K是M的弱半素子模当且仅当C(K)=M\K是n′-系;(ii)设K是左R-模M的子模,则K是M的弱半素子模当且仅当对任意f∈HomR(R,M),及任意A R,若f(A2)K,就有f(A)K.     

环的B-稳定正则理想

研究了正则理想是B-稳定的充分和必要条件,并且证明环R的正则理想I是B-稳定的当且仅当对任意的有限生成投射右R-模A,如果A₁和A₂是A的有限生成子模且满足A₁≌A₂,A₁=A₁I以及A₂=A₂I,则存在一个有限生成子模B,使得A=A₁（?）B=A₂（?）B;当且仅当对任意的幂等元e,f∈I,eR≌fR蕴含eR/（eR∩fR）≌fR/（eR∩fR）;当且仅当对任意的a∈1+I,存在一个幂等元e∈I,使得a-e∈∪（R）并且aR∩eR=0.进而构造了相关的例子.     

Gorenstein dimensions of unbounded complexes and change of base (with an appendix by Driss Bennis)

For a commutative ring R and a faithfully flat R-algebra S we prove,under mild extra assumptions,that an R-module M is Gorenstein flat if and only if the left S-module S?_R M is Gorenstein flat,and that an R-module N is Gorenstein injective if and only if it is cotorsion and the left S-module Hom R(S,N)is Gorenstein injective.We apply these results to the study of Gorenstein homological dimensions of unbounded complexes.In particular,we prove two theorems on stability of these dimensions under faithfully flat (co-)base change.     

Gorenstein projective and flat complexes over noetherian rings

We give sufficient conditions on a class of R‐modules $\mathcal {C}We give sufficient conditions on a class of R‐modules $\mathcal {C}$ in order for the class of complexes of $\mathcal {C}$‐modules, $dw \mathcal {C}$, to be covering in the category of complexes of R‐modules. More precisely, we prove that if $\mathcal {C}$ is precovering in R ? Mod and if $\mathcal {C}$ is closed under direct limits, direct products, and extensions, then the class $dw \mathcal {C}$ is covering in Ch(R). Our first application concerns the class of Gorenstein flat modules. We show that when the ring R is two sided noetherian, a complex C is Gorenstein flat if and only if each module C_n is Gorenstein flat. If moreover every direct product of Gorenstein flat modules is a Gorenstein flat module, then the class of Gorenstein flat complexes is covering. We consider Gorenstein projective complexes as well. We prove that if R is a commutative noetherian ring of finite Krull dimension, then the class of Gorenstein projective complexes coincides with that of complexes of Gorenstein projective modules. We also show that if R is commutative noetherian with a dualizing complex then every right bounded complex has a Gorenstein projective precover.     

The connes spectrums of certain finite non-abelian groups

In this article, we investigate when every simple module has a projective (pre)envelope. It is proven that (1) every simple right R-module has a projective preenvelope if and only if the left annihilator of every maximal right ideal of R is finitely generated; (2) every simple right R-module has an epic projective envelope if and only if R is a right PS ring; (3) Every simple right R-module has a monic projective preenvelope if and only if R is a right Kasch ring and the left annihilator of every maximal right ideal of R is finitely generated.     

分圆域Q（ζ33）的幂元整基

伽罗华数域L称有一个幂元整基,如果其代数整数环具有形式Ζα,其中α∈L.此时称α是L的幂元整基生成元.设α,β是L的两个幂元整基生成元,若β=m±σ（α）,m∈Z,σ∈Gal（L/Q）,则称α与β等价.本文主要研究分圆域Q（ζ33）的幂元整基问题.分圆域Q（ζ33）的代数整环是Z[ζ33],所以ζ33是Q（ζ33）的幂元整基生成元.设α是Q（ζ33）的幂元整基生成元,证明了当α＋ā Z时,α与ζ33等价.从而给出在此条件下分圆域Q（ζ33）的所有幂元整基生成元.     

分圆域Q(ζ_(33))的幂元整基

伽罗华数域L称有一个幂元整基,如果其代数整数环具有形式Ζα,其中α∈L.此时称α是L的幂元整基生成元.设α,β是L的两个幂元整基生成元,若β=m±σ(α),m∈Z,σ∈Gal(L/Q),则称α与β等价.本文主要研究分圆域Q(ζ33)的幂元整基问题.分圆域Q(ζ33)的代数整环是Z[ζ33],所以ζ33是Q(ζ33)的幂元整基生成元.设α是Q(ζ33)的幂元整基生成元,证明了当α+ā■Z时,α与ζ33等价.从而给出在此条件下分圆域Q(ζ33)的所有幂元整基生成元.