相对于半对偶化模的Gorenstein同调维数与Auslander范畴

设$R$是一个局部noether环. 我们在本文中研究了相对于半对偶化模$C$的Gorenstein投射, 内射与平坦模. 给出了$C$-Gorenstein同调维数与$\hat{R}$的Auslander范畴之间的关系.     

关于余挠三元组的余星子范畴和余倾斜子范畴

设$\mathcal{A}$ 是一个Abel范畴,且 $(\mathcal{X}, \mathcal{Z},\mathcal{Y})$ 是一个完全遗传余挠三元组.介绍 $\mathcal{A}$ 的 $n$-$\mathcal{Y}$-余倾斜子范畴的定义,并给出 $n$-$\mathcal{Y}$-余倾斜子范畴的一个刻画,类似于 $n$-余倾斜模的 Bazzoni 刻画.作为应用,证明了在一个几乎 Gorenstein 环 $R$ 上, 如果 $\mathcal{GP}$ 是 $n$-$\mathcal{GI}$-余倾斜的, 那么 $R$ 是一个 $n$-Gorenstein 环, 其中 $\mathcal{GP}$ 表示 Gorenstein 投射 $R$-模组成的子范畴且 $\mathcal{GI}$ 表示 Gorenstein 内射 $R$-模组成的子范畴. 进而, 研究 任意环$R$上的$n$-余星子范畴, 以及关于余挠三元组 $(\mathcal{P}, R$-Mod, $\mathcal{I})$ 的 $n$-$\mathcal{I}$-子范畴与 $n$-余星子范畴之间的关系, 其中 $\mathcal{P}$ 表示投射左 $R$-模组成的子范畴且 $\mathcal{I}$ 表示内射左 $R$-模组成的子范畴.     

纯奇点范畴中的Buchweitz定理

我们定义纯奇点范畴D_(psg)~b(R)为有界纯导出范畴D_(pur)~b(R)与纯投射模构成的有界同伦范畴K~b(■)的Verdier商,得到了纯奇点范畴D_(psg)~b(R)三角等价于相对纯投射模的Gorenstein范畴的稳定范畴■的一个充分必要条件.同时,还给出三角等价D_(psg)~b(R)≌D_(psg)~b(S)的充分条件,这里R和S都是环.     

Gorenstein环上的Gorenstein平坦和内射

设R是一个Gorenstein环. 证明了, 如果I是R的一个理想且使得R/I是一个半单环, 则R/I作为右R-模的Gorenstein平坦维数与R/I作为左R-模的Gorenstein内射维数是相等的. 另外证明了, 如果R→S是一个环同态且_SE是左S-模范畴的一个内射余生成元, 则S作为右R-模的Gorenstein平坦维数与E作为左R-模的Gorenstein内射维数是相等的. 同时给出了这些结果的一些应用.     

Gorenstein AC亏范畴*

作者定义了Gorenstein AC导出范畴 D^b_gac(R)并且和导出范畴作了一些比较.作者定义了Gorenstein AC奇点范畴 D^b_gacsg(R),在这个范畴中具有有限Gorenstein AC- 投射维数的模都是零对象.同时, 作者给出了由Gorenstein AC- 投射模构成的稳定范畴到奇点范畴的三角嵌入 F : GAC → D^b_sg(R) .通过作函子 F 的商引入Gorenstein AC亏范畴 D^b_gacd(R),并且给出三角等价 D^b_gacd(R) = D^b_gacsg(R)     

Gorenstein正则环、奇点范畴和Ding模

本文证明了任意环的整体Ding投射维数和整体Ding内射维数一致,研究了奇点范畴和相对于Ding模的稳定范畴间的关系,并刻画了Gorenstein (正则)环以及环的整体维数的有限性.     

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■.     

正交类和相对Gorenstein模

在这篇论文中,我们研究了$\mathcal{A}$-Gorenstein投射模类和$\mathcal{A}$的左正交模类之间的关系,以及$\mathcal{A}$-Gorenstein内射模类和A的右正交模类之间的关系.我们得到了$\mathcal{A}$-Gorenstein投射模和$\mathcal{A}$-Gorenstein内射模的一些函子刻画.以完备对偶对为工具,我们讨论了$\mathcal{A}$-Gorenstein投射模和$\mathcal{B}$-Gorenstein平坦模之间的关系,并推广了一些已知结论.     

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.     

三角矩阵代数上奇点范畴和Gorenstein亏范畴的2-粘合

设■是三角矩阵代数,其中A和B是Artin代数,_AM_B是A-B-双模.本文研究了T上奇点范畴和Gorenstein亏范畴的2-粘合结构.在恰当的假设下,我们给出了T上奇点范畴和Gorenstein亏范畴的2-粘合存在的充分必要条件.     

Tilting Theory and Functor Categories II. Generalized Tilting

In this paper we continue the project of generalizing tilting theory to the category of contravariant functors $\mathrm{Mod}(\mathcal{C})$ , from a skeletally small preadditive category $\mathcal{C}$ to the category of abelian groups, initiated in [15]. We introduce the notion of a generalized tilting category $\mathcal{T}$ , and we concentrate here on extending Happel’s theorem to $\mathrm{Mod}(\mathcal{C})$ ; more specifically, we prove that there is an equivalence of triangulated categories $\mathcal{D}^{b}( \mathrm{Mod}(\mathcal{C}))\cong \mathcal{D}^{b}(\mathrm{Mod}(\mathcal{T}))$ . We then add some restrictions on our category $\mathcal{C}$ , in order to obtain a version of Happel’s theorem for the categories of finitely presented functors. We end the paper proving that some of the theorems for artin algebras, relating tilting with contravariantly finite categories proved in Auslander and Reiten (Adv Math 12(3):306–366, 1974; Adv Math 86(1):111–151, 1991), can be extended to the category of finitely presented functors $\mathrm{mod}(\mathcal{C})$ , with $\mathcal{C}$ a dualizing variety.     

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.     

7c-Gorenstein Projective, ;Cc-Gorenstein Injectiveand Wcr-Gorenstei丨丨 Flat Modules

The authors introduce and investigate the Tc-Gorenstein projective, Lc- Gorenstein injective and Hc-Gorenstein flat modules with respect to a semidualizing module C which shares the common properties with the Gorenstein projective, injective and flat modules, respectively. The authors prove that the classes of all the Tc-Gorenstein projective or the Hc-Gorenstein flat modules are exactly those Gorenstein projective or flat modules which are in the Auslander class with respect to C, respectively, and the classes of all the Lc-Gorenstein ＇injective modules are exactly those Gorenstein injective modules which are in the Bass class, so the authors get the relations between the Gorenstein projective, injective or flat modules and the C-Gorenstein projective, injective or flat modules. Moreover, the authors consider the Tc（R）-projective and Lc（R）-injective dimensions and Tc（R）-precovers and Lc（R）-preenvelopes. Fiually, the authors study the Hc-Gorenstein flat modules and extend the Foxby equivalences.     

Additive Maps Preserving the Star Partial Order on $\mathcal{B}(\mathcal{H})$

Let $\mathcal{B}(\mathcal{H})$ be the $C^∗$-algebra of all bounded linear operators on a complex Hilbert space $\mathcal{H}$. It is proved that an additive surjective map $φ$ on $\mathcal{B}(\mathcal{H})$ preserving the star partial order in both directions if and only if one of the following assertions holds. (1) There exist a nonzero complex number $α$ and two unitary operators $\boldsymbol{U}$and$\boldsymbol{V}$ on $\mathcal{H}$ such that $φ(\boldsymbol{X}) = α\boldsymbol{UXV}$or $φ(\boldsymbol{X}) = α\boldsymbol{UX}^∗\boldsymbol{V}$ for all $X ∈ \mathcal{B}(\mathcal{H})$. (2) There exist a nonzero $α$ and two anti-unitary operators$\boldsymbol{U}$and$\boldsymbol{V}$on $\mathcal{H}$ such that $φ(\boldsymbol{X}) = α\boldsymbol{UXV}$ or $φ(\boldsymbol{X}) = α\boldsymbol{UX}^∗\boldsymbol{V}$ for all $X ∈ \mathcal{B}(\mathcal{H})$.     

Rigid objects, triangulated subfactors and abelian localizations

We show that the abelian category $\mathsf{mod}\text{-}\mathcal{X }$ of coherent functors over a contravariantly finite rigid subcategory $\mathcal{X }$ in a triangulated category $\mathcal{T }$ is equivalent to the Gabriel–Zisman localization at the class of regular maps of a certain factor category of $\mathcal{T }$ , and moreover it can be calculated by left and right fractions. Thus we generalize recent results of Buan and Marsh. We also extend recent results of Iyama–Yoshino concerning subfactor triangulated categories arising from mutation pairs in $\mathcal{T }$ . In fact we give a classification of thick triangulated subcategories of a natural pretriangulated factor category of $\mathcal{T }$ and a classification of functorially finite rigid subcategories of $\mathcal{T }$ if the latter has Serre duality. In addition we characterize $2$ -cluster tilting subcategories along these lines. Finally we extend basic results of Keller–Reiten concerning the Gorenstein and the Calabi–Yau property for categories arising from certain rigid, not necessarily cluster tilting, subcategories, as well as several results of the literature concerning the connections between $2$ -cluster tilting subcategories of triangulated categories and tilting subcategories of the associated abelian category of coherent functors.     

Schur–Weyl Theory for C*‐algebras

To each irreducible infinite dimensional representation $(\pi ,\mathcal {H})$ of a C*‐algebra $\mathcal {A}$, we associate a collection of irreducible norm‐continuous unitary representations $\pi _{\lambda }^\mathcal {A}$ of its unitary group ${\rm U}(\mathcal {A})$, whose equivalence classes are parameterized by highest weights in the same way as the irreducible bounded unitary representations of the group ${\rm U}_\infty (\mathcal {H}) = {\rm U}(\mathcal {H}) \cap (\mathbf {1} + K(\mathcal {H}))$ are. These are precisely the representations arising in the decomposition of the tensor products $\mathcal {H}^{\otimes n} \otimes (\mathcal {H}^*)^{\otimes m}$ under ${\rm U}(\mathcal {A})$. We show that these representations can be realized by sections of holomorphic line bundles over homogeneous Kähler manifolds on which ${\rm U}(\mathcal {A})$ acts transitively and that the corresponding norm‐closed momentum sets $I_{\pi _\lambda ^\mathcal {A}}^{\bf n} \subseteq {\mathfrak u}(\mathcal {A})^{\prime }$ distinguish inequivalent representations of this type.     

2×2阶上三角型算子矩阵的Moore-Penrose谱

设$H_{1}$和$H_{2}$是无穷维可分Hilbert空间. 用$M_{C}$表示$H_{1}\oplusH_{2}$上的2$\times$2阶上三角型算子矩阵$\left(\begin{array}{cc} A & C \\ 0 & B \\\end{array}\right)$. 对给定的算子$A\in{\mathcal{B}}(H_{1})$和$B\in{\mathcal{B}}(H_{2})$,描述了集合$\bigcap\limits_{C\in{\mathcal{B}}(H_{2},H_{1})}\!\!\!\sigma_{M}(M_{C})$与$\bigcup\limits_{C\in{\mathcal{B}}(H_{2},H_{1})}\!\!\!\sigma_{M}(M_{C})$,其中$\sigma_{M}(\cdot)$表示Moore-Penrose谱.