N-复形范畴上的模型结构与粘合(英文)

我们阐述了从完备遗传的余挠对得到N-复形范畴的导出范畴D_N(C)模型结构的方法,并且用模型范畴的理论推广了Krause和Verdier关于N-复形范畴的导出范畴D_N(C)的粘合性质.相应地,我们在任意环的N-复形范畴上定义了奇点范畴和Gorenstein亏范畴,并且借助于三角矩阵代数,得到了这些三角范畴的一些粘合.     

单边挠对

由Beligiannis和Marmaridis引入的单边三角范畴(即左或右三角范畴)是三角范畴的自然推广. 挠理论在范畴的结构方面, 特别在导出范畴或更一般的三角范畴的研究中有着非常重要的作用. 本文引入单边三角范畴中的单边挠对概念, 讨论了这样一种单边挠理论与预三角范畴、稳定范畴以及三角范畴中相应概念的关系, 最后通过Abel范畴上的有界导出范畴D^b (A)给出例子表明存在非平凡单边挠对.     

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.     

相对遗传环

本文给出了左相对遗传环的概念及刻划.讨论了左相对遗传环与左遗传环,左半遗传环的关系,并给出了左遗传环的几个新的刻划.最后,证明了左相对遗传环(相对于左模范畴的生成元)的左理想的自同态环是左半遗传的;左相对遗传环(相对于左模范畴的内射生成元)上的有限生成相对投射模的自同态环是左半遗传的等结果.     

有限群分次环上的余挠对研究

本文研究了余挠对在有限群分次和非分次情况下的联系.利用分次理论以及相对同调,我们首先研究了R是任意环G是有限群的情况下,余挠对在R-模范畴以及斜群环S=R*G-模范畴之间的关系;然后我们研究了R-gr范畴中刚性余挠对的等价刻画,同时给出了余挠对在R-gr范畴与R-模范畴之间的关系,其中R是G分次环,群G是有限群且|G|^-1∈R.     

有限群分次环上的余挠对研究（英文）

本文研究了余挠对在有限群分次和非分次情况下的联系.利用分次理论以及相对同调,我们首先研究了R是任意环G是有限群的情况下,余挠对在R-模范畴以及斜群环S=R*G-模范畴之间的关系;然后我们研究了R-gr范畴中刚性余挠对的等价刻画,同时给出了余挠对在R-gr范畴与R-模范畴之间的关系,其中R是G分次环,群G是有限群且|G|~(-1)∈R.     

粘合、倾斜同调维数与高维Auslander-Reiten理论

本文主要研究阿贝尔范畴粘合$(\mathscr{A}, \mathscr{B}, \mathscr{C})$中$\mathscr{A}$, $\mathscr{B}$与$\mathscr{C}$之间的倾斜同调维数关系. 特别地,对遗传的阿贝尔范畴$\mathscr{B}$,给出了粘合$(\mathscr{A}, \mathscr{B}, \mathscr{C})$中的范畴之间的$n$-几乎可裂序列间的联系.     

相关于遗传挠理论的平坦模和ML模

本文引入了相关于遗传挠理论的平坦模和 ML 模,利用它们刻划了相关 Coherent环,相关 noether 环以及半遗传环,并使得[3]中主要定理和命题有了更完美的形式,此外,我们还给出了平坦模是τ—平坦模、fg τ—平坦模是投射模的条件。     

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.     

相对于g(X)的导出范畴

设A是有足够投射对象的Abel范畴,(x,y)是A中的完备遗传余挠对.本文引入Gorenstein x-导出范畴,记作Dg(x)(A),并且从不同的角度研究D_g(x)(A).当(x,y)取作特殊的余挠对时,得到一些已知的导出范畴,如Gorenstein导出范畴和Gorenstein平坦导出范畴等.通过与g(X)有关的同伦范畴K~(-,gxb)(g(x))描述有界Gorenstein x-导出范畴D~b_(g(x))(A)和有界导出范畴D~b(A),并给出一些三角等价.     

On one-sided torsion pair

Motivated by the concept of a torsion pair in a pre-triangulated category induced by Beligiannis and Reiten, the notion of a left (right) torsion pair in the left (right) triangulated category is introduced and investigated. We provide new connections between different aspects of torsion pairs in one-sided triangulated categories, pre-triangulated categories, stable categories and derived categories.     

Torsion pairs in cluster tubes

We give a complete classification of torsion pairs in the cluster categories associated to tubes of finite rank. The classification is in terms of combinatorial objects called Ptolemy diagrams which already appeared in our earlier work on torsion pairs in cluster categories of Dynkin type A. As a consequence of our classification we establish closed formulae enumerating the torsion pairs in cluster tubes, and find that the torsion pairs in cluster tubes exhibit a cyclic sieving phenomenon.     

Torsion Pairs and Quasi-abelian Categories

Algebras and Representation Theory - We define torsion pairs for quasi-abelian categories and give several characterisations. We show that many of the torsion theoretic concepts translate from...     

TTF Triples in Functor Categories

We characterize the hereditary torsion pairs of finite type in the functor category of a ring R that are associated to tilting torsion pairs in the category of R-modules. Moreover, we determine a condition under which they give rise to TTF triples.     

Extending Ring Derivations to Right and Symmetric Rings and Modules of Quotients

We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie, and perfect torsion theories are differential.

We also study conditions under which a derivation on a right or symmetric module of quotients extends to a right or symmetric module of quotients with respect to a larger torsion theory. Using these results, we study extensions of ring derivations to maximal, total, and perfect right and symmetric rings of quotients.     

Mutation of Torsion Pairs in Triangulated Categories and its Geometric Realization

We introduce and study mutation of torsion pairs, as a generalization of mutation of cluster tilting objects, rigid objects and maximal rigid objects. It is proved that any mutation of a torsion pair is again a torsion pair. A geometric realization of mutation of torsion pairs in the cluster category of type A _n or A _∞ is given via rotation of Ptolemy diagrams.     

Affine analog of the proper base change theorem

We prove a rigidity property for the étale cohomology with torsion coefficients of affine Hensel pairs.