广义Comma范畴的Recollement

本文研究广义Comma范畴上Recollement问题.利用Abel范畴上Recollement及其伴随函子,诱导出广义Comma范畴,并利用比较函子构造出广义Comma范畴上的Recollement.这些结果推广了一般Abel范畴上的Recollement,丰富了Comma范畴研究.     

From recollement of triangulated categories to recollement of abelian categories

In this paper,we prove that if a triangulated category D admits a recollement relative to triangulated categories D' and D″,then the abelian category D/T admits a recollement relative to abelian categories D'/i(T) and D″/j(T) where T is a cluster tilting subcategory of D and satisfies i i (T)  T,j j (T) T.     

三角范畴的coherent函子范畴的recollement

文章研究了三角范畴D及其coherent函子范畴A(D)的recollement之间的关系.利用D的recollement可以诱导A(D)的prerecollement,文章证明了该prerecollement是recollement的充分必要条件是D的recollement是可裂的;并且D的recollement可以诱导A(D)的prerecollement.     

自由函子及其伴随函子

本文证明了由集合范畴到f-模范畴的自由函子的存在性，构造了自由函子的伴随函子。     

模范畴Recollement的Koenig定理

Koenig定理描述了环的导出范畴允许recollement的一个充分必要条件.本文给出环的模范畴版本的Koenig定理及其应用.应用一是可以导出Morita等价定理,应用二是可以描述三角矩阵环与模范畴的recollement之间的密切联系.     

A recollement construction of Gorenstein derived categories

We first give an equivalence between the derived category of a locally finitely presented category and the derived category of contravariant functors from its finitely presented subcategory to the category of abelian groups, in the spirit of Krause’s work [Math. Ann., 2012, 353: 765–781]. Then we provide a criterion for the existence of recollement of derived categories of functor categories, which shows that the recollement structure may be induced by a proper morphism defined in finitely presented subcategories. This criterion is then used to construct a recollement of derived category of Gorenstein injective modules over CM-finite 2-Gorenstein artin algebras.     

单子和余单子的缠绕结构

研究单子和余单子的缠绕结构和缠绕模以及与代数和余代数的缠绕结构和缠绕模之间的关系,定义了余单子的类群元,得到了一些有意义的结论.最后构造了缠绕模范畴上的两个函子,并证明了它们是伴随函子.     

函子范畴上加性函子的一个表示

研究函子范畴ModC上加性函子的表示,把一个Abel群作成范畴ModC上的一个左C-模,构造出一个Hom函子和一个函子态射,证明了从函子范畴ModC到范畴Ab的任意变和为积的反变左正合可加函子都与某个Hom函子自然等价.所得结论在函子范畴上,推广了Watts定理.     

Models for homotopy categories of injectives and Gorenstein injectives

A natural generalization of locally noetherian and locally coherent categories leads us to define locally type FP_∞ categories. They include not just all categories of modules over a ring, but also the category of sheaves over any concentrated scheme. In this setting we generalize and study the absolutely clean objects recently introduced in [5 Bravo, D., Gillespie, J., Hovey, M. The stable module category of a general ring (arXiv:1405.5768). [Google Scholar]]. We show that 𝒟(𝒜𝒞), the derived category of absolutely clean objects, is always compactly generated and that it is embedded in K(Inj), the chain homotopy category of injectives, as a full subcategory containing the DG-injectives. Assuming the ground category 𝒢 has a set of generators satisfying a certain vanishing property, we also show that there is a recollement relating 𝒟(𝒜𝒞) to the (also compactly generated) derived category 𝒟(𝒢). Finally, we generalize the Gorenstein AC-injectives of [5 Bravo, D., Gillespie, J., Hovey, M. The stable module category of a general ring (arXiv:1405.5768). [Google Scholar]], showing that they are the fibrant objects of a cofibrantly generated model structure on 𝒢.     

分次环与模范畴上的伴随函子

设 G是有限群 ,R是强 G-分次环 .本文证明了 R Re-与 Hom Re(R,- )都是从模范畴 R - mod到 Re- mod的“纯量”限制函子 F的伴随函子 ,并且两个函子 R Re-和Hom Re(R,- )是自然同构的 .     

Adjoint Functors and Representation Dimensions

We study the global dimensions of the coherent functors over two categories that are linked by a pair of adjoint functors. This idea is then exploited to compare the representation dimensions of two algebras. In particular, we show that if an Artin algebra is switched from the other, then they have the same representation dimension.     

THE TWO FUNCTORS IN THE FUZZY MODULAR CATEGORY

1 IntroductionWe have iIltroduced the fuzzy nlodular category Pm in the reference [1] alld thell cOI1-structed son1e kinds of fuzzy quotiellt n1odule in the reference [2l and [3l. Furtllernlore wediscussed the concepts of tlie exact sequeIlces of Linear maps in Fm and obtained 1naIly iInpor-taIlt conclusions in the reference [4]. It is a bright idea to illvestigate the algebric structure ofFln by means of homological algebra. We illtroduce Honl functors and Tensor fu1lctors iIl Fm i11thi…     

Track Extensions of Categories and Cohomology

We introduce the notion of n-fold track extensions of a category C by a natural system D and prove that such extensions represent classes in the cohomology of C with coefficients in D introduced by Baues–Wirsching. This generalizes a result of Huebschmann on the cohomology of groups.     

Algebraic logoi

We introduce normal cores, as well as the more general action cores, in the context of a semi-abelian category, and further generalise those to split extension cores in the context of a homological category. We prove that, if the category is moreover well-powered with (small) joins, then the existence of split extension cores is equivalent to the condition that the change-of-base functors in the fibration of points are geometric. We call a finitely complete category that satisfies this condition an algebraic logos. We give examples of such categories, compare them with algebraically coherent ones, and study equivalent conditions as well as stability under common categorical operations.     

One-point extensions of generalized hexagons and octagons

In this note, we prove the uniqueness of the one-point extension S of a generalized hexagon of order 2 and prove the non-existence of such an extension S of any other finite generalized hexagon of classical order, different from the one of order 2, and of the known finite generalized octagons provided the following property holds: for any three points x, y and z of S, the graph-theoretic distance from y to z in the derived generalized hexagon S_x is the same as the distance from x to z in S_y.     

Stable t-structures and Homotopy Category of Strongly Copure Projective Modules

In this paper, we study the homotopy category of unbounded complexes of strongly copure projective modules with bounded relative homologies K~(∞,bscp)(SCP).We show that the existence of a right recollement of K~(∞,bscp)(SCP) with respect to K~(-,bscp)(SCP), K_(scpac)(SCP) and K~(∞,bscp)(SCP) has the homotopy category of strongly copure projective acyclic complexes as a triangulated subcategory in some case.     

Balanced pairs induce recollements

Let 𝒜 be an abelian category with arbitrary (set-indexed) coproducts and exact products. Let (𝒫,?) be a complete balanced pair. Then as in the classical case, we prove that there exists a recollement with the middle term K(𝒜), the homotopy category of 𝒜. In particular, this implies that the relative derived category exists. Two applications are given.     

Kan Extensions of Internal Functors. Algebraic Approach

We investigate the question of the Kan extension in the case when all categories and functors are internal in the category of groups. Under certain assumptions we establish the necessary and sufficient conditions for the existence of internal Kan extensions. Questions related to this problem are also discussed.     

Separable Groupoid Rings

We show that groupoid rings are separable over their ring of coefficients if and only if the groupoid is finite and the orders of the associated principal groups are invertible in the ring of coefficients. We use this to show that if we are given a finite groupoid, then the associated groupoid ring is semisimple (or hereditary) if and only if the ring of coefficients is semisimple (or hereditary) and the orders of the principal groups are invertible in the ring of coefficients. To this end, we extend parts of the theory of graded rings and modules from the group graded case to the category graded, and, hence, groupoid graded situation. In particular, we show that strongly groupoid graded rings are separable over their principal components if and only if the image of the trace map contains the identity.