One-point extension and recollement

This paper is devoted to studying the recollement of the categories of finitely generated modules over finite dimensional algebras. We prove that for algebras A, B and C, if A-mod admits a recollement relative to B-mod and C-mod, then A[R]-mod admits a recollement relative to B[S]-mod and C-mod, where A[R]and B[S]are the one-point extensions of A by R and of B by S.     

Lifting and Restricting Recollement Data

We study the problem of lifting and restricting TTF triples (equivalently, recollement data) for a certain wide type of triangulated categories. This, together with the parametrizations of TTF triples given in Nicolás and Saorín (Parametrizing recollement data for triangulated categories. To appear in J. Algebra), allows us to show that many well-known recollements of right bounded derived categories of algebras are restrictions of recollements in the unbounded level, and leads to criteria to detect recollements of general right bounded derived categories. In particular, we give in Theorem 1 necessary and sufficient conditions for a right bounded derived category of a differential graded (=dg) category to be a recollement of right bounded derived categories of dg categories. Theorem 2 considers the case of dg categories with cohomology concentrated in non-negative degrees. In Theorem 3 we consider the particular case in which those dg categories are just ordinary algebras.     

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.     

一类代数扩张与其模范畴

对一已知代数进行扩张,并研究扩张代数、重复代数与其模范畴之间的关系.首先利用代数A的双边理想I构造扩张代数T(A,I)和重复代数T(A,I),并研究其模范畴;其次研究范畴T(A,I)-Mod与T(A,I)-Mod的关系,得到于T^v(A,I)-Mod同构于T(A,I)-mod;另外证明存在T(A,I)-mod到T(A,I)-mod的覆盖函子;最后研究商代数A/I的平凡扩张代数T(A/I),得出T(A/I)/I与扩张代数T(A,I)同构.     

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.     

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.     

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.     

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.     

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.     

Abel范畴的右双-Giraud粘合

我们在本文中引入了Abel范畴的右双-Giraud粘合定义.我们证明了右双-Giraud粘合与余遗传和遗传的挠对存在着双射对应.此外,我们通过模范畴中特定的幂等理想刻画了这类挠对.     

Properties of abelian categories via recollements

A recollement is a decomposition of a given category (abelian or triangulated) into two subcategories with functorial data that enables the glueing of structural information. This paper is dedicated to investigating the behaviour under glueing of some basic properties of abelian categories (well-poweredness, Grothendieck's axioms AB3, AB4 and AB5, existence of a generator) in the presence of a recollement. In particular, we observe that in a recollement of a Grothendieck abelian category the other two categories involved are also Grothendieck abelian and, more significantly, we provide an example where the converse does not hold and explore multiple sufficient conditions for it to hold.     

Recollements of Module Categories

We establish a correspondence between recollements of abelian categories up to equivalence and certain TTF-triples. For a module category we show, moreover, a correspondence with idempotent ideals, recovering a theorem of Jans. Furthermore, we show that a recollement whose terms are module categories is equivalent to one induced by an idempotent element, thus answering a question by Kuhn.     

Stable model categories are categories of modules

A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard's work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R.     

κ-线性双扩张范畴(Ⅰ)

κ-线性范畴是有限维κ-代数的自然推广.对应于双扩张代数,定义了κ-线性双扩张范畴■,并且证明了■Mod等价于四元组范畴■,推广了双扩张代数的模范畴理论.     

Stable Projective Homotopy Theory of Modules,Tails, and Koszul Duality

A contravariant functor is constructed from the stable projective homotopy theory of finitely generated graded modules over a finite-dimensional algebra to the derived category of its Yoneda algebra modulo finite complexes of modules of finite length. If the algebra is Koszul with a noetherian Yoneda algebra, then the constructed functor is a duality between triangulated categories. If the algebra is self-injective, then stable homotopy theory specializes trivially to stable module theory. In particular, for an exterior algebra the constructed duality specializes to (a contravariant analog of) the Bernstein–Gelfand–Gelfand correspondence.     

A functorial approach to categorical resolutions

Using a relative version of Auslander's formula, we give a functorial approach to show that the bounded derived category of every Artin algebra admits a categorical resolution. This, in particular, implies that the bounded derived categories of Artin algebras of finite global dimension determine bounded derived categories of all Artin algebras. Hence, this paper can be considered as a typical application of functor categories,introduced in representation theory by Auslander(1971), to categorical resolutions.     

Jordan–Hölder theorems for derived module categories of piecewise hereditary algebras

A Jordan–Hölder theorem is established for derived module categories of piecewise hereditary algebras, in particular for representations of quivers and for hereditary abelian categories of a geometric nature. The resulting composition series of derived categories are shown to be independent of the choice of bounded or unbounded derived module categories, and also of the choice of finitely generated or arbitrary modules.     

Relative derived equivalences and relative homological dimensions

Let A be a small abelian category. For a closed subbifunctor F of Ext_A¹(-,-), Buan has generalized the construction of Verdier's quotient category to get a relative derived category, where he localized with respect to F-acyclic complexes. In this paper, the homological properties of relative derived categories are discussed, and the relation with derived categories is given. For Artin algebras, using relative derived categories, we give a relative version on derived equivalences induced by F-tilting complexes. We discuss the relationships between relative homological dimensions and relative derived equivalences.     

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.