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.     

Gorenstein global dimension of recollements of abelian categories

Abstract

In this article, we investigate the Gorenstein global dimension with respect to the recollements of abelian categories. With the invariants spli and silp of the categories, we give some upper bounds of Gorenstein global dimensions of the categories involved in a recollement of abelian categories. We apply our results to some rings and artin algebras, especially to the triangular matrix artin algebras.     

Derived equivalences between triangular matrix algebras

In this paper, we study derived equivalences between triangular matrix algebras using certain classical recollements. We show that special properties of these recollements actually characterize triangular matrix algebras and describe methods to construct tilting modules and tilting complexes inducing derived equivalences between them.     

Upper recollements and triangle expansions

By using the stable t-structure induced by an adjoint pair, we extend several results concerning recollements to upper (resp. lower) recollements. These include the fundamental results of Parshall and Scott on comparisons of recollements, Wiedemann’s result on the global dimension and Happel’s result on the finitistic dimension, occurring in a recollement (D ^b (A′),D ^b (A),D ^b (A″)) of bounded derived categories of Artin algebras. We introduce and describe a triangle expansion of a triangulated category and illustrate it by examples.     

Comparisons of Left Recollements

Let (F′, F, F″) be a comparison of left recollements of triangulated categories such that F′ and F″ are equivalences. We prove that if F is full then F is an equivalence; and on the other hand, we construct a class of examples via the derived categories of Morita rings, showing that there really exists such a comparison (F′, F, F″) so that F is not an equivalence. This is in contrast to the case of a recollement. We also give a class of examples of left recollements of homotopy categories, which can not sit in recollements.     

Triangulated categories and Kac-Moody algebras

By using the Ringel-Hall algebra approach, we find a Lie algebra arising in each triangulated category with T ²=1, where T is the translation functor. In particular, the generic form of the Lie algebras determined by the root categories, the 2-period orbit categories of the derived categories of finite dimensional hereditary associative algebras, gives a realization of all symmetrizable Kac-Moody Lie algebras. Oblatum 4-XII-1998 & 11-XI-1999?Published online: 21 February 2000     

KK-Theories for topological algebras

We propose a unified construction of generalizedKK-functors on various categories of topological algebras ranging between the two extreme cases: the categories of discrete algebras and the category ofC ^*-algebras. Usual functorial properties of these functors are derived.Supported in part by an NSF grant.     

On the Recollements of Functor Categories

This paper is devoted to the study of recollements of functor categories in different levels. In the first part of the paper, we start with a small category \(\mathcal {S}\) and a maximal object s of \(\mathcal {S}\) and construct a recollement of \(\text {Mod-}\mathcal {S}\) in terms of \(\text {Mod-End}_{\mathcal {S}}(s)\) and \(\text {Mod-}(\mathcal {S}\setminus \{s\})\) in four different levels. In case \(\mathcal {S}\) is a finite directed category, by iterating this argument, we get chains of recollements having some interesting applications. In the second part, we start with a recollement of rings and construct a recollement of their path rings, with respect to a finite quiver. Third part of the paper presents some applications, including recollements of triangular matrix rings, an example of a recollement in Gorenstein derived level and recollements of derived categories of N-complexes.     

Gorensteinness,homological invariants and Gorenstein derived categories

Relations between Gorenstein derived categories, Gorenstein defect categories and Gorenstein stable categories are established. Using these, the Gorensteinness of an algebra A and invariants with respect to recollements of the bounded Gorenstein derived category D _gp ^b (A-mod) of A are investigated. Specifically, the Gorensteinness of A is characterized in terms of recollements of D _gp ^b (A-mod) and Gorenstein derived equivalences. It is also shown that Cohen-Macaulay-finiteness is invariant with respect to the recollements of D _gp ^b (A-mod).     

Derived Azumaya algebras and generators for twisted derived categories

We introduce a notion of derived Azumaya algebras over ring and schemes generalizing the notion of Azumaya algebras of Grothendieck (Le groupe de Brauer. I. Algèbres d’Azumaya et interprétations diverses. Dix Exposés sur la Cohomologie des Schémas, pp. 46–66, North-Holland, Amsterdam, 1968). We prove that any such algebra B on a scheme X provides a class ϕ(B) in . We prove that for X a quasi-compact and quasi-separated scheme ϕ defines a bijective correspondence, and in particular that any class in , torsion or not, can be represented by a derived Azumaya algebra on X. Our result is a consequence of a more general theorem about the existence of compact generators in twisted derived categories, with coefficients in any local system of reasonable dg-categories, generalizing the well known existence of compact generators in derived categories of quasi-coherent sheaves of Bondal and Van Den Bergh (Mosc. Math. J. 3(1):1–36, 2003). A huge part of this paper concerns the treatment of twisted derived categories, as well as the proof that the existence of compact generator locally for the fppf topology implies the existence of a global compact generator. We present explicit examples of derived Azumaya algebras that are not represented by classical Azumaya algebras, as well as applications of our main result to the localization for twisted algebraic K-theory and to the stability of saturated dg-categories by direct push-forwards along smooth and proper maps.     

Weakly group-theoretical and solvable fusion categories

We introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups – weakly group-theoretical categories and solvable categories. These are fusion categories that are Morita equivalent to iterated extensions (in the world of fusion categories) of arbitrary, respectively solvable finite groups. Weakly group-theoretical categories have integer dimension, and all known fusion categories of integer dimension are weakly group-theoretical. Our main results are that a weakly group-theoretical category C has the strong Frobenius property (i.e., the dimension of any simple object in an indecomposable C-module category divides the dimension of C), and that any fusion category whose dimension has at most two prime divisors is solvable (a categorical analog of Burnside's theorem for finite groups). This has powerful applications to classification of fusion categories and semsisimple Hopf algebras of a given dimension. In particular, we show that any fusion category of integer dimension <84 is weakly group-theoretical (i.e. comes from finite group theory), and give a full classification of semisimple Hopf algebras of dimensions pqr and pq², where p,q,r are distinct primes.     

Dimensions of triangulated categories via Koszul objects

Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin algebras.     

半粘合诱导的t-结构

本文研究了对于给定的一个三角范畴的上（下）粘合（C'',C,C"）,如何由C的一个t-结构诱导C''和C"的t-结构的问题.利用左（右）t-正合函子的概念,给出了由C的一个t-结构可诱导出C''和C"的t-结构的充分条件.将粘合的一些相关结果推广到了上（下）粘合的情形.     

On Quiver Grassmannians and Orbit Closures for Gen-Finite Modules

We show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A-module, meaning there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras of suitable cogenerators associated to M, and the resulting recollements, we construct desingularisations of the orbit closure and quiver Grassmannians of M, thus generalising all results from previous work of Crawley-Boevey and the second author in 2017. We provide dual versions of the key results, in order to also treat cogen-finite modules.

     

Blocks of group algebras are derived simple

A derived version of Maschke’s theorem for finite groups is proved: the derived categories, bounded or unbounded, of all blocks of the group algebra of a finite group are simple, in the sense that they admit no nontrivial recollements. This result is independent of the characteristic of the base field.     

On the Derived Categories and Quasitilted Algebras

In this paper, we present some results on the bounded derived category of Artin algebras, and in particular on the indecomposable objects in these categories, using homological properties. Given a complex X ^*, we consider the set and we define the application . We give relationships between some homological properties of an algebra and the respective application l. On the other hand, using homological properties again, we determine two subcategories of the bounded derived category of an algebra, which turn out to be the bounded derived categories of quasi-tilted algebras. As a consequence of these results we obtain new characterizations of quasi-tilted and quasi-tilted gentle algebras. Presented by Raymundo Bautista.     

Reducing Homological Conjectures by <Emphasis Type="Italic">n</Emphasis>-Recollements

n-recollements of triangulated categories and n-derived-simple algebras are introduced. The relations between the n-recollements of derived categories of algebras and the Cartan determinants, homological smoothness and Gorensteinness of algebras respectively are clarified. As applications, the Cartan determinant conjecture is reduced to 1-derived-simple algebras, and the Gorenstein symmetry conjecture is reduced to 2-derived-simple algebras.     

Rota–Baxter Hom-Lie–Admissible Algebras

We study Hom-type analogs of Rota–Baxter and dendriform algebras, called Rota–Baxter G-Hom–associative algebras and Hom-dendriform algebras. Several construction results are proved. Free algebras for these objects are explicitly constructed. Various functors between these categories, as well as an adjunction between the categories of Rota–Baxter Hom-associative algebras and of Hom-(tri)dendriform algebras, are constructed.     

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.     

Relative global dimensions of extensions

An extension of algebras is a homomorphism of algebras preserving identities. Given an extension f:B→A, the relative global dimension of f is defined to be the supremum of relative projective dimensions of all A-modules. In this paper, we compare relative homological dimensions of two extensions of ordinary algebras under certain conditions. As an application, for any natural number n, we present a general method for constructing non-trivial extensions of Artin algebras of relative global dimension at least n.