When the heart of a faithful torsion pair is a module category

An abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category (Mitchell (1964) [17]). A tilting object in an abelian category is a natural generalization of a small projective generator. Moreover, any abelian category with a tilting object admits arbitrary coproducts (Colpi et al. (2007) [8]). It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By Colpi et al. (2007) [8], the problem simplifies in understanding when, given an associative ring R and a faithful torsion pair (X,Y) in the category of right R-modules, the heartH(X,Y)of the t-structure associated with (X,Y) is equivalent to a category of modules. In this paper, we give a complete answer to this question, proving necessary and sufficient conditions on (X,Y) for H(X,Y) to be equivalent to a module category. We analyze in detail the case when R is right artinian.     

Recollements and tilting objects

We study connections between recollements of the derived category D(Mod R) of a ring R and tilting theory. We first provide constructions of tilting objects from given recollements, recovering several different results from the literature. Secondly, we show how to construct a recollement from a tilting module of projective dimension one. By Nicolás and Saorín (2009) [31], every recollement of D(Mod R) is associated to a differential graded homological epimorphism λ:R→S. We will focus on the case where λ is a homological ring epimorphism or even a universal localization. Our results will be employed in a forthcoming paper in order to investigate stratifications of D(Mod R).     

Perpendicular categories of infinite dimensional partial tilting modules and transfers of tilting torsion classes

Let R be a ring and P be an (infinite dimensional) partial tilting module. We show that the perpendicular category of P is equivalent to the full module category where and ?R is the Bongartz complement of P modulo its P-trace. Moreover, there is a ring epimorphism φ:R→S. We characterize the case when φ is a perfect localization. By [Riccardo Colpi, Alberto Tonolo, Jan Trlifaj, Partial cotilting modules and the lattices induced by them, Comm. Algebra 25 (10) (1997) 3225-3237], there exist mutually inverse isomorphisms μ^′ and ν^′ between the interval in the lattice of torsion classes in , and the lattice of all torsion classes in . We provide necessary and sufficient conditions for μ^′ and ν^′ to preserve tilting torsion classes. As a consequence, we show that these conditions are always satisfied when R is a Dedekind domain, and if P is finitely presented and R is an artin algebra, then the conditions reduce to the trivial ones, namely that each value of μ^′ and ν^′ contains all injectives.     

An elementary construction of tilting complexes

Let A be an artin algebra and e∈A an idempotent with add(eA_A)=add(D(_AAe)). Then a projective resolution of Ae_eAe gives rise to tilting complexes for A, where P(l)^• is of term length l+1. In particular, if A is self-injective, then is self-injective and has the same Nakayama permutation as A. In case A is a finite dimensional algebra over a field and eAe is a Nakayama algebra, a projective resolution of eAe over the enveloping algebra of eAe gives rise to two-sided tilting complexes {T(2l)^•}_l?1 for A, where T(2l)^• is of term length 2l+1. In particular, if eAe is of Loewy length two, then we get tilting complexes {T(l)^•}_l?1 for A, where T(l)^• is of term length l+1.     

Derived equivalences and Gorenstein algebras

We introduce a notion of Gorenstein R-algebras over a commutative Gorenstein ring R. Then we provide a necessary and sufficient condition for a tilting complex over a Gorenstein R-algebra A to have a Gorenstein R-algebra B as the endomorphism algebra and a construction of such a tilting complex. Furthermore, we provide an example of a tilting complex over a Gorenstein R-algebra A whose endomorphism algebra is not a Gorenstein R-algebra.     

Cluster tilting objects in generalized higher cluster categories

We prove the existence of an m-cluster tilting object in a generalized m-cluster category which is (m+1)-Calabi-Yau and Hom-finite, arising from an (m+2)-Calabi-Yau dg algebra. This is a generalization of the result for the m=1 case in Amiot’s Ph.D. thesis. Our results apply in particular to higher cluster categories associated to Ginzburg dg categories coming from suitable graded quivers with superpotential, and higher cluster categories associated to suitable finite-dimensional algebras of finite global dimension.     

Classification of split torsion torsionfree triples in module categories

A TTF-triple (C,T,F) in an abelian category is one-sided split in case either (C,T) or (T,F) is a split torsion theory. In this paper we classify one-sided split TTF-triples in module categories, thus completing Jans’ classification of two-sided split TTF-triples and answering a question that has remained open for almost 40 years.     

Artin-Schelter regular algebras and categories

Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin-Schelter regular algebras of dimension n to algebras and categories to include Auslander algebras and a graded analogue for infinite representation type. A generalized Artin-Schelter regular algebra or a category of dimension n is shown to have common properties with the classical Artin-Schelter regular algebras. In particular, when they admit a duality, then they satisfy Serre duality formulas and the -category of nice sets of simple objects of maximal projective dimension n is a finite length Frobenius category.     

Tilting theory and cluster combinatorics

We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin-Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.     

Pseudosymmetric braidings, twines and twisted algebras

A laycle is the categorical analogue of a lazy cocycle. Twines (introduced by Bruguières) and strong twines (as introduced by the authors) are laycles satisfying some extra conditions. If c is a braiding, the double braiding c² is always a twine; we prove that it is a strong twine if and only if c satisfies a sort of modified braid relation (we call such cpseudosymmetric, as any symmetric braiding satisfies this relation). It is known that the category of Yetter-Drinfeld modules over a Hopf algebra H is symmetric if and only if H is trivial; we prove that the Yetter-Drinfeld category _HYD^H over a Hopf algebra H is pseudosymmetric if and only if H is commutative and cocommutative. We introduce as well the Hopf algebraic counterpart of pseudosymmetric braidings under the name pseudotriangular structures and prove that all quasitriangular structures on the 2ⁿ⁺¹-dimensional pointed Hopf algebras E(n) are pseudotriangular. We observe that a laycle on a monoidal category induces a so-called pseudotwistor on every algebra in the category, and we obtain some general results (and give some examples) concerning pseudotwistors, inspired by the properties of laycles and twines.     

Exactness of direct limits for abelian categories with an injective cogenerator

We prove that the exactness of direct limits in an abelian category with products and an injective cogenerator J is equivalent to a condition on J which is well-known to characterize pure-injectivity in module categories, and we describe an application of this result to the tilting theory. We derive our result as a consequence of a more general characterization of when inverse limits in the Eilenberg–Moore category of a monad on the category of sets preserve regular epimorphisms.     

Classifying subcategories of finitely generated modules over a Noetherian ring

For R a commutative Noetherian ring, wide and Serre subcategories of finitely generated R-modules have been classified by their support. This paper studies general torsion classes and introduces narrow subcategories. These are closed under fewer operations than wide and Serre subcategories, but still for finitely generated R-modules both narrow subcategories and torsion classes are classified using the same support data. Although for finitely generated R-modules all four kinds of subcategories coincide, they do not coincide in the larger category of all R-modules.     

Abelian categories of modules over a (lax) monoidal functor

Crane and Yetter (Deformations of (bi)tensor categories, Cahier de Topologie et Géometrie Differentielle Catégorique, 1998) introduced a deformation theory for monoidal categories. The related deformation theory for monoidal functors introduced by Yetter (in: E. Getzler, M. Kapranov (Eds.), Higher Category Theory, American Mathematical Society Contemporary Mathematics, Vol. 230, American Mathematical Society, Providence, RI, 1998, pp. 117-134.) is a proper generalization of Gerstenhaber's deformation theory for associative algebras (Ann. Math. 78(2) (1963) 267; 79(1) (1964) 59; in: M. Hazewinkel, M. Gerstenhaber (Eds.), Deformation Theory of Algebras and Structure and Applications, Kluwer, Dordrecht, 1988, pp. 11-264). In the present paper we solidify the analogy between lax monoidal functors and associative algebras by showing that under suitable conditions, categories of functors with an action of a lax monoidal functor are abelian categories. The deformation complex of a monoidal functor is generalized to an analogue of the Hochschild complex with coefficients in a bimodule, and the deformation complex of a monoidal natural transformation is shown to be a special case. It is shown further that the cohomology of a monoidal functor F with coefficients in an F,F-bimodule is given by right derived functors.     

The derived category of quasi-coherent sheaves and axiomatic stable homotopy

We prove in this paper that for a quasi-compact and semi-separated (nonnecessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(Aqc(X)), is a stable homotopy category in the sense of Hovey, Palmieri and Strickland, answering a question posed by Strickland. Moreover we show that it is unital and algebraic. We also prove that for a noetherian semi-separated formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies Dqct(X) is a stable homotopy category. It is algebraic but if the formal scheme is not a usual scheme, it is not unital, therefore its abstract nature differs essentially from that of the derived category Dqc(X) (which is equivalent to D(Aqc(X))) in the case of a usual scheme.     

Bounds on the global dimension of certain piecewise hereditary categories

We give bounds on the global dimension of a finite length, piecewise hereditary category in terms of quantitative connectivity properties of its graph of indecomposables.We use this to show that the global dimension of a finite-dimensional, piecewise hereditary algebra A cannot exceed 3 if A is an incidence algebra of a finite poset or more generally, a sincere algebra. This bound is tight.     

On the cohomology rings of small categories

Let C be a small category and R a commutative ring with identity. The cohomology ring of C with coefficients in R is defined as the cohomology ring of the topological realization of its nerve. First we give an example showing that this ring modulo nilpotents is not finitely generated in general, even when the category is finite EI. Then we study the relationship between the cohomology ring of a category and those of its subcategories and extensions. The main results generalize certain theorems in group cohomology theory.     

The bar derived category of a curved dg algebra

Curved A_∞-algebras appear in nature as deformations of dg algebras. We develop the basic theory of curved A_∞-algebras and, in particular, curved dg algebras. We investigate their link with a suitable class of dg coalgebras via the bar construction and produce Quillen model structures on their module categories. We define the analogue of the relative derived category for a curved dg algebra.     

Composite of irreducible morphisms in the bounded derived category

We study necessary and sufficient conditions for the existence of n irreducible morphisms in the bounded derived category of an Artin algebra, with non-zero composite in the n+1-power of the radical. In the case of , the bounded derived category of an Ext-finite hereditary k-category with tilting object, such irreducible morphisms exist if and only if H is derived equivalent to a wild hereditary algebra or to a wild canonical algebra. We also characterize the cluster tilted algebras having such irreducible morphisms.     

Cluster-tilted algebras are Gorenstein and stably Calabi-Yau

We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen-Macaulay modules is 3-Calabi-Yau. We deduce in particular that cluster-tilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press]. In addition, we prove a general result about relative 3-Calabi-Yau duality over non-stable endomorphism rings. This strengthens and generalizes the Ext-group symmetries obtained in [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press] for simple modules. Finally, we generalize the results on relative Calabi-Yau duality from 2-Calabi-Yau to d-Calabi-Yau categories. We show how to produce many examples of d-cluster tilted algebras.