No abstract

Derived equivalences and t-structures are closely related. We use realisation functors associated to t-structures in triangulated categories to establish a derived Morita theory for abelian categories with a projective generator or an injective cogenerator. For this purpose we develop a theory of (non-compact, or large) tilting and cotilting objects that generalises the preceding notions in the literature. Within the scope of derived Morita theory for rings we show that, under some assumptions, the realisation functor is a derived tensor product. This fact allows us to approach a problem by Rickard on the shape of derived equivalences. Finally, we apply the techniques of this new derived Morita theory to show that a recollement of derived categories is a derived version of a recollement of abelian categories if and only if there are tilting or cotilting t-structures glueing to a tilting or a cotilting t-structure. As a further application, we answer a question by Xi on a standard form for recollements of derived module categories for finite dimensional hereditary algebras.

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. IntroductionA recollement of abelian categories is an exact sequence of abelian categories where both the inclusion and the quotient functors admit left and right adjoints. They first appeared in the construction of the category of perverse sheaves on a singular space by Beilinson, Bernstein and Deligne ([5]), arising from recollements of triangulated categories with additional structures (compatible t-structures). Properties of recollements of abelian categories were more recently studied by Franjou and Pirashvilli in [12], motivated by the MacPherson-Vilonen construction for the category of perverse sheaves ([23]).Recollements of abelian categories were used by Cline, Parshall and Scott to study module categories of finite dimensional algebras over a field (see [28]). Later, Kuhn used them in the study of polynomial functors ([20]), which arise not only in representation theory but also in algebraic topology and algebraic K-theory. Recollements of triangulated categories have appeared in the work of Angeleri Hügel, Koenig and Liu in connection with tilting theory, homological conjectures and stratifications of derived categories of rings ([1], [2], [3]). In particular, Jordan-Hölder theorems for recollements of derived module categories were obtained for some classes of algebras ([2], [3]). Also, Chen and Xi have investigated recollements in relation with tilting theory ([7]) and algebraic K-theory ([8]). Homological properties of recollements of abelian and triangulated categories have also been studied in [29].Recollements and TTF-triples of triangulated categories are well-known to be in bijection ([5], [24], [25]). We will show that such a bijection holds for Mod-A (see Proposition 5.2), where Mod-A denotes the category of right A-modules, for a unitary ring A. Similar considerations in Mod-A were made for split TTF-triples in [26]. More generally, we show that recollements of an abelian category A (up to equivalence) are in bijection with bilocalising TTF-classes (see Theorem 4.3).Examples of recollements are easily constructed for the module category of triangular matrix rings (see [9], [17], [22]) or, more generally, using idempotent elements of a ring (see Example 2.9). In fact, we will see that there is a correspondence between idempotent ideals of A and recollements of Mod-A, recovering Jans' bijection ([19]) between TTF-triples in Mod-A and idempotent ideals. Moreover, Kuhn conjectured in [20] that if the categories of a recollement are equivalent to categories of modules over finite dimensional algebras over a field, then it is equivalent to one arising from an idempotent element. In our main result, we prove this ...

Abstract. Given an artin algebra Λ with an idempotent element a we compare the algebras Λ and aΛa with respect to Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology. In particular, we identify assumptions on the idempotent element a which ensure that Λ is Gorenstein if and only if aΛa is Gorenstein, that the singularity categories of Λ and aΛa are equivalent and that Fg holds for Λ if and only if Fg holds for aΛa. We approach the problem by using recollements of abelian categories and we prove the results concerning Gorensteinness and singularity categories in this general setting. The results are applied to stable categories of Cohen-Macaulay modules and classes of triangular matrix algebras and quotients of path algebras.

We study Morita rings Λ (φ,ψ) in the context of Artin algebras from various perspectives. First we study covariantly finite, contravariantly finite, and functorially finite subcategories of the module category of a Morita ring when the bimodule homomorphisms φ and ψ are zero. Further we give bounds for the global dimension of a Morita ring Λ (0,0) , as an Artin algebra, in terms of the global dimensions of A and B in the case when both φ and ψ are zero. We illustrate our bounds with some examples. Finally we investigate when a Morita ring is a Gorenstein Artin algebra and then we determine all the Gorenstein-projective modules over the Morita ring Λ φ,ψ in case A = N = M = B and A an Artin algebra.The Morita ring of a Morita context, not to be confused with the notion of a (right or left) Morita ring appearing in Morita duality, has been studied explicitly by various authors in ring, module, or