Abstract:
We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their cluster tilting objects are related by a simple combinatorial construction, which we call a flip-flop. We deduce that the posets of cluster tilting objects of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent. In particular, all Cambrian lattices corresponding to the various orientations of the same Dynkin diagram are universally derived equivalent.

Abstract:
Given a set of 'simple-minded' objects in a derived category, Rickard constructed a complex, which over a symmetric algebra provides a derived equivalence sending the 'simple-minded' objects to simple ones. We characterise in terms of t-structures, when this complex is a tilting complex, show that there is an associated natural $t$-structure and we provide an alternative construction of this complex in terms of A-infinity-structures. Our approach is similar to that of Keller--Nicol\'as.

Abstract:
The Nakayama conjecture states that an algebra of infinite dominant dimension should be self-injective. Motivated by understanding this conjecture in the context of derived categories, we study dominant dimensions of algebras under derived equivalences induced by tilting modules, specifically, the infinity of dominant dimensions under tilting procedure. We first give a new method to produce derived equivalences from relatively exact sequences, and then establish relationships and lower bounds of dominant dimensions for derived equivalences induced by tilting modules. Particularly, we show that under a sufficient condition the infinity of dominant dimensions can be preserved by tilting, and get not only a class of derived equivalences between two algebras such that one of them is a Morita algebra in the sense of Kerner-Yamagata and the other is not, but also the first counterexample to a question whether generalized symmetric algebras are closed under derived equivalences.

Abstract:
We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their tilting modules are related by a simple combinatorial construction, which we call flip-flop. We deduce that the posets of tilting modules of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent.

Abstract:
A triangular matrix ring A is defined by a triplet (R,S,M) where R and S are rings and M is an S-R-bimodule. In the main theorem of this paper we show that if T is a tilting S-module, then under certain homological conditions on M as an S-module, one can extend T to a tilting complex over A inducing a derived equivalence between A and another triangular matrix ring specified by (S',R,M'), where the ring S' and the R-S'-bimodule M' depend only on T and M, and S' is derived equivalent to S. Note that no conditions on the ring R are needed. These conditions are satisfied when S is an Artin algebra of finite global dimension and M is finitely generated as an S-module. In this case, $(S',R,M')=(S,R,DM)$ where D is the duality on the category of finitely generated S-modules. They are also satisfied when S is arbitrary, M has a finite projective resolution as an S-module, and $\Ext^n_S(M,S)=0$ for all $n>0$. In this case, $(S',R,M')=(S,R,\Hom_S(M,S))$.

Abstract:
We generalize Brenner and Butler's Theorem as well as Happel's Theorem on the equivalences induced by a finitely generated tilting module over artin algebras, to the case of an infinitely generated tilting module over an arbitrary associative ring establishing the equivalences induced between subcategories of module categories and also at the level of derived categories.

Abstract:
We consider the problem of comparing t-structures under the derived McKay correspondence and for tilting equivalences. We relate the t-structures using certain natural torsion theories. As an application, we give a criterion for rationality for surfaces with a tilting bundle. In particular we show that every smooth projective surface which admits a full, strong, exceptional collection of line bundles is rational.

Abstract:
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.

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 (noncompact, 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.

Abstract:
We consider recollements of derived categories of dg-algebras induced by self orthogonal compact objects obtaining a generalization of Rickard's Theorem. Specializing to the case of partial tilting modules over a ring, we extend the results on triangle equivalences proved in [B2] and [BMT]. In the end we focus on the connection between recollements of derived categories of rings, bireflective subcategories and generalized universal localizations".