We review the basic definitions of derived categories and derived functors. We illustrate them on simple but non trivial examples. Then we explain Happel’s theorem which states that each tilting triple yields an equivalence between derived categories. We establish its link with Rickard’s theorem which characterizes derived equivalent algebras. We then examine invariants under derived equivalenc...

We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.

We put cluster tilting in a general framework by showing that any quotient of a triangulated category modulo a tilting subcategory (that is, a maximal one-orthogonal subcategory) carries an induced abelian structure. These abelian quotients turn out to be module categories of Gorenstein algebras of dimension at most one.

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.

BACKGROUND Despite advances in neonatal intensive care, germinal matrix-intraventricular hemorrhage (GMH-IVH) remains a frequent, serious complication of premature birth. Neutral head position and head tilting have been suggested to reduce the risk of GMH-IVH in preterm infants during the first 72 h of life. OBJECTIVE The aim of this study was to provide a systematic review of the effect of n...

BY Rickard's fundamental theorem [8], the rings which are derived equivalent to a ring A are precisely the endomorphism rings of tilting complexes over A. A tilting complex T is a finitely generated complex of finitely generated projective modules, which does not admit selfextensions and which has the property that the smallest triangulated subcategory of D(A) which contains T also contains all...

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.

cumulus parameterization in numerical weather prediction models can significantly affect severe weather forecasts, such as hurricanes, flash floods, and winter storms. the role of convection is essential in stabilizing an unstable atmosphere through vertically transferring moisture, energy, chemical species and momentum. cumulus parameterization schemes use simple one-dimensional convective clo...

The body's ability to rapidly and appropriately regulate blood pressure in response to changing physiological demand is a key feature of a healthy cardiovascular system. Passively tilting the body, thereby changing central blood volume, is a well-recognized and controlled method of evaluating this ability. However, such studies usually involve single tilt angles, or intermittent tilting separat...

25 In the physiological condition, mammalian oocytes and embryos appears to be stimulated not only chemically but also mechanically, such as compression, shear stress and/or friction force in the follicle and female reproductive tract. The present study was undertaken to examine the effects of kinetic culture with a tilting device in chemically defined media during in vitro maturation (IVM) of ...