Reflexivity in derived categories (original) (raw)

Introduction to abelian and derived categories

Representations of Reductive Groups, 1998

This is an account of three 1-hour lectures given at the Instructional Conference on Representation Theory of Algebraic Groups and Related Finite Groups, Isaac Newton Institute, Cambridge, 6-11 January 1997.

Derived categories and their applications

Revista de la Unión Matemática …, 2007

In this notes we start with the basic definitions of derived categories, derived functors, tilting complexes and stable equivalences of Morita type. Our aim is to show via several examples that this is the best framework to do homological algebra, We also exhibit their usefulness for getting new proofs of well known results. Finally we consider the Morita invariance of Hochschild cohomology and other derived functors.

On relative derived categories

arXiv (Cornell University), 2013

The paper is devoted to study some of the questions arises naturally in connection to the notion of relative derived categories. In particular, we study invariants of recollements involving relative derived categories, generalise two results of Happel by proving the existence of AR-triangles in Gorenstein derived categories, provide situations for which relative derived categories with respect to Gorenstein projective and Gorenstein injective modules are equivalent and finally study relations between the Gorenstein derived category of a quiver and its image under a reflection functor. Some interesting applications are provided.

On abelian 2-categories and derived 2-functors

arXiv (Cornell University), 2010

." Several new facts added, including the material on the derived 2-functors and the proof of the Gabriel-Mitchel theorem and the Morita theory for 2-rings.

Derived Categories and Tilting

2004

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 equiv- alence between derived categories. We establish its link with Rickard's theorem which characterizes derived equivalent algebras. We then examine invariants under derived equivalences. Using t-structures we

Derived categories of N-complexes

Journal of the London Mathematical Society, 2017

We study the homotopy category K N (B) of N-complexes of an additive category B and the derived category D N (A) of an abelian category A. First we show that both K N (B) and D N (A) have natural structures of triangulated categories. Then we establish a theory of projective (resp., injective) resolutions and derived functors. Finally, under some conditions of an abelian category A, we show that D N (A) is triangle equivalent to the ordinary derived category D(Mor N−2 (A)) where Mor N−2 (A) is the category of sequential N − 2 morphisms of A.