Francesca Mantese

Derived equivalence induced by infinitely generated n-tilting modules

Let TR be a right n-tilting module over an arbitrary associative ring R. In this paper we prove that there exists a n-tilting module T ′ R equivalent to TR which induces a derived equivalence between the unbounded derived category D(R) and a triangulated subcategory E⊥ of D(End(T ′)) equivalent to the quotient category of D(End(T ′)) modulo the kernel of the total left derived functor − ⊗L S′ T ′. If TR is a classical n-tilting module, we have that T = T ′ and the subcategory E⊥ coincides with D(End(T )), recovering the classical case.

Reflexivity in derived categories

Abstract An adjoint pair of contravariant functors between abelian categories can be extended to the adjoint pair of their derived functors in the associated derived categories. We describe the reflexive complexes and interpret the achieved results in terms of objects of the initial abelian categories. In particular we prove that, for functors of any finite cohomological dimension, the objects of the initial abelian categories which are reflexive as stalk complexes form the largest class where a Cotilting Theorem in the sense of Colby and Fuller [CbF1, Ch. 5] works.

Leavitt path algebras are Bézout

Let E be a directed graph, K any field, and let LK(E) denote the Leavitt path algebra of E with coefficients in K. We show that LK(E) is a Bézout ring, i.e., that every finitely generated one-sided ideal of LK(E) is principal.