We introduce the continuous Frobenius category. This category is constructed using representations of the circle over a discrete valuation ring. We show that it is Krull-Schmidt with one indecomposable object for each pair of not necessarily distinct points on the circle. By putting restrictions on these points we obtain various subquotient categories with good properties. The main purpose of o...

We study the K0 and K1-groups of exact and triangulated categories of perfect complexes, and we apply the results to show how determinant functors on triangulated categories can be used for the construction of Euler characteristics in relative algebraic K0-groups.

Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin algebras.

This is an introduction to some aspects of Fomin-Zelevinsky’s cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture fo...