{A∞-bimodules A-infinity-bimodules and Serre {A∞-functors [A-infinity-functors] [Elektronische Ressource] / Oleksandr Manzyuk

A -bimodules and Serre A -functors∞ ∞Oleksandr ManzyukVom Fachbereich Mathematik der Technischen Universit¨at Kaiserslauternzur Verleihung des akademischen Grades Doktor der Naturwissenschaften(Doctor rerum naturalium, Dr. rer. nat.) genehmigte Dissertation1. Gutachter: Prof. Dr. Gert-Martin Greuel2. Gutachter: Prof. Dr. Bernhard KellerVollzug der Promotion 25.10.2007D 386AbstractThis dissertation is intended totransport the theory ofSerre functors into the contextof A -categories. We begin with an introduction to multicategories and closed multi-∞categories, which form a framework in which the theory of A -categories is developed.∞We prove that (unital) A -categories constitute a closed symmetric multicategory. We∞define the notion of A -bimodule similarly to Tradler and show that it is equivalent to∞anA -functor of two arguments which takes values in the differential graded category of∞complexes ofk-modules, wherek is a commutative ground ring. Serre A -functors are∞defined via A -bimodules following ideas of Kontsevich and Soibelman. We prove that∞a unital closed under shifts A -category A over a fieldk admits a Serre A -functor if∞ ∞0and only if its homotopy categoryH A admits an ordinary Serre functor. The proof usescategories enriched in K, the homotopy category of complexes ofk-modules, and SerreK-functors. Another important ingredient is an A -version of the Yoneda Lemma.