Fe b 20 08 A ∞ - bimodules and Serre A ∞ - functors

We define A∞-bimodules similarly to Tradler and show that this notion is equivalent to an A∞-functor with two arguments which takes values in the differential graded category of complexes of k-modules, where k is a ground commutative ring. Serre A∞-functors are defined via A∞-bimodules likewise Kontsevich and Soibelman. We prove that a unital closed under shifts A∞-category A over a field k admits a Serre A∞-functor if and only if its homotopy category H 0A admits a Serre k-linear functor. The proof uses categories enriched in K, the homotopy category of complexes of k-modules, and Serre K-functors. Also we use a new A∞-version of the Yoneda Lemma generalizing the previously obtained result. Serre–Grothendieck duality for coherent sheaves on a smooth projective variety was reformulated by Bondal and Kapranov in terms of Serre functors [BK89]. Being an abstract category theory notion Serre functors were discovered in other contexts as well, for instance, in Kapranov’s studies of constructible sheaves on stratified spaces [Kap90]. Reiten and van den Bergh showed that Serre functors in categories of modules are related to Auslander–Reiten sequences and triangles [RvdB02]. Often Serre functors are considered in triangulated categories and it is reasonable to lift them to their origin – pretriangulated dg-categories or A∞-categories. Soibelman defines Serre A∞-functors in [Soi04], based on Kontsevich and Soibelman work which is a sequel to [KS06]. In the present article we consider Serre A∞-functors in detail. We define them via A∞-bimodules in Section 6 and use enriched categories to draw conclusions about existence of Serre A∞-functors. A∞-modules over A∞-algebras are introduced by Keller [Kel01]. A∞-bimodules over A∞-algebras are defined by Tradler [Tra01, Tra02]. A∞-modules and A∞-bimodules over A∞-categories over a field were first considered by Lefèvre-Hasegawa [LH03] under the Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska st., Kyiv-4, 01601 MSP, Ukraine; [email protected] Fachbereich Mathematik, Postfach 3049, 67653 Kaiserslautern, Germany; [email protected]