Functors given by kernels, adjunctions and duality

Original · EN

Let X₁ and X₂ be schemes of finite type over a field of characteristic 0. Let Q be an object in the category D-mod(X₁× X₂) and consider the functor F:D-mod(X₁)->Dmod(X₂) defined by Q. Assume that F admits a right adjoint also defined by an object P in D-mod(X₁× X₂). The question that we pose and answer in this paper is how P is related to the Verdier dual of Q. We subsequently generalize this question to the case when X₁ and X₂ are no longer schemes but Artin stacks, where the situation becomes much more interesting.