Functors given by kernels, adjunctions and duality (Q2804213)

The paper under review concerns a generalisation of a well-known property of a Fourier-Mukai transform between the derived categories of smooth projective varieties: it automatically has a right (and left) adjoint, it is again of Fourier-Mukai type, and its kernel is related to the original kernel using Serre duality.NEWLINENEWLINEFirst, the case of smooth separated schemes of finite type over a field of characteristic zero is treated, for (a dg enhancement of) their derived categories of D-modules. Given a continuous functor between such categories it can always be described as a Fourier-Mukai transform in the context of D-modules, and it automatically has a right adjoint. Now if this right adjoint is itself continuous it has a description as a Fourier-Mukai transform, and the kernel is nothing but the shift of the Verdier dual. The result is a corollary to an abstract result regarding duality in a monoidal compactly generated dg category.NEWLINENEWLINEThe second part generalises this to Artin stacks. It turns out that by the same techniques one can show that for a quasicompact Artin stack (whose derived category of D-modules is compactly generated) the functor associated to the Verdier dual of the original kernel is the composition of the original functor and an endofunctor whose kernel is the pushforward of the constant sheaf. In the smooth and separated case this endofunctor is nothing but the shift.NEWLINENEWLINEThe main application that the author has in mind is for the non-quasicompact Artin stack of \(G\)-bundles on a smooth projective curve, an object central to the geometric Langlands programme. He identifies the appropriate dg categories for which there is an explicit description of the right adjoint as before. The techniques in this paper are mostly categorical, and might be applied in other situations. Unfortunately the paper does not give any references to standard works regarding supposedly basic properties of D-modules, building only on earlier works of the author and his co-authors.