The compactness locus of a geometric functor and the formal construction   of the Adams isomorphism
  We introduce the compactness locus of a geometric functor between rigidly-compactly generated tensor-triangulated categories, and describe it for several examples arising in equivariant homotopy theory and algebraic geometry. It is a subset of the tensor-triangular spectrum of the target category which, crudely speaking, measures the failure of the functor to satisfy Grothendieck-Neeman duality (or equivalently, to admit a left adjoint). We prove that any geometric functor --- even one which does not admit a left adjoint --- gives rise to a Wirthm\"uller isomorphism once one passes to a colocalization of the target category determined by the compactness locus. When applied to the inflation functor in equivariant stable homotopy theory, this produces the Adams isomorphism. 
Beren Sanders
math.CT math.AG math.AT