The compactness locus of a geometric functor and the formal construction of the Adams isomorphism

TL;DR

This paper introduces the concept of the compactness locus for geometric functors in tensor-triangulated categories, linking it to duality failures and deriving the Adams isomorphism via colocalization.

Contribution

It defines the compactness locus and demonstrates its role in establishing the Adams isomorphism through colocalization, even without a left adjoint.

Findings

The compactness locus measures duality failure in geometric functors.

Any geometric functor yields a Wirthmüller isomorphism after colocalization.

Application to equivariant homotopy theory produces the Adams isomorphism.

Abstract

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.