Conditions ensuring the projection formula isomorphism for a general morphism of sites

Determine sufficient and explicit conditions on a morphism of sites f: X -> Y under which the projection morphism Rf_*(C ⊗ C') -> Rf_*(C) ⊗ f^*(C') is an isomorphism for all C in Db(S(Y)) and C' in D(S(X)), by extending the telescope/filtered-colimit reduction from perfect complexes used in the proof of Theorem 1 to the general case.

Background

The paper establishes a sheaf-theoretic universal coefficient theorem via a projection formula for derived categories of sheaves on sites. For a morphism of sites f: X -> Y, with inverse image functor f* and right adjoint Rf_, the author defines a projection morphism Rf_(C ⊗ C') -> Rf_(C) ⊗ f^(C'), where C ∈ Db(S(Y)) and C' ∈ D(S(X)). Theorem 1 proves this morphism is an isomorphism in two cases: (i) when C is a perfect complex, and (ii) in the special situation Y is the point (so Rf_* = RΓ(X, -)) and RΓ commutes with infinite direct sums.

The proof of Case (ii) relies on expressing bounded complexes of abelian groups as filtered colimits (telescopes) of perfect complexes in D(Ab), allowing the isomorphism to be deduced from Case (i). The author explicitly notes uncertainty about how to generalize this method to an arbitrary morphism of sites f, motivating the open problem of identifying conditions on f that make the reduction via telescopes and colimits work and thereby yield the projection formula isomorphism beyond the stated special cases.