Ferrand’s Hom-based normic polynomial law beyond projectivity

Establish whether, for a finite locally free ring extension R → R′ and arbitrary R′-modules M1 and M2 (without assuming M1 is finitely generated projective), there exists a normic polynomial law W(Hom_{R′}(M1,M2)) → W(Hom_R(N_{R′/R}(M1), N_{R′/R}(M2)) that induces an R-linear map N_{R′/R}(Hom_{R′}(M1,M2)) → Hom_R(N_{R′/R}(M1), N_{R′/R}(M2)) as claimed in Ferrand [Fe, 3.2.5], and verify its functoriality on S-points for all R-algebras S. Determine precise conditions under which the canonical horizontal maps used in the construction are isomorphisms, thereby removing the projectivity hypothesis invoked in Lemma 3.12 and clarifying the general validity of Ferrand’s construction.

Background

In Section 3.12 the authors discuss a construction inspired by Ferrand’s norm functor that should produce, from an R′-linear map f: M1 → M2, a normic polynomial law between Hom-modules leading to a natural R-linear map N_{R′/R}(Hom_{R′}(M1,M2)) → Hom_R(N_{R′/R}(M1), N_{R′/R}(M2)). This relies on certain canonical identifications to define the law on S-points for all R-algebras S.

They point out that these canonical maps are isomorphisms when M1 is finitely generated projective (so that the Hom-modules behave well under base change), which enables their development of Lemma 3.12. However, in the absence of this projectivity assumption, they were unable to verify Ferrand’s suggested construction, leaving the general case unresolved.