Rigorous use of the Auslander–Gruson–Jensen (AGJ) transformation for tensor-copresented functors

Investigate how to apply the Auslander–Gruson–Jensen transformation to tensor-copresented additive functors on module categories in a rigorous, nonheuristic manner, notwithstanding that the AGJ transformation is not exact on the category of tensor-copresented functors and does not yield a duality with finitely presented functors.

Background

The paper develops fundamental sequences and universal coefficient theorems for (co)homology by working with additive functors on module categories, emphasizing finitely presented functors for cohomology and tensor-copresented functors for homology. The Auslander–Gruson–Jensen (AGJ) transformation interchanges tensor product and Hom functors and maps finitely presented functors to tensor-copresented functors.

However, the AGJ transformation is not exact on tensor-copresented functors and, as shown via counterexamples in Section 7, the tensor product is not injective in the category of tensor-copresented functors. Consequently, AGJ does not furnish a duality between finitely presented and tensor-copresented functors. The authors note that, beyond heuristic guidance, a rigorous method for employing AGJ in their setting is unclear, motivating a direct construction of quot-stabilization for tensor-copresented functors and leaving open how AGJ might be systematically used.