Dual of Theorem 1.1 for the category of all right R-modules (Mod-R)

Establish a one-to-one correspondence between the equivalence classes of Wakamatsu tilting right R-modules under the equivalence relation ∼ and the precovering resolving subcategories of Mod-R that admit an Ext-injective cogenerator and are maximal among those with the same Ext-injective cogenerator, thereby providing the dual of Theorem 1.1 in the category of all right R-modules.

Background

Theorem 1.1 in the paper proves a bijection between equivalence classes of Wakamatsu tilting right R-modules (under the relation ∼) and certain preenveloping coresolving subcategories of Mod-R that have an Ext-projective generator and satisfy a maximality condition.

Theorem 1.2 provides the dual-type correspondence for resolving subcategories but only in the finitely generated setting (mod-R), while Theorem 1.3 gives a partial dual in Mod-R restricted to product-complete Wakamatsu cotilting modules.

The authors explicitly note that extending the dual of Theorem 1.1 to the full category Mod-R remains open, highlighting a gap between the established results and the desired symmetric duality at the level of all modules. They also indicate partial progress via a surjective map (Corollary 6.8).