Representing ICE-closed subcategories as relative ICE-closed subcategories

Determine whether, for any artin algebra Λ, every ICE-closed subcategory M of mod-Λ (closed under extensions, admissible cokernels, and admissible images) can be realized as a P-ICE-closed subcategory relative to some projective Λ-module P as defined in Definition 6.1, i.e., M is closed under P-right exact sequences of type 0 and type 1 and under the specified admissible factorizations.

Background

Section 6 develops a framework to relate ICE-closed subcategories in the morphism category P(Λ) to subcategories of mod-Λ. The authors introduce ICE-closed subcategories of mod-Λ relative to a projective module P (Definition 6.1) and prove a bijection between ICE-closed subcategories of P(Λ) and these relative ICE-closed subcategories (Proposition 6.4). This enables transferring structural results between the two settings.

They then raise the general question of whether an ICE-closed subcategory of mod-Λ (in the original, non-relative sense) can always be represented as a relative ICE-closed subcategory for some projective module. The authors provide partial progress: they show this representation holds when the subcategory has enough Ext-projectives and offer characterizations via CE-closed subcategories (Lemma 6.11 and Corollary 6.12), and in the hereditary case they recover Enomoto’s bijection (Theorem 6.14). The unrestricted general case, however, remains unresolved.