Separated monic representations II: Frobenius subcategories and RSS equivalences

Source abstract

This paper looks for Frobenius subcategories, via the separated monomorphism category smon ⁡ ( Q , I , X ) \operatorname {smon}(Q, I, \mathscr {X}) , and on the other hand, aims to establish an RSS equivalence from smon ⁡ ( Q , I , X ) \operatorname {smon}(Q, I, \mathscr {X}) to its dual sepi ⁡ ( Q , I , X ) \operatorname {sepi}(Q, I, \mathscr {X}) . For a bound quiver ( Q , I ) (Q, I) and an algebra A A , where Q Q is acyclic and I I is generated by monomial relations, let Λ = A ⊗ k k Q / I \Lambda =A\otimes _k kQ/I . For any additive subcategory X \mathscr {X} of A A -mod, we introduce smon ⁡ ( Q , I , X ) \operatorname {smon}(Q, I, \mathscr {X}) combinatorially. It describes Gorenstein-projective Λ \Lambda -modules as G P ( Λ ) = smon ⁡ ( Q , I , G P ( A ) ) \mathcal {GP}(\Lambda ) = \operatorname {smon}(Q, I, \mathcal {GP}(A)) . It admits a homological interpretation and enjoys a reciprocity smon ⁡ ( Q , I , ⊥ T ) = ⊥ ( T ⊗ k Q / I ) \operatorname {smon}(Q, I, \ ^\bot T)= \ ^\bot (T\otimes kQ/I) for a cotilting A A -module T T . As an application, smon ⁡ ( Q , I , X ) \operatorname {smon}(Q, I, \mathscr {X}) has Auslander-Reiten sequences if X \mathscr {X} is resolving and contravariantly finite with X ^ = \widehat {\mathscr {X}}= 𝐴-mod . In particular, smon ⁡ ( Q , I , A ) \operatorname {smon}(Q, I, A) has Auslander-Reiten sequences. It also admits a filtration interpretation as smon ⁡ ( Q , I , X ) = Fil ⁡ ( X ⊗ P ( k Q / I ) ) \operatorname {smon}(Q, I, \mathscr {X})=\operatorname {Fil}(\mathscr {X}\otimes \mathcal P(kQ/I)) , provided that X \mathscr {X} is extension-closed. As an application, smon ⁡ ( Q , I , X ) \operatorname {smon}(Q, I, \mathscr {X}) is an extension-closed Frobenius subcategory if and only if so is X \mathscr {X} . This gives “new” Frobenius subcategories of Λ \Lambda -mod in the sense that they may not be G P ( Λ ) \mathcal {GP}(\Lambda ) . Ringel-Schmidmeier-Simson equivalence smon ⁡ ( Q , I , X ) ≅ sepi ⁡ ( Q , I , X ) \operatorname {smon}(Q, I, \mathscr {X})\cong \operatorname {sepi}(Q, I, \mathscr {X}) is introduced and the existence is proved for arbitrary extension-closed subcategories X \mathscr {X} . In particular, the Nakayama functor N Λ \mathcal N_\Lambda gives an RSS equivalence smon ⁡ ( Q , I , A ) ≅ sepi ⁡ ( Q , I , A ) \operatorname {smon}(Q, I, A)\cong \operatorname {sepi}(Q, I, A) if and only if A A is Frobenius. For a chain Q Q with arbitrary I I , an explicit formula of an RSS equivalence is found for arbitrary additive subcategories X \mathscr {X} .