On strongly FP-injective accessibility and weakly FP-projective periodicity

Accessibility results are established for (Cartan-Eilenberg) strongly FP-injective complexes, by first proving the accessibility of Strongly FP-injective modules and then utilizing the pseudopullback theorem. We also prove that every acyclic complex of weakly FP-projective modules has weakly FP-projective cycles over left $<\aleph_m$-coherent rings for a non-negative integer $m$. This is achieved by introducing the cotorsion pairs $\bigl(\mathcal{C}\aleph_mP\text{-}proj,\mathcal{C}\aleph_mP\text{-}inj\bigr)$, cogenerated by all $\aleph_m$-presentable weakly FP-projective modules, and proving that acyclic complexes of modules from the class $\mathcal{C}\aleph_mP\text{-}proj$ have weakly FP-projective cycles for every non-negative integer $m$.