Generalized periodicity theorems

Let $R$ be a ring and $\mathsf{S}$ be a class of strongly finitely presented (FP${}_{\mathrm{\infty }}$) $R$-modules closed under extensions, direct summands, and syzygies. Let $(\mathsf{A},\mathsf{B})$ be the (hereditary complete) cotorsion pair generated by $\mathsf{S}$ in $\mathsf{\text{Mod-}}R$, and let $(\mathsf{C},\mathsf{D})$ be the (also hereditary complete) cotorsion pair in which $\mathsf{C}=\mathrm{}\mathsf{A}=\mathrm{}\mathsf{S}$. We show that any $\mathsf{A}$-periodic module in $\mathsf{C}$ belongs to $\mathsf{A}$, and any $\mathsf{D}$-periodic module in $\mathsf{B}$ belongs to $\mathsf{D}$. Further generalizations of both results are obtained, so that we get a common generalization of the flat/projective and fp-projective periodicity theorems, as well as a common generalization of the fp-injective/injective and cotorsion periodicity theorems. Both are applicable to modules over an arbitrary ring, and in fact, to Grothendieck categories.