Generalized periodicity theorems

Let R be a ring and $S$ be a class of strongly finitely presented (${\mathrm{FP}}_{\infty }$) R-modules closed under extensions, direct summands, and syzygies. Let $(A,B)$ be the (hereditary complete) cotorsion pair generated by $S$ in $\mathrm{Mod--}R$, and let $(C,D)$ be the (also hereditary complete) cotorsion pair in which $C=\underrightarrow{\lim }A=\underrightarrow{\lim }S$. We show that any $A$-periodic module in $C$ belongs to $A$, and any $D$-periodic module in $B$ belongs to $D$. 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.