Coderived and contraderived categories for a cotorsion pair, flat-type cotorsion pairs, and relative periodicity

Abstract: Given a hereditary complete cotorsion pair $(\mathsf A,\mathsf B)$ generated by a set of objects in a Grothendieck category $\mathsf K$, we construct a natural equivalence between the Becker coderived category of the left-hand class $\mathsf A$ and the Becker contraderived category of the right-hand class $\mathsf B$. We show that a nested pair of cotorsion pairs $$(\mathsf A_1,\mathsf B_1)\le(\mathsf A_2,\mathsf B_2)$$ provides an adjunction between the related co/contraderived categories, which is induced by a Quillen adjunction between abelian model structures. Then we specialize to the cotorsion pairs $(\mathsf F,\mathsf C)$ sandwiched between the projective and the flat cotorsion pairs in a module category, and prove that the related co/contraderived categories for $(\mathsf F,\mathsf C)$ are the same as for the projective and flat cotorsion pairs if and only if two periodicity properties hold for $\mathsf F$ and $\mathsf C$. The same applies to the cotorsion pairs sandwiched between the very flat and the flat cotorsion pairs in the category of quasi-coherent sheaves over a quasi-compact semi-separated scheme. The motivating examples of the classes of flaprojective modules and relatively cotorsion modules for a ring homomorphism are discussed, and periodicity conjectures formulated for them.