Coderived and contraderived categories for a cotorsion pair, flat-type cotorsion pairs, and relative periodicity
Leonid Positselski
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. More generally, we define and discuss cotorsion pairs of the very flat type and of the flat type in Grothendieck categories (as well as exact categories of the flat type), and work with a cotorsion pair sandwiched between one of the very flat type and one of the flat type. 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.