Contraherent cosheaves of contramodules on Noetherian formal schemes

Abstract

We define the exact category of contraherent cosheaves of contramodules on a locally Noetherian formal scheme, as well as the exact categories of locally contraherent cosheaves of contramodules (with respect to a given open covering). We also construct the direct image and inverse image functors of locally contraherent cosheaves of contramodules under morphisms of locally Noetherian formal schemes, and discuss the functors of contraherent $\mathfrak{Hom}$ and contratensor product of quasi-coherent torsion sheaves and contraherent cosheaves of contramodules. At the end, we have a discussion of projective, antilocally flat, coflasque, and flat contraherent cosheaves of contramodules. The exposition in the section of preliminaries in adic commutative algebra is worked out in the greater generality of arbitrary commutative rings with adic topologies (of finitely generated ideals).