Contraherent cosheaves of contramodules on Noetherian formal schemes

TL;DR

This paper develops the theory of contraherent cosheaves of contramodules on Noetherian formal schemes, defining categories, functors, and algebraic structures in a broad adic algebra context.

Contribution

It introduces the exact categories of contraherent cosheaves, constructs key functors, and extends adic algebra preliminaries to general commutative rings.

Findings

Defined exact categories of contraherent cosheaves on formal schemes.

Constructed direct and inverse image functors for these cosheaves.

Extended adic algebra concepts to arbitrary commutative rings.

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. 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).