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Contraherent cosheaves of contramodules on Noetherian formal schemes

Leonid Positselski

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). 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).

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). 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).

Paper Structure

This paper contains 23 sections, 101 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries in Adic Commutative Algebra
  3. Adic topological rings and their maps
  4. Torsion modules
  5. Adic completions
  6. Contramodules
  7. Injective torsion modules, projective and flat contramodules
  8. Change of scalars
  9. Tensor products of adic topological rings
  10. Formal open immersions and formal open coverings
  11. Very flat and contraadjusted contramodules
  12. Cotorsion contramodules
  13. The coextension of scalars is quotseparated
  14. Colocalization of contraadjustedness and colocality of cotorsion
  15. Colocality of exactness
  16. ...and 8 more sections

Key Result

Lemma 2.1.2

Let $I$ and $J$ be two ideals of definition of an adic topological ring $R$. Then the quotient ring $R/I$ is Noetherian if and only if the quotient ring $R/J$ is Noetherian.

Theorems & Definitions (206)

  • Lemma 2.1.2
  • proof
  • Lemma 2.1.3
  • proof
  • Lemma 2.1.4
  • proof
  • Remark 2.1.5
  • Lemma 2.1.6
  • proof
  • Lemma 2.1.8
  • ...and 196 more