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The finite presentation of the stable Hom functors, the Bass torsion, and the cotorsion coradical

Alex Martsinkovsky

Abstract

We provide necessary and/or sufficient conditions for the stable Hom functors to be finitely presented. When the covariant Hom functor modulo projectives is finitely presented, its defect is isomorphic to the Bass torsion of the fixed argument. When the contravariant Hom functor modulo injectives is finitely presented, its defect is isomorphic to the cotorsion of the fixed argument. We also give a sufficient condition for the sub-stabilization of the tensor product to be finitely presented. A finite presentation of the tensor product leads to an unexpected application.

The finite presentation of the stable Hom functors, the Bass torsion, and the cotorsion coradical

Abstract

We provide necessary and/or sufficient conditions for the stable Hom functors to be finitely presented. When the covariant Hom functor modulo projectives is finitely presented, its defect is isomorphic to the Bass torsion of the fixed argument. When the contravariant Hom functor modulo injectives is finitely presented, its defect is isomorphic to the cotorsion of the fixed argument. We also give a sufficient condition for the sub-stabilization of the tensor product to be finitely presented. A finite presentation of the tensor product leads to an unexpected application.
Paper Structure (8 sections, 25 theorems, 23 equations)

This paper contains 8 sections, 25 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. The category of finitely presented functors
  3. Finite presentation of the stable Hom functors
  4. The functor $(\overline{A, \space \space })$ is finitely presented
  5. The functor $(\underline{A, \space \space })$
  6. The functor $(\underline{\space \space , A})$ is finitely presented
  7. The functor $(\overline{\space \space , A})$
  8. The sub-stabilization of the tensor product

Key Result

Proposition 2.2

Let $F$ and $G$ be functors. If $F$ is finitely generated, then the class of all natural transformations $g : F \to G$, denoted by $\mathrm{Nat}(F, G)$, is a set.

Theorems & Definitions (51)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 41 more