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Cotorsion pairs and Tor-pairs over commutative noetherian rings

Dolors Herbera, Michal Hrbek, Giovanna Le Gros

Abstract

For a commutative noetherian ring $R$, we classify all the hereditary cotorsion pairs cogenerated by pure-injective modules of finite injective dimension. The classification is done in terms of integer-valued functions on the spectrum of the ring. Each such function gives rise to a system of local depth conditions which describes the left-hand class in the corresponding cotorsion pair. Furthermore, we show that these cotorsion pairs correspond by explicit duality to hereditary Tor-pairs generated by modules of finite flat dimension.

Cotorsion pairs and Tor-pairs over commutative noetherian rings

Abstract

For a commutative noetherian ring , we classify all the hereditary cotorsion pairs cogenerated by pure-injective modules of finite injective dimension. The classification is done in terms of integer-valued functions on the spectrum of the ring. Each such function gives rise to a system of local depth conditions which describes the left-hand class in the corresponding cotorsion pair. Furthermore, we show that these cotorsion pairs correspond by explicit duality to hereditary Tor-pairs generated by modules of finite flat dimension.

Paper Structure

This paper contains 22 sections, 37 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Homological formulae
  4. Pure-injective modules
  5. Cotorsion pairs.
  6. $\mathop{\mathrm{\mathsf{Tor}}}\nolimits$-pairs.
  7. The lattice of $\mathop{\mathrm{\mathsf{Tor}}}\nolimits$-pairs and cotorsion pairs
  8. Hereditary cotorsion pairs cogenerated by pure-injectives
  9. Generation of $\mathop{\mathrm{\mathsf{Tor}}}\nolimits$-pairs
  10. Cotilting classes.
  11. Commutative rings: grade, depth and Koszul complexes.
  12. Cotilting classes over commutative rings.
  13. Homological properties of commutative noetherian rings.
  14. Preliminary results
  15. Hereditary cotorsion pairs cogenerated by pure-injectives
  16. ...and 7 more sections

Key Result

Lemma 2.1

Let $R$ be a ring, $F \in \mathsf{Mod}\hbox{-}R$ a flat $R$-module, $Y\in R\hbox{-}\mathsf{Mod}\hbox{-}S$ and $N \in \mathsf{Mod}\hbox{-}S$ a pure-injective module. Then for every $j\geq0$, there is the following natural isomorphism of abelian groups.

Theorems & Definitions (84)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Corollary 2.3
  • Remark 2.4
  • Theorem 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Remark 2.8
  • ...and 74 more