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0-Dimensional Ideal Approximation Theory

H. Y. Zhu, X. H. Fu, I. Herzog, K. Schlegel

TL;DR

The paper develops a rigorous framework for $0$-dimensional ideal approximation theory in additive categories by introducing weak kernel-cokernel structures and weak exact categories, then defines complete ideal torsion pairs via Hom-special precovers/preenvelopes. It proves a zero-dimensional Salce-type lemma and Christensen's lemma within this setting, and situates the theory in extriangulated and Frobenius categories, revealing deep links between $0$- and $1$-dimensional approximation theories through stable categories. In abelian categories, complete ideal torsion pairs correspond to preradicals, with image-maximal ideals yielding monic coverings and broad completeness results under natural finiteness conditions; these ideas extend to Frobenius categories where a bijection with stable-category torsion pairs clarifies the structure of almost split sequences and cotorsion phenomena. Collectively, the results provide a unifying, axiomatized approach to ideal approximation theory with concrete implications for module categories and representation theory of Artin algebras.

Abstract

We propose axioms for 0-dimensional ideal approximation theory and note that extriangulated categories satisfy these axioms.

0-Dimensional Ideal Approximation Theory

TL;DR

The paper develops a rigorous framework for -dimensional ideal approximation theory in additive categories by introducing weak kernel-cokernel structures and weak exact categories, then defines complete ideal torsion pairs via Hom-special precovers/preenvelopes. It proves a zero-dimensional Salce-type lemma and Christensen's lemma within this setting, and situates the theory in extriangulated and Frobenius categories, revealing deep links between - and -dimensional approximation theories through stable categories. In abelian categories, complete ideal torsion pairs correspond to preradicals, with image-maximal ideals yielding monic coverings and broad completeness results under natural finiteness conditions; these ideas extend to Frobenius categories where a bijection with stable-category torsion pairs clarifies the structure of almost split sequences and cotorsion phenomena. Collectively, the results provide a unifying, axiomatized approach to ideal approximation theory with concrete implications for module categories and representation theory of Artin algebras.

Abstract

We propose axioms for 0-dimensional ideal approximation theory and note that extriangulated categories satisfy these axioms.

Paper Structure

This paper contains 21 sections, 35 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Ideal Torsion Pairs
  3. $\hbox{\rm Hom}$-orthogonality of morphisms
  4. Ideal annihilators
  5. Operations on ideals
  6. Complete Ideal Torsion Pairs
  7. Weak kernels and cokernels
  8. Weak kernel-cokernel structures
  9. Weak extensions of ideals
  10. Ideal approximations
  11. Weak kernel-cokernel categories with enough weak inflations and weak deflations
  12. Weak Exact Categories
  13. Elementary properties
  14. Minimal Salce's Lemma
  15. Christensen's Lemma
  16. ...and 6 more sections

Key Result

Theorem 3

Let $\mathcal{I}$ be an ideal of $\mathcal{A}$. Then $\mathcal{I}^{\perp}=\ell(\mathcal{I})$ and $^{\perp}\mathcal{I}=r(\mathcal{I})$. A pair $(\mathcal{I}, \mathcal{J})$ of ideals in $\mathcal{A}$ is therefore an ideal torsion pair if and only if $\mathcal{I}=r(\mathcal{J})$ and $\mathcal{J}=\ell(\

Theorems & Definitions (76)

  • Example 1
  • Definition 2
  • Theorem 3
  • proof
  • Corollary 4
  • proof
  • Example 5
  • Corollary 6
  • Remark 7
  • Example 8
  • ...and 66 more