The construction of tilting cotorsion pairs for hereditary abelian categories

Rongmin Zhu; Tiwei Zhao; Rongmin Zhu; Tiwei Zhao

doi:10.3934/era.2025120

Electronic Research Archive

2025, Volume 33, Issue 5: 2719-2735. doi: 10.3934/era.2025120

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Research article

The construction of tilting cotorsion pairs for hereditary abelian categories

Rongmin Zhu ¹,
Tiwei Zhao ^{2
,
,}

1.
School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China
2.
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

Received: 04 March 2025 Revised: 22 April 2025 Accepted: 25 April 2025 Published: 06 May 2025

Through tilting objects, we construct complete cotorsion pairs for specific hereditary abelian categories, such as the category of modules that are finitely generated over a finite-dimensional hereditary algebra as well as the category of coherent sheaves over weighted projective lines. We prove that a complete cotorsion pair exists in the category of coherent sheaves over a weighted projective curve $ \mathbb{X} $ if and only if $ \mathbb{X} $ is a weighted projective line. We also characterize the canonical tilting cotorsion pair for any weighted projective line and obtain Hovey triples in the category of vector bundles over a weighted projective line.
- cotorsion pair,
- tilting object,
- hereditary abelian category,
- coherent sheave,
- weighted projective line
Citation: Rongmin Zhu, Tiwei Zhao. The construction of tilting cotorsion pairs for hereditary abelian categories[J]. Electronic Research Archive, 2025, 33(5): 2719-2735. doi: 10.3934/era.2025120

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Abstract

Through tilting objects, we construct complete cotorsion pairs for specific hereditary abelian categories, such as the category of modules that are finitely generated over a finite-dimensional hereditary algebra as well as the category of coherent sheaves over weighted projective lines. We prove that a complete cotorsion pair exists in the category of coherent sheaves over a weighted projective curve $ \mathbb{X} $ if and only if $ \mathbb{X} $ is a weighted projective line. We also characterize the canonical tilting cotorsion pair for any weighted projective line and obtain Hovey triples in the category of vector bundles over a weighted projective line.

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