Consistent pairs of $ \mathfrak{s} $-torsion pairs in extriangulated categories with negative first extensions

Limin Liu; Hongjin Liu; Limin Liu; Hongjin Liu

doi:10.3934/math.2024073

AIMS Mathematics

2024, Volume 9, Issue 1: 1494-1508. doi: 10.3934/math.2024073

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Research article

Consistent pairs of $ \mathfrak{s} $-torsion pairs in extriangulated categories with negative first extensions

Limin Liu ,
Hongjin Liu ^,

School of Mathematics and Information Engineering, Longyan University, Longyan 364012, China

Received: 03 November 2023 Revised: 24 November 2023 Accepted: 30 November 2023 Published: 11 December 2023
MSC : 18G25, 18G80

As a generalization of a consistent pair of $ t $-structures on triangulated categories, we introduced the notion of a consistent pair of $ \mathfrak{s} $-torsion pairs in the extriangulated setup. Let $ (\mathcal{T}_{i}, \mathcal{F}_{i}) $ be an $ \mathfrak{s} $-torsion pair in an extriangulated category with a negative first extension for any $ i = 1, 2 $. By using the consistent pair, we gave a criterion for $ (\mathcal{T}_{1}\ast\mathcal{T}_{2}, \mathcal{F}_{1}\cap\mathcal{F}_{2}) $ to be an $ \mathfrak{s} $-torsion pair. Our results were then applied to the torsion theory induced by $ \tau $-rigid modules.
- extriangulated categories,
- $ \mathfrak{s} $-torsion pairs,
- consistent pairs,
- $ \tau $-rigid modules
Citation: Limin Liu, Hongjin Liu. Consistent pairs of $ \mathfrak{s} $-torsion pairs in extriangulated categories with negative first extensions[J]. AIMS Mathematics, 2024, 9(1): 1494-1508. doi: 10.3934/math.2024073

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Abstract

As a generalization of a consistent pair of $ t $-structures on triangulated categories, we introduced the notion of a consistent pair of $ \mathfrak{s} $-torsion pairs in the extriangulated setup. Let $ (\mathcal{T}_{i}, \mathcal{F}_{i}) $ be an $ \mathfrak{s} $-torsion pair in an extriangulated category with a negative first extension for any $ i = 1, 2 $. By using the consistent pair, we gave a criterion for $ (\mathcal{T}_{1}\ast\mathcal{T}_{2}, \mathcal{F}_{1}\cap\mathcal{F}_{2}) $ to be an $ \mathfrak{s} $-torsion pair. Our results were then applied to the torsion theory induced by $ \tau $-rigid modules.

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