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arXiv · 2403.00475

Torsion pairs via the Ziegler spectrum

Lidia Angeleri Hügel·Rosanna Laking·Francesco Sentieri
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Abstract

We establish a bijection between torsion pairs in the category of finite-dimensional modules over a finite-dimensional algebra A and pairs (Z, I) formed by a closed rigid set Z in the Ziegler spectrum of A and a set I of indecomposable injective A-modules. This can be regarded as an extension of a result from $\tau$-tilting theory which parametrises the functorially finite torsion pairs over A. We also obtain a one-one-correspondence between finite-dimensional bricks and certain (possibly infinite-dimensional) indecomposable modules satisfying a rigidity condition. Our results also hold when A is an artinian ring.

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Lidia Angeleri Hügel, Rosanna Laking, Francesco Sentieri. 2024-03-01. Torsion pairs via the Ziegler spectrum. https://arxiv.org/abs/2403.00475

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