Torsion pairs via the Ziegler spectrum

Abstract

We establish a bijection between torsion pairs in the category of finite‐dimensional modules over a finite‐dimensional algebra and pairs formed by a closed rigid set in the Ziegler spectrum of and a set of indecomposable injective ‐modules. This is as an extension of a result from ‐tilting theory [T. Adachi, O. Iyama, I. Reiten, Compos. Math. 150 (2014), no. 3, 415–452] which parametrises the functorially finite torsion pairs over . We also extend a map from [L. Demonet, O. Iyama, I. Reiten, Int. Math. Res. Not. IMRN 2019 (2019), no. 3, 852–892] which assigns a brick to any indecomposable ‐rigid module and obtain a bijection between finite‐dimensional bricks and certain (possibly infinite‐dimensional) indecomposable modules satisfying a rigidity condition. We use this to give a new characterisation of brick finiteness. The majority of our results also holds when is an artinian ring.