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Inspired by work of Ladkani and Groth–Šťovíček, we explain how to construct generalisations of the classical reflection functors of Bernšteĭn, Gel′fand, and Ponomarev by means of the Grothendieck construction. [ABSTRACT FROM AUTHOR]

The class of support τ-tilting modules was introduced to provide a completion of the class of tilting modules from the point of view of mutations. In this article, we study τ-tilting finite algebras, that is, finite dimensional algebras A with finitely many isomorphism classes of indecomposable τ-rigid modules. We show that A is τ-tilting finite if and only if every torsion class in mod, A is functorially finite. Moreover we give a bijection between indecomposable τ-rigid A-modules and bricks of A satisfying a certain finiteness condition, which is automatic for τ-tilting finite algebras. We observe that cones generated by g-vectors of indecomposable direct summands of each support τ-tilting module form a simplicial complex Δ(A). We show that if A is τ-tilting finite, then Δ(A) is homeomorphic to an (n−1)-dimensional sphere, and moreover the partial order on support τ-tilting modules can be recovered from the geometry of Δ(A). [ABSTRACT FROM AUTHOR]

The class of support τ-tilting modules was introduced recently by Adachi et al. These modules complete the class of tilting modules from the point of view of mutations. Given a finite-dimensional algebra A, we study all basic support τ-tilting A-modules which have a given basic τ-rigid A-module as a direct summand. We show that there exist an algebra C such that there exists an order-preserving bijection between these modules and all basic support τ-tilting C-modules; we call this passage τ-tilting reduction. An important step in our proof is the formation of τ-perpendicular categories which are analogs of ordinary perpendicular categories. Finally, we show that τ-tilting reduction is compatible with silting reduction and 2-Calabi-Yau reduction in appropriate triangulated categories. [ABSTRACT FROM AUTHOR]