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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]

This article consists of an introduction to Iyama's higher Auslander–Reiten theory for Artin algebras from the viewpoint of higher homological algebra. We provide alternative proofs of the basic results in higher Auslander–Reiten theory, including the existence of d‐almost‐split sequences in d‐cluster‐tilting subcategories, following the approach to classical Auslander–Reiten theory due to Auslander, Reiten, and Smalø. We show that Krause's proof of Auslander's defect formula can be adapted to give a new proof of the defect formula for d‐exact sequences. We use the defect formula to establish the existence of morphisms determined by objects in d‐cluster‐tilting subcategories. [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]

Let R be a commutative artinian ring. We extend higher Auslander correspondence from Artin R-algebras of finite representation type to dualizing R-varieties. More precisely, for a positive integer d, we show that a dualizing R-variety is d-abelian if and only if it is a d-Auslander dualizing R-variety if and only if it is equivalent to a d-cluster-tilting subcategory of the category of finitely presented modules over a dualizing R-variety. [ABSTRACT FROM AUTHOR]

We introduce n-abelian and n-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that n-cluster-tilting subcategories of abelian (resp. exact) categories are n-abelian (resp. n-exact). These results allow to construct several examples of n-abelian and n-exact categories. Conversely, we prove that n-abelian categories satisfying certain mild assumptions can be realized as n-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable category of a Frobenius n-exact category has a natural $$(n+2)$$ -angulated structure in the sense of Geiß-Keller-Oppermann. We give several examples of n-abelian and n-exact categories which have appeared in representation theory, commutative algebra, commutative and non-commutative algebraic geometry. [ABSTRACT FROM AUTHOR]

We show that the category of finitely generated free modules over certain local rings is nangulated for every n ⩾ 3. In fact, we construct several classes of n-angles, parameterized by equivalence classes of units in the local rings. Finally, we show that for odd values of n some of these n-angulated categories are not algebraic. [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]