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-clustertilting 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 ncluster-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. 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, i.e. 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. We observe that cones generated by g-vectors of indecomposable direct summands of each support τ -tilting module form a simplicial complex ∆(A). We show that if A is τ -tilting finite, then ∆(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 ∆(A). Finally 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.

The class of support τ -tilting modules was introduced recently by Adachi, Iyama and Reiten. 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 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 process τ -tilting reduction. An important step in this process 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 appropiate triangulated categories.2010 Mathematics Subject Classification. 16G10.

Abstract. Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the automorphism group of the corresponding cluster category and second by giving explicit sequences of mutations.