Higher Auslander algebras of type A\mathbb{A} and the higher Waldhausen S⁡\operatorname{S}-constructions

These notes are an expanded version of my talk at the ICRA 2018 in Prague, Czech Republic; they are based on joint work with Tobias Dyckerhoff and Tashi Walde. In them we relate Iyama's higher Auslander algebras of type $\mathbb{A}$ to Eilenberg--Mac Lane spaces in algebraic topology and to higher-dimensional versions of the Waldhausen $\operatorname{S}$-construction from algebraic $K$-theory.Comment: 16 pages. The author's contribution to the Proceedings of the ICRA 2018, v.2 minor edits following referee repor

nn-abelian and nn-exact categories

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 Gei\ss-Keller-Oppermann. We give several examples of $n$-abelian and $n$-exact categories which have appeared in representation theory, commutative ring theory, commutative and non-commutative algebraic geometry.Comment: 58 pages. Corrected an error in Thm 3.20 and Lemma 3.22 and several typos. Accepted for publication in Mathematische Zeitschrif

τ\tau-tilting finite algebras, bricks and gg-vectors

The class of support $\tau$-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 $\tau$-tilting finite algebras, i.e. finite dimensional algebras $A$ with finitely many isomorphism classes of indecomposable $\tau$-rigid modules. We show that $A$ is $\tau$-tilting finite if and only if very torsion class in $\mod A$ is functorially finite. We observe that cones generated by $g$-vectors of indecomposable direct summands of each support $\tau$-tilting module form a simplicial complex $\Delta(A)$. We show that if $A$ is $\tau$-tilting finite, then $\Delta(A)$ is homeomorphic to an $(n-1)$-dimensional sphere, and moreover the partial order on support $\tau$-tilting modules can be recovered from the geometry of $\Delta(A)$. Finally we give a bijection between indecomposable $\tau$-rigid $A$-modules and bricks of $A$ satisfying a certain finiteness condition, which is automatic for $\tau$-tilting finite algebras.Comment: 29 pages. Changed title. Added Theorem 6.5 and Proposition 6.

Derived equivalences of upper-triangular ring spectra via reflection functors

We use the generalised Bernstein-Gelfand-Ponomarev reflection functors constructed in joint work with Dyckerhoff and Walde to extend a theorem of Ladkani concerning derived equivalences between upper-triangular matrix rings from ordinary rings to ring spectra. Our result also extends an analogous theorem of Maycock for differential graded algebras.Comment: 5 pages. Changed title. Added reference to previous work of Maycock and other minor edit