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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., 5 pages. Changed title. Added reference to previous work of Maycock and other minor edits

We provide an outline of the proof of the Donovan--Wemyss Conjecture in the context of the Homological Minimal Model Program for threefolds. The proof relies on results of August, of Hua and the second-named author, Wemyss, and on the Triangulated Auslander--Iyama Correspondence -- a recent result by the first- and third-named authors., 25 pages. The authors' contribution to the proceedings of the Abel Symposium 2022: Triangulated categories in representation theory and beyond

We work over a perfect field. Recent work of the third-named author established a Derived Auslander Correspondence that relates finite-dimensional self-injective algebras that are twisted $3$-periodic to algebraic triangulated categories of finite type. Moreover, the aforementioned work also shows that the latter triangulated categories admit a unique differential graded enhancement. In this article we prove a higher-dimensional version of this result that, given an integer $d\geq1$, relates twisted $(d+2)$-periodic algebras to algebraic triangulated categories with a $d\mathbb{Z}$-cluster tilting object. We also show that the latter triangulated categories admit a unique differential graded enhancement. Our result yields recognition theorems for interesting algebraic triangulated categories, such as the Amiot cluster category of a self-injective quiver with potential in the sense of Herschend and Iyama and, more generally, the Amiot-Guo-Keller cluster category associated with a $d$-representation finite algebra in the sense of Iyama and Oppermann. As an application of our result, we obtain infinitely many triangulated categories with a unique differential graded enhancement that is not strongly unique. In the appendix, B. Keller explains how - combined with crucial results of August and Hua-Keller - our main result yields the last key ingredient to prove the Donovan-Wemyss Conjecture in the context of the Homological Minimal Model Program for threefolds., Appendix by Bernhard Keller. 117 pages, 2 figures. v2: added references; corrected minor typos and other editorial changes; new subsection 5.5.4 on the comparison with topological enhancements. v3: corrected minor typos; new subsection 4.6 on an explicit example of a universal Massey product. v4: new title; added further details to the proof of Thm B; several minor edits and some typos corrected

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., 16 pages. The author's contribution to the Proceedings of the ICRA 2018, v.2 minor edits following referee report

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 Smal{\o}. 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., Comment: 25 pages, final version

Let $k$ be a field. In this short note we give an example of a $2$-abelian $k$-category, realized as a $2$-cluster-tilting subcategory of the category $\operatorname{mod}\,A$ of finite dimensional (right) $A$-modules over a finite dimensional $k$-algebra $A$, for which the naive idea for constructing its "bounded derived category" as $2$-cluster-tilting subcategory of the bounded derived category of $\operatorname{mod}\,A$ cannot work., 4 pages. This note is not intended for publication

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 6.6

The class of support $��$-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 $��$-tilting finite algebras, i.e. finite dimensional algebras $A$ with finitely many isomorphism classes of indecomposable $��$-rigid modules. We show that $A$ is $��$-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 $��$-tilting module form a simplicial complex $��(A)$. We show that if $A$ is $��$-tilting finite, then $��(A)$ is homeomorphic to an $(n-1)$-dimensional sphere, and moreover the partial order on support $��$-tilting modules can be recovered from the geometry of $��(A)$. Finally we give a bijection between indecomposable $��$-rigid $A$-modules and bricks of $A$ satisfying a certain finiteness condition, which is automatic for $��$-tilting finite algebras., 29 pages. Changed title. Added Theorem 6.5 and Proposition 6.6

Let $\mathbb{X}$ be a weighted projective line and $\operatorname{coh}\mathbb{X}$ the associated categoy of coherent sheaves. We classify the tilting complexes $T$ in $D^b(\operatorname{coh}\mathbb{X})$ such that $\tau^2 T\cong T$, where $\tau$ is the Auslander-Reiten translation in $D^b(\operatorname{coh}\mathbb{X})$. As an application of this result, we classify the 2-representation-finite algebras which are derived-equivalent to a canonical algebra. This complements Iyama-Oppermann's classification of the iterated tilted 2-representation-finite algebras. By passing to 3-preprojective algebras, we obtain a classification of the selfinjective cluster-tilted algebras of canonical-type. This complements Ringel's classification of the selfinjective cluster-tilted algebras., Comment: 28 pages, corrected typos, minor edits

Let $\mathbb{X}$ be a weighted projective line and $\operatorname{coh}\mathbb{X}$ the associated categoy of coherent sheaves. We classify the tilting complexes $T$ in $D^b(\operatorname{coh}\mathbb{X})$ such that $��^2 T\cong T$, where $��$ is the Auslander-Reiten translation in $D^b(\operatorname{coh}\mathbb{X})$. As an application of this result, we classify the 2-representation-finite algebras which are derived-equivalent to a canonical algebra. This complements Iyama-Oppermann's classification of the iterated tilted 2-representation-finite algebras. By passing to 3-preprojective algebras, we obtain a classification of the selfinjective cluster-tilted algebras of canonical-type. This complements Ringel's classification of the selfinjective cluster-tilted algebras., 28 pages, corrected typos, minor edits

The class of support $��$-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 $��$-tilting $A$-modules which have given basic $��$-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 $��$-tilting $C$-modules; we call this process $��$-tilting reduction. An important step in this process is the formation of $��$-perpendicular categories which are analogs of ordinary perpendicular categories. Finally, we show that $��$-tilting reduction is compatible with silting reduction and 2-Calabi-Yau reduction in appropiate triangulated categories., 32 pages. Shortened abstract, corrected typos in references [1] and [9]