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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 $��^2 T\cong T$, 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]