On solvability of the first Hochschild cohomology of a finite-dimensional algebra

For an arbitrary finite-dimensional algebra $A$, we introduce a general approach to determining when its first Hochschild cohomology ${\rm HH}^1(A)$, considered as a Lie algebra, is solvable. If $A$ is moreover of tame or finite representation type, we are able to describe ${\rm HH}^1(A)$ as the direct sum of a solvable Lie algebra and a sum of copies of $\mathfrak{sl}_2$. We proceed to determine the exact number of such copies, and give an explicit formula for this number in terms of certain chains of Kronecker subquivers of the quiver of $A$. As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll and Solotar

Derived categories of noncommutative quadrics and Hilbert squares

A noncommutative deformation of a quadric surface is usually described by a three-dimensional cubic Artin-Schelter regular algebra. In this paper we show that for such an algebra its bounded derived category embeds into the bounded derived category of a commutative deformation of the Hilbert scheme of two points on the quadric. This is the second example in support of a conjecture by Orlov. Based on this example, we formulate an infinitesimal version of the conjecture, and provide some evidence in the case of smooth projective surfaces.Comment: 21 pages. Small corrections + expanded proof of Lemma 2.

A reduction theorem for tau -rigid modules

We prove a theorem which gives a bijection between the support τ -tilting modules over a given finite-dimensional algebra A and the support τ -tilting modules over A / I, where I is the ideal generated by the intersection of the center of A and the radical of A. This bijection is both explicit and well-behaved. We give various corollaries of this, with a particular focus on blocks of group rings of finite groups. In particular we show that there are τ -tilting-finite wild blocks with more than one simple module. We then go on to classify all support τ -tilting modules for all algebras of dihedral, semidihedral and quaternion type, as defined by Erdmann, which include all tame blocks of group rings. Note that since these algebras are symmetric, this is the same as classifying all basic two-term tilting complexes, and it turns out that a tame block has at most 32 different basic two-term tilting complexes. We do this by using the aforementioned reduction theorem, which reduces the problem to ten different algebras only depending on the ground field k, all of which happen to be string algebras. To deal with these ten algebras we give a combinatorial classification of all τ -rigid modules over (not necessarily symmetric) string algebras

Embeddings of algebras in derived categories of surfaces

By a result of Orlov there always exists an embedding of the derived category of a finite-dimensional algebra of finite global dimension into the derived category of a high-dimensional smooth projective variety. In this article we give some restrictions on those algebras whose derived categories can be embedded into the bounded derived category of a smooth projective surface. This is then applied to obtain explicit results for hereditary algebras.Comment: 13 pages; revised versio