On the geometry of lattices and finiteness of Picard groups

Let (K, O, k) be a p-modular system with k algebraically closed and O unramified, and let Λ be an O-order in a separable K-algebra. We call a Λ-lattice L rigid if Ext1Λ(L, L) = 0, in analogy with the definition of rigid modules over a finite-dimensional algebra. By partitioning the Λ-lattices of a given dimension into “varieties of lattices”, we show that there are only finitely many rigid Λ-lattices L of any given dimension. As a consequence we show that if the first Hochschild cohomology of Λ vanishes, then the Picard group and the outer automorphism group of Λ are finite. In particular the Picard groups of blocks of finite groups defined over O are always finite

The Picard group of an order and Külshammer reduction

Let (K, O, k) be a p-modular system and assume k is algebraically closed. We show that if Λ is an O-order in a separable K-algebra, then PicO(Λ) carries the structure of an algebraic group over k. As an application to the modular representation theory of finite groups, we show that a reduction theorem by Kulshammer concerned with Donovan’s conjecture remains valid over O

Blocks with a generalized quaternion defect group and three simple modules over a 2-adic ring

We show that two blocks of generalized quaternion defect with three simple modules over a sufficiently large 2-adic ring O are Morita-equivalent if and only if the corresponding blocks over the residue field of O are Morita-equivalent. As a corollary we show that any two blocks defined over O with three simple modules and the same generalized quaternion defect group are derived equivalent

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