1,993 research outputs found
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 Î be an O-order in a separable K-algebra. We call a Î-lattice L rigid if Ext1Î(L, L) = 0, in analogy with the definition of rigid modules over a finite-dimensional algebra. By partitioning the Î-lattices of a given dimension into âvarieties of latticesâ, we show that there are only finitely many rigid Î-lattices L of any given dimension. As a consequence we show that if the first Hochschild cohomology of Î vanishes, then the Picard group and the outer automorphism group of Î are finite. In particular the Picard groups of blocks of finite groups defined over O are always finite
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The Picard group of an order and KĂŒlshammer reduction
Let (K, O, k) be a p-modular system and assume k is algebraically closed. We show that if Î is an O-order in a separable K-algebra, then PicO(Î) 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 Donovanâs 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 , we introduce a general approach to determining when its first Hochschild cohomology , considered as a Lie algebra, is solvable. If is moreover of tame or finite representation type, we are able to describe as the direct sum of a solvable Lie algebra and a sum of copies of . 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 . As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll and Solotar
A counterexample to the first Zassenhaus conjecture
Hans J. Zassenhaus conjectured that for any unit u of finite order in the integral group ring of a finite group G there exists a unit a in the rational group algebra of G such that aâ1· u · a = ±g for some g â G. We disprove this conjecture by first proving general results that help identify counterexamples and then providing an infinite number of examples where these results apply. Our smallest example is a metabelian group of order 27·32·5·72·192 whose integral group ring contains a unit of order 7 · 19 which, in the rational group algebra, is not conjugate to any element of the form ±g
A reduction theorem for tau -rigid modules
We prove a theorem which gives a bijection between the support Ï -tilting modules over a given finite-dimensional algebra A and the support Ï -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 Ï -tilting-finite wild blocks with more than one simple module. We then go on to classify all support Ï -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 Ï -rigid modules over (not necessarily symmetric) string algebras
Donovanâs conjecture, blocks with abelian defect groups and discrete valuation rings
We give a reduction to quasisimple groups for Donovanâs conjecture for blocks with abelian defect groups defined with respect to a suitable discrete valuation ring O. Consequences are that Donovanâs conjecture holds for O-blocks with abelian defect groups for the prime two, and that, using recent work of Farrell and Kessar, for arbitrary primes Donovanâs conjecture for O-blocks with abelian defect groups reduces to bounding the Cartan invariants of blocks of quasisimple groups in terms of the defect. A result of independent interest is that in general (i.e. for arbitrary defect groups) Donovanâs conjecture for O-blocks is a consequence of conjectures predicting bounds on the O-Frobenius number and on the Cartan invariants, as was proved by Kessar for blocks defined over an algebraically closed field
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Basic Orders for Defect Two Blocks of â€pÎŁn
We show how basic orders for defect two blocks of symmetric groups over the ring of p-adic integers can be constructed by purely combinatorial means
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The p-adic group ring of
In the present article we show that the Zp[ζpfâ1]-order Zp[ζpfâ1]SL2(pf) can be recognized among those orders whose reduction modulo p is isomorphic to FpfSL2(pf) using only ring-theoretic properties. In other words we show that FpfSL2(pf) lifts uniquely to a Zp[ζpfâ1]-order, provided certain reasonable conditions are imposed on the lift. This proves a conjecture made by Nebe in [8] concerning the basic order of Z2[ζ2fâ1]SL2(2f)
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