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Equivalences of derived categories for symmetric algebras
 J. Algebra
"... It is about a decade since Broué made his celebrated conjecture [2] on equivalences of derived categories in block theory: that the module categories of a block algebra A of a finite group algebra and its Brauer correspondent B should have equivalent derived categories if their defect group is abeli ..."
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Cited by 32 (3 self)
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It is about a decade since Broué made his celebrated conjecture [2] on equivalences of derived categories in block theory: that the module categories of a block algebra A of a finite group algebra and its Brauer correspondent B should have equivalent derived categories if their defect group is abelian. Since then, charactertheoretic evidence for the conjecture
The Group of EndoPermutation Modules
, 1998
"... The group D(P ) of all endopermutation modules for a finite pgroup P is a finitely generated abelian group. We prove that its torsionfree rank is equal to the number of conjugacy classes of noncyclic subgroups of P . We also obtain partial results on its torsion subgroup. We determine next the s ..."
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Cited by 27 (11 self)
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The group D(P ) of all endopermutation modules for a finite pgroup P is a finitely generated abelian group. We prove that its torsionfree rank is equal to the number of conjugacy classes of noncyclic subgroups of P . We also obtain partial results on its torsion subgroup. We determine next the structure of Q#D() viewed as a functor, which turns out to be a simple functor SE,Q , indexed by the elementary group E of order p and the trivial Out(E)module Q . Finally we describe a rather strange exact sequence relating Q#D(P ) , Q#B(P ) , and Q#R(P ) , where B(P ) is the Burnside ring and R(P ) is the Grothendieck ring of QP modules.
Block Theory via Stable and Rickard Equivalences
, 2000
"... This paper owes a lot to M. Collins for his renewed encouragements. 2. Symmetric algebras, functors and equivalences ..."
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Cited by 26 (3 self)
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This paper owes a lot to M. Collins for his renewed encouragements. 2. Symmetric algebras, functors and equivalences
Geometry of chain complexes and outer automorphisms under derived equivalence
 Transactions of the American Mathematical Society
"... The authors wish to dedicate this paper to Idun Reiten on the occasion of her sixtieth birthday. Abstract. The two main theorems proved here are as follows: If A is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms o ..."
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Cited by 18 (1 self)
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The authors wish to dedicate this paper to Idun Reiten on the occasion of her sixtieth birthday. Abstract. The two main theorems proved here are as follows: If A is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of A is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, given an appropriate geometrization CompA d of the family of finite Amodule complexes with fixed sequence d of dimensions and an “almost projective ” complex X ∈ CompA d, there exists a canonical vector space embedding TX(Comp A d)/TX(G.X) −→
Constructions of stable equivalences of Morita type for finitedimensional algebras I
, 2005
"... In the representation theory of finite groups, the stable equivalence of Morita type plays an important role. For general finitedimensional algebras, this notion is still of particular interest. However, except for the class of selfinjective algebras, one does not know much on the existence of s ..."
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Cited by 18 (10 self)
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In the representation theory of finite groups, the stable equivalence of Morita type plays an important role. For general finitedimensional algebras, this notion is still of particular interest. However, except for the class of selfinjective algebras, one does not know much on the existence of such equivalences between two finitedimensional algebras; in fact, even a nontrivial example is not known. In this paper, we provide two methods to produce new stable equivalences of Morita type from given ones. The main results are Corollary 1.2 and Theorem 1.3. Here the algebras considered are not necessarily selfinjective. As a consequence of our constructions, we give an example of a stable equivalence of Morita type between two algebras of global dimension 4, such that one of them is quasihereditary and the other is not. This shows that stable equivalences of Morita type do not preserve the quasiheredity of algebras. As another byproduct, we construct a Morita equivalence inside each given stable equivalence of Morita type between algebras A and B. This leads not only to a general formulation of a result by Linckelmann (1996), but also to a nice correspondence of some torsion pairs in Amod with those in Bmod if both A and B are symmetric algebras. Moreover, under the assumption of symmetric algebras we can get a new stable equivalence of Morita type. Finally, we point out that stable equivalences of Morita type are preserved under separable extensions of ground fields.