Stable Equivalences Of Morita Type Research Articles

Invariance of the restricted p-power map on integrable derivations under stable equivalences

We show that the p-power maps in the first Hochschild cohomology space of finite-dimensional selfinjective algebras over a field of prime characteristic p commute with stable equivalences of Morita type on the subgroup of classes represented by integrable derivations. We show, by giving an example, that the p-power maps do not necessarily commute with arbitrary transfer maps in the Hochschild cohomology of symmetric algebras.

On equivalences for cohomological Mackey functors

By results of Rognerud, a source algebra equivalence between two p p -blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories of cohomological Mackey functors. We prove this by giving an intrinsic description of cohomological Mackey functors of a block in terms of the source algebras of the block, and then using this description to construct explicit two-sided tilting complexes realising the above mentioned derived equivalence. We show further that a splendid stable equivalence of Morita type between two blocks induces an equivalence between the categories of cohomological Mackey functors which vanish at the trivial group. We observe that the module categories of a block, the category of cohomological Mackey functors, and the category of cohomological Mackey functors which vanish at the trivial group arise in an idempotent recollement. Finally, we extend a result of Tambara on the finitistic dimension of cohomological Mackey functors to blocks.

Singular equivalence of Morita type with level

We generalize the notion of stable equivalence of Morita type and define what is called “singular equivalence of Morita type with level”. Such an equivalence induces an equivalence between singular categories. We will also prove that a derived equivalence of standard type induces a singular equivalence of Morita type with level.

ON THE BRAUER INDECOMPOSABILITY OF SCOTT MODULES

Let k be an algebraically closed field of prime characteristic p, and let P be a p-subgroup of a finite group G. We give sufficient conditions for the kG-Scott module Sc(G,P) with vertex P to remain indecomposable under the Brauer construction with respect to any subgroup of P. This generalizes similar results for the case where P is abelian. The background motivation for this note is the fact that the Brauer indecomposability of a p-permutation bimodule is a key step towards showing that the module under consideration induces a stable equivalence of Morita type, which then may possibly be lifted to a derived equivalence.

On stable equivalences with endopermutation source

We show that a bimodule between block algebras which has a fusion stable endopermutation module as a source and which induces Morita equivalences between centralisers of nontrivial subgroups of a defect group induces a stable equivalence of Morita type; this is the converse to a theorem of Puig. The special case where the source is trivial has long been known by many authors. The earliest instance of a result deducing a stable equivalence of Morita type from local Morita equivalences with possibly nontrivial endopermutation source is due to Puig, in the context of blocks with abelian defect groups with a Frobenius inertial quotient. The present note is motivated by an application, due to Biland, to blocks of finite groups with structural properties known to hold for hypothetical minimal counterexamples to the Zp⁎-Theorem.

Classification of Symmetric Special Biserial Algebras With At Most One Non-Uniserial Indecomposable Projective

AbstractWe consider a natural generalization of symmetric Nakayama algebras, namely, symmetric special biserial algebras with at most one non-uniserial indecomposable projective module. We describe the basic algebras explicitly by quiver and relations, then classify them up to derived equivalence and up to stable equivalence of Morita type. This includes the weakly symmetric algebras of Euclidean type n, as studied by Bocian et al., as well as some algebras of dihedral type.

On Stable Equivalence of Tensor Products of Algebras

The paper investigates the following problem. Let bimodules N, M yield a stable equivalence of Morita type between self-injective K-algebras A and E. Further, let bimodules S, T yield a stable equivalence of Morita type between self-injective K-algebras B and F. Then we want to know whether the functor M ⊗ A − ⊗ B S: mod(A ⊗ K B op ) → mod(E ⊗ K F op ) induces a stable equivalence between A ⊗ K B op and E ⊗ K F op . There is given a reduction of this problem to some smaller subcategories for self-injective algebras. Moreover, new invariants of stable equivalences of Morita type are constructed in a general case of arbitrary finite-dimensional algebras over a field.

Simple-minded systems, configurations and mutations for representation-finite self-injective algebras

Simple-minded systems of objects in a stable module category are defined by common properties with the set of simple modules, whose images under stable equivalences do form simple-minded systems. Over a representation-finite self-injective algebra, it is shown that all simple-minded systems are images of simple modules under stable equivalences of Morita type, and that all simple-minded systems can be lifted to Nakayama-stable simple-minded collections in the derived category. In particular, all simple-minded systems can be obtained algorithmically using mutations.

Strong fusion control and stable equivalences

This article is dedicated to the proof of the following theorem. Let G be a finite group, p be a prime number, and e be a p-block of G. Assume that the centraliser CG(P) of an e-subpair (P,eP) “strongly” controls the fusion of the block e, and that a defect group of e is either abelian or (for odd p) has a non-cyclic centre. Then there exists a stable equivalence of Morita type between the block algebras OGe and OCG(P)eP, where O is a complete discrete valuation ring of residual characteristic p. This stable equivalence is constructed by gluing together a family of local Morita equivalences, which are induced by bimodules with fusion-stable endo-permutation sources.Broué had previously obtained a similar result for principal blocks, in relation with the search for a modular proof of the odd Zp⁎-theorem. Thus our theorem points towards a block-theoretic analogue of the Zp⁎-theorem, which we state in terms of fusion control and Morita equivalences.

Orbit algebras that are invariant under stable equivalences of Morita type

AbstractIn this note we show that there are a lot of orbit algebras that are invariant under stable equivalences of Morita type between self-injective algebras. There are also indicated some links between Auslander-Reiten periodicity of bimodules and noetherianity of their orbit algebras.

On singular equivalences of Morita type

Stable equivalences of Morita type preserve many interesting properties and are proved to be the appropriate concept for studying equivalences between stable categories. Recently the singularity category attained much attraction and Xiao-Wu Chen and Long-Gang Sun gave an appropriate definition of singular equivalence of Morita type. We shall show that under some conditions singular equivalences of Morita type have some biadjoint functor properties and preserve positive degree Hochschild homology.

On Iterated Almost ν-Stable Derived Equivalences

In this article, by characterizing iterated almost ν-stable derived equivalences, we give several sufficient conditions for a derived equivalence between general finite-dimensional algebras to induce a stable equivalence of Morita type. In particular, we prove the following: Let A and B be two finite-dimensional algebras over a field. Suppose that there is a derived equivalence between A and B induced by a tilting complex T • over A. If each indecomposable projective A-module P without the property “ is projective for all i ≥ 0” occurs only in the 0-degree term T 0 of T • with multiplicity 1, then A and B are stably equivalent of Morita type.

Invariants under stable equivalences of morita type

The aim of this article is to study some invariants of associative algebras under stable equivalences of Morita type. First of all, we show that, if two finite-dimensional self-injective k-algebras are stably equivalent of Morita type, then their orbit algebras are isomorphic. Secondly, it is verified that the quasitilted property of an algebra is invariant under stable equivalences of Morita type. As an application of this result, it is obtained that if an algebra is of finite representation type, then its tilted property is invariant under stable equivalences of Morita type; the other application to partial tilting modules is given in Section 4. Finally, we prove that when two finite-dimensional k-algebras are stably equivalent of Morita type, their repetitive algebras are also stably equivalent of Morita type under certain conditions.

On blocks stably equivalent to a quantum complete intersection of dimension 9 in characteristic 3 and a case of the Abelian defect group conjecture

Using a stable equivalence due to Rouquier, we prove that Broue's abelian defect group conjecture holds for 3-blocks of defect 2 whose Brauer correspondent has a unique isomorphism class of simple modules. The proof makes use of the fact, also due to Rouquier, that a stable equivalence of Morita type between self-injective algebras induces an isomorphism between the connected components of the outer automorphism groups of the algebras.

Module-relative-Hochschild (co)homology under stable equivalences of Morita type

Based on the theory of relative right derived functors, module-relative-Hochschild cohomology was introduced by Ardizzoni, Brzezin′ski and Menini when they studied the formal smoothness. In this paper, we introduce the module-relative-Hochschild homology using relative left derived functors, discuss the relations of Hochschild (co)homology, relative Hochschild (co)homology and module-relative-Hochschild (co)homology under stable equivalences of Morita type, and prove that module-relative-Hochschild homology and cohomology are invariants under stable equivalences of Morita type. As a consequence, we get a method of constructing formally smooth bimodules and separable bimodules, and give a new proof to the known fact that the stable equivalence of Morita type preservers the ordinary Hochschild (co)homology.

Deformed preprojective algebras of type L: Külshammer spaces and derived equivalences

Białkowski, Erdmann and Skowroński classified those indecomposable selfinjective algebras for which the Nakayama shift of every (non-projective) simple module is isomorphic to its third syzygy. It turned out that these are precisely the deformations, in a suitable sense, of preprojective algebras associated to the simply laced ADE Dynkin diagrams and of another graph L n , which also occurs in the Happel–Preiser–Ringel classification of subadditive but not additive functions. In this paper we study these deformed preprojective algebras of type L n via their Külshammer spaces, for which we give precise formulae for their dimensions. These are known to be invariants of the derived module category, and even invariants under stable equivalences of Morita type. As main application of our study of Külshammer spaces we can distinguish many (but not all) deformations of the preprojective algebra of type L n up to stable equivalence of Morita type, and hence also up to derived equivalence.

Inductions and restrictions for stable equivalences of Morita type

In this paper, we present two methods, induction and restriction procedures, to construct new stable equivalences of Morita type. Suppose that a stable equivalence of Morita type between two algebras A and B is defined by a B - A -bimodule N . Then, for any finite admissible set Φ and any generator X of the category of A -modules, the Φ -Auslander–Yoneda algebras of X and N ⊗ A X are stably equivalent of Morita type. Moreover, under certain conditions, we transfer stable equivalences of Morita type between A and B to ones between e A e and f B f , where e and f are idempotent elements in A and B , respectively. Consequently, for self-injective algebras A and B over a field without semisimple direct summands, and for any A -module X and B -module Y , if the Φ -Auslander–Yoneda algebras of A ⊕ X and B ⊕ Y are stably equivalent of Morita type for one finite admissible set Φ , then so are the Ψ -Auslander–Yoneda algebras of A ⊕ X and B ⊕ Y for every finite admissible set Ψ . Moreover, two representation-finite algebras over a field without semisimple direct summands are stably equivalent of Morita type if and only if so are their Auslander algebras. As another consequence, we construct an infinite family of algebras of the same dimension and the same dominant dimension such that they are pairwise derived-equivalent, but not stably equivalent of Morita type. This answers a question by Thorsten Holm.

Transfer maps in Hochschild (co)homology and applications to stable and derived invariants and to the Auslander–Reiten conjecture

Derived equivalences and stable equivalences of Morita type, and new (candidate) invariants thereof, between symmetric algebras will be investigated, using transfer maps as a tool. Close relationships will be established between the new invariants and the validity of the Auslander–Reiten conjecture, which states the invariance of the number of non-projective simple modules under stable equivalence. More precisely, the validity of this conjecture for a given pair of algebras, which are stably equivalent of Morita type, will be characterized in terms of data refining Hochschild homology (via Külshammer ideals) being invariant and also in terms of cyclic homology being invariant. Thus, validity of the Auslander–Reiten conjecture implies a whole set of ring theoretic and cohomological data to be invariant under stable equivalence of Morita type, and hence also under derived equivalence. We shall also prove that the Batalin–Vilkovisky algebra structure of Hochschild cohomology for symmetric algebras is preserved by derived equivalence. The main tools to be developed and used are transfer maps and their properties, in particular a crucial compatibility condition between transfer maps in Hochschild homology and Hochschild cohomology via the duality between them.

Classifying tame blocks and related algebras up to stable equivalences of Morita type

We contribute to the classification of finite dimensional algebras under stable equivalence of Morita type. More precisely we give a classification of Erdmann’s algebras of dihedral, semi-dihedral and quaternion type and obtain as byproduct the validity of the Auslander–Reiten conjecture for stable equivalences of Morita type between two algebras, one of which is of dihedral, semi-dihedral or quaternion type.

Auslander–Reiten conjecture for symmetric algebras of polynomial growth

This paper studies self-injective algebras of polynomial growth. We prove that two indecomposable weakly symmetric algebras of domestic type are derived equivalent if and only if they are stably equivalent. Furthermore we prove that for indecomposable weakly symmetric non-domestic algebras of polynomial growth, up to some scalar problems, the derived equivalence classification coincides with the classification up to stable equivalences of Morita type. As a consequence, we get the validity of the Auslander–Reiten conjecture for stable equivalences between weakly symmetric algebras of domestic type and for stable equivalences of Morita type between weakly symmetric algebras of polynomial growth.