Construction of stable equivalences of Morita type for finite-dimensional algebras I

Abstract

In the representation theory of finite groups, the stable equivalence of Morita type plays an important role. For general finite-dimensional algebras, this notion is still of particular interest. However, except for the class of self-injective algebras, one does not know much on the existence of such equivalences between two finite-dimensional algebras; in fact, even a non-trivial 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 self-injective. 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 quasi-hereditary and the other is not. This shows that stable equivalences of Morita type do not preserve the quasi-heredity of algebras. As another by-product, 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 $A$-mod with those in $B$-mod 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.