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Let A be an indecomposable basic artin algebra and TA a basic tilting module with B=End(TA). Let us denote by R and S the trivialextension selfinjectivealgebras Am DA and BmDB, respectively. In the papers [24] and [22], H. Tachikawa and the author have proved that there is a stably equivalent functor S: mod-i?―>mod-S and the restrictionof S to the tilting torsion class 3:={Xemod-A\ExtA(T, X)―Q} coincides with that of the tilting functor Horn^T, ?). D. Hughes and J. Waschbiisch [18] introduced the following doubly infinite matrix algebra:

Stable equivalence between universal covers of trivial extension self-injective algebras