Derived and stable equivalence classification of twisted multifold extensions of piecewise hereditary algebras of tree type
Let \(k\) be an algebraically closed field and let \(\Lambda\) and \(\Pi\) be basic connected selfinjective finite-dimensional \(k\)-algebras. If \(\Lambda\) and \(\Pi\) are derived equivalent, then they are stably equivalent [see \textit{B. Keller} and \textit{D. Vossieck}, C. R. Acad. Sci., Paris, Sér. I 305, 225-228 (1987; Zbl 0628.18003) or \textit{J. Rickard}, J. Pure Appl. Algebra 61, No. 3, 303-317 (1989; Zbl 0685.16016)]. In an earlier paper the author showed that the converse holds if both \(\Lambda\) and \(\Pi\) are of finite representation type [\textit{H. Asashiba}, J. Algebra 214, No. 1, 182-221 (1999; Zbl 0949.16013)]. The question of when the converse holds is related to Broué's conjecture [\textit{M. Broué}, Astérisque 181-182, 61-92 (1990; Zbl 0704.20010)] stating that if \(G\) is a finite group, then a block of the group algebra \(kG\) with Abelian defect is derived equivalent to its Brauer correspondent. The relationship is that in some important cases it is known that the block and its Brauer correspondent are stably equivalent. In the present paper the author shows that the converse holds for a certain class of algebras that need not be of finite representation type. The main result is that if \(\Lambda\) and \(\Pi\) are twisted multifold extensions of piecewise hereditary algebras of tree type, then \(\Lambda\) and \(\Pi\) are derived equivalent if and only if they are stably equivalent.
scientific article; zbMATH DE number 1750359
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