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| The notion of derived equivalence and that of stable equivalence is mutually independent for general algebras. However, iterated almost $
u$-stable derived equivalences always induce stable equivalences between algebras. In this note, we give a new proof that an iterated almost $
u$-stable derived equivalence always induce a stable equivalence. First, we prove that the stable category modulo $
u$-stably projective modules can be embedded into the singularity category of the same algebra; Second, we show that the embedding is an equivalence if and only if the algebra is self-injective. This generalizes a result of Rickard. Finally, it is shown that an iterated almost $
u$-stable derived equivalence gives rise to an equivalence between the stable categories modulo $
u$-stably projective modules, and consequently provide a stable equivalence. |
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| Keywords:Algebra; derived equivalence; stable equivalence |
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