Abstract
We prove that an associated graded algebra R G of a finite dimensional algebra R is QF (= selfinjective) if and only if R is QF and Loewy coincident. Here R is said to be Loewy coincident if, for every primitive idempotent e, the upper Loewy series and the lower Loewy series of Re and eR coincide. QF-3 algebras are an important generalization of QF algebras; note that Auslander algebras form a special class of these algebras. We prove that for a Loewy coincident algebra R, the associated graded algebra R G is QF-3 if and only if R is QF-3.
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Tachikawa, H. Loewy coincident algebra and QF-3 associated graded algebra. Czech Math J 59, 583–589 (2009). https://doi.org/10.1007/s10587-009-0050-2
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DOI: https://doi.org/10.1007/s10587-009-0050-2