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Homological domino effects and the first finitistic dimension conjecture. (English) Zbl 0792.16011
For any field \(k\) and any integer \(m\geq 2\), it is shown that there exist finite dimensional \(k\)-algebras \(\Lambda\) such that the little finitistic dimension of \(\Lambda\) is \(m\), while the big finitistic dimension is \(m+1\). This provides a negative solution to the first of the following Finitistic Dimension Conjectures.
For a finite dimensional algebra \(\Lambda\), (1) \(\text{fin}\dim \Lambda = \text{Fin}\dim \Lambda\) and (2) \(\text{fin}\dim \Lambda < \infty\), where \(\text{fin}\dim\Lambda = \sup\{\text{p}\dim M\mid M\) a finitely generated left \(\Lambda\)-module with \(\text{p}\dim M < \infty\}\) is the (left) little finitistic dimension of \(\Lambda\), and \(\text{Fin}\dim \Lambda = \sup\{\text{p}\dim M \mid M\) an arbitrary left \(\Lambda\)-module with \(\text{p}\dim M < \infty\}\) is the (left) big finitistic dimension of \(\Lambda\).
Reviewer: Y.Kurata (Ube)

16E10 Homological dimension in associative algebras
16P10 Finite rings and finite-dimensional associative algebras
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
Full Text: DOI EuDML
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