Homological domino effects and the first finitistic dimension conjecture (Q1207410)

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 \(\Lambda\).