Integral versions of the Nakayama and finitistic dimension conjectures
The author proves the (generalized) Nakayama conjecture and the finitistic dimension conjecture for lattices over classical orders in special cases. These cases are somehow the integral analogue of the cases which where considered for artinian algebras by \textit{P. Dräxler} and \textit{D. Happel} [J. Pure Appl. Algebra 78, No. 2, 161-164 (1992; Zbl 0794.16007)]. More precisely, let \(R\) be a complete Dedekind domain with field of fractions \(K\), and let \(\Lambda\) be an \(R\)-order in a separable \(K\)-algebra \(A\). For a left \(\Lambda\)-lattice \(M\), the dual \(\text{Hom}_ R(M,R)\) is denoted by \(M^*\). As for algebras, the generalized Nakayama conjecture is equivalent to the following statement: For each simple \(\Lambda\)-module \(S\), there exists an integer \(i \geq 0\) such that \(\text{Ext}^ i_ \Lambda(\Lambda^*,S) \neq 0\). To avoid technicalities, let me just state a corollary of one of the main results: The generalized Nakayama conjecture holds for \(\Lambda\) unless there exists a proper chain of over-orders \(\Lambda \subset \Lambda_ 1 \subset \Lambda_ 2\) such that \(\Lambda_ 2\) is of infinite lattice type. The author describes a class of orders \(\Lambda\), where the ring of multipliers of the radical is a special Bäckström order, for which the finitistic dimension conjecture holds. He also gives illuminating examples where his results can be applied to prove the finitistic dimension conjecture.
scientific article; zbMATH DE number 703225