Stacks of algebras and their finitistic dimensions

For monomial algebras, Huisgen-Zimmermann proved that the big and little finitistic dimensions differ by at most one, and are bounded by quantities which can be calculated from the quiver and relations of the algebra. We apply these results to more general path algebras modulo relations. In particular, we introduce the peninsular partition, a way of breaking a path algebra modulo relations, &Lgr;, down into two simpler algebras, &Lgr; ′ and &Lgr;″. Bounds for the finitistic dimensions of &Lgr; are given in terms of the finitistic dimensions of &Lgr; ′ and &Lgr;″ for all path algebras. When &Lgr; ′ is a monomial algebra, Huisgen-Zimmermann's results can be applied. This yields useful bounds on the finitistic dimensions of &Lgr; in the case where certain vertices in &Lgr;′ are homogeneous, meaning that all arrows starting in those vertices end in &Lgr;′. Bounds on the finitistic dimensions of &Lgr; which can be calculated from the quivers and relations of &Lgr;′ and &Lgr;″ are given in the case when both &Lgr; ′ and &Lgr;″ are monomial algebras. In this case we call &Lgr; a stack of monomial algebras.; When &Lgr; is a monomial algebra with Jacobson radical J such that J3 = 0, Huisgen-Zimmermann has shown that the big and little finitistic dimensions of &Lgr; coincide. We present a family of non-monomial algebras having J3 = 0 and little finitistic dimension one less than the big finitistic dimension. Another family of examples, with n-generated finitistic dimension one less than their n + 1-generated finitistic dimension, is our ultimate result.