Tilting modules for the Cummings construction

Abstract

Finiteness of the right little finitistic dimension of a finite dimensional algebra $\La$ is known to be equivalent to existence of a (possibly infinite dimensional) tilting right $\La$-module $T_f$ whose tilting class is $\{T_f{\}}^{{\perp }_{\infty }}=({\mathcal{P}}^{<\infty }{)}^{\perp }$, \cite{AT}. We use this equivalence to interpret the recent surprising results of Cummings \cite{C} concerning the asymmetry of left and right finitistic dimensions of the triangular matrix algebras $\tilde{A}$ built from arbitrary basic finite dimensional algebras $A$. In particular, we determine the structure of the tilting right $\tilde{A}$-module $T_f$ in the case when the algebra $A$ has finite right global dimension.