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Abstract
We introduce the notion of a directed stratification for a finite-dimensional algebra. For algebras that admit such a stratification, we characterize the projective resolutions of finitely generated modules and obtain a result for the finitistic dimension, which is an inductive version of a result of Fossum, Griffith, and Reiten. With the developed techniques, which are adopted from the theory of EI-category algebras, we gain deeper insight in the combinatorial nature of this result. A characterization of algebras that do not admit a directed stratification is given in terms of the Ext-quiver.
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ACKNOWLEDGEMENT
This article is based on parts of my Ph.D. thesis under the supervision of Henning Krause. I would like to thank him for his guidance and inspiring discussions on the topic.
The first version of this article contained a proof of the finiteness of the finitistic dimension for EI-category algebras. I am grateful to Jesper Grodal for pointing out the relevance of Lueck's work [Citation17], which already contains a proof of this fact.
Finally, I would like to thank Birge Huisgen-Zimmermann and Steffen Koenig for their helpful remarks, which led me in the right direction.
Notes
Communicated by D. Zacharia.