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The aim of this paper is to provide a criterion to determine, by quivers withrelations, when an algebra has global dimension at most two. In order to dothat, we introduce a new class of algebras of global dimension three, and wecall them critical algebras. Furthermore we give a characterization of criticalalgebras by quivers with relations. Our main theorem states that if a stronglysimply connected schurian algebra does not contain a critical algebra as a fullsubcategory, then it has global dimension at most two.

A Criterion for global dimension two for strongly simply connected schurian algebras