Finitistic Dimension Conjectures for representations of quivers

Authors: SERGIO ESTRADA, SALAHATTİN ÖZDEMİR

Abstract: Let R be a ring and Q be a quiver. We prove the first Finitistic Dimension Conjecture to be true for RQ, the path ring of Q over R, provided that R satisfies the conjecture. In fact, we prove that if the little and the big finitistic dimensions of R coincide and equal n <\infty, then this is also true for RQ and, both the little and the big finitistic dimensions of RQ equal n + 1 when Q is non-discrete and n when Q is discrete. We also prove that RQ is a quasi-Frobenius ring if and only if R is quasi-Frobenius and Q is discrete.

Keywords: Finitistic dimension conjecture, path ring, quasi-Frobenius ring, quiver representation

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