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Finitistic dimension conjectures for representations of Quivers

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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 < ∞, 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. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2013) 37: 585 – 591 ¨ ITAK ˙ c TUB ⃝ doi:10.3906/mat-1106-13 Finitistic Dimension Conjectures for representations of quivers ¨ ˙ 2,∗ Sergio ESTRADA,1 Salahattin OZDEM IR Department of Applied Mathematics, University of Murcia, Murcia, Spain 2 ˙ Department of Mathematics, Faculty of Sciences, Dokuz Eyl¨ ul University, Buca, Izmir, Turkey 1 Received: 09.06.2011 • Accepted: 08.07.2012 • Published Online: 12.06.2013 • Printed: 08.07.2013 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 as R[x1 ] - modules, where X = {x1 , x2 , . . .}, is the free monoid of words on X , and x1 is the submonoid of of words which does not start by x1 (notice that if p is the empty word, then we set R[x1 ] · p = R[x1 ]). Since R[x1 ] · p ∼ = R[x1 ] as R[x1 ]-modules, then ⊕p∈ R[x1 ] · p is R[x1 ] -projective. So x1 P is R[x1 ] -projective as a direct summand, or equivalently P is a projective representation in (Q2 , R-Mod) . 2 Lemma 3.2 Let Q1 = (V1 , E1 ) be a subquiver of Q = (V, E). Assume that every projective representation over Q is also projective when it is restricted to Q1 . Let M be any representation of Q1 . If pdQ1 (M ) = n , then f) ≥ n , where M f is the following representation of Q: M f(v) = M (v) ∀v ∈ V1 , M f(v) = 0 ∀v ∈ V − V1 , pdQ (M f(a) = M (a) ∀a ∈ E1 , M f(a) = 0 ∀a ∈ E − E1 . and M Proof f) < n . Then there exists an exact sequence in (Q, R-Mod) Suppose for the contrary that r = pdQ (M f → 0, 0 → Pr → · · · → P1 → P0 → M 587 ¨ ˙ ESTRADA and OZDEM IR/Turk J Math with Pi , 0 ≤ i ≤ r being projective representations over Q . Now by the assumption, f |Q = M → 0 0 → Pr |Q1 → · · · → P1 |Q1 → P0 |Q1 → M 1 is a projective