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Homological dimensions of gentle algebras via geometric models
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Let A=kQ/I be a finite-dimensional basic algebra over an algebraically closed field k,which is a gentle algebra with the marked ribbon surface(SA,M A,ГA).It is known that SA can be divided into some elementary polygons {Δi | 1 ≤ i ≤ d} by ГA,which has exactly one side in the boundary of SA.Let (c)(Δi)be the number of sides of Δi belonging to ГA if the unmarked boundary component of SA is not a side of Δi;otherwise,(c)(Δi)=∞,and let f-Δ be the set of all the non-∞-elementary polygons and FA(resp.f-FA)be the set of all the forbidden threads(resp.of finite length).Then we have(1)the global dimension of A is max1≤i≤d(c)(Δi)—1=maxп∈FA l(Π),where l(Π)is the length of Π;(2)the left and right self-injective dimensions of A are{0,if Q is either a point or an oriented cycle with full relations,maxΔi∈f-Δ{1,(c)(Δi)-1}=maxΠ∈f-FA l(Π),otherwise.As a consequence,we get that the finiteness of the global dimension of gentle algebras is invariant under Avella-Geiss(AG)-equivalence.In addition,we get that the number of indecomposable non-projective Gorenstein projective modules over gentle algebras is also invariant under AG-equivalence.
global dimensionself-injective dimensiongentle algebrasmarked ribbon surfacesgeometric modelsAG-equivalence
Yu-Zhe Liu、Hanpeng Gao、Zhaoyong Huang
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Department of Mathematics,Nanjing University,Nanjing 210093,China
School of Mathematical Sciences,Anhui University,Hefei 230601,China
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