Homological dimensions of gentle algebras via geometric models  

作　　者：Yu-Zhe Liu Hanpeng Gao Zhaoyong Huang 

机构地区：[1]Department of Mathematics,Nanjing University,Nanjing 210093,China [2]School of Mathematical Sciences,Anhui University,Hefei 230601,China

出　　处：《Science China Mathematics》2024年第4期733-766,共34页中国科学（数学）（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.11971225 and 12171207)。

摘　　要：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,MA,ΓA).It is known that SAcan be divided into some elementary polygons{Δi|1≤i≤d}byΓA,which has exactly one side in the boundary of SA.Let■(Δi)be the number of sides ofΔibelonging toΓAif the unmarked boundary component of SAis not a side ofΔi;otherwise,■(Δi)=∞,and let f-Δbe the set of all the non-co-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■(Δi)-1=maxΠ∈FAl(Π),where l(Π)is the length ofΠ;(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.masΔi∈f-Δ{1,■(Δi)-1}=max n∈f-F_(A)l(П),otherwise,As a consequence,we get that the finiteness of the global dimension of gentle algebras is invariant under AvellaGeiss(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.

关 键 词：global dimension self-injective dimension gentle algebras marked ribbon surfaces geometric models AG-equivalence 

分 类 号：O15[理学—数学]