2022/08/28 by Liu, Yu-Zhe, Gao, Hanpeng, Huang, Zhaoyong · 1 citation
#16E10 #16G10 #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2208.13180
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 SA can be divided into some elementary polygons \Δi| 1≤ i≤ d\ by ΓA which has exactly one side in the boundary of SA. Let \mathfrakC(Δi) be the number of sides of Δi belonging to ΓA if the unmarked boundary component of SA is not a side of Δi; otherwise, \mathfrakC(Δi)=∞, and let f-Δ be the set of all non-∞-elementary polygons and FA (respectively, f-FA) the set of all forbidden threads (respectively, of finite length). Then we have \beginenumerate \item[\rm (1)] The global dimension of A=max1≤ i≤ d\mathfrakC(Δi)-1 =max\mathitΠ\inFA l(\mathitΠ), where l(\mathitΠ) is the length of \mathitΠ. \item[\rm (2)] The left and right self-injective dimensions of A= \begincases 0, \mbox\textif \it Q is either a point or an oriented cycle with full relations; max_Δi∈f-Δ\1, \mathfrakC(Δi)-1 \= max_\mathitΠ∈f-FA l(\mathitΠ), otherwise. \endcases \endenumerate As a consequence, we get that the finiteness of the global dimension of gentle algebras is invariant under 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.