Gentle代数的矩阵模型及其整体维数

The Matrix Model of Gentle Algebra and the Global Dimension

该文利用Gentle代数的矩阵模型刻画了Gentle代数上的单模和投射模,给出了单模的投射分解的矩阵表示.由此指出Gentle代数的整体维数可以由它的矩阵模型所诱导的一类特殊子矩阵序列进行刻画.并进一步指出这一类特殊子矩阵序列对应于Gentle代数的箭图上的极大非平凡Forbidden路,从而得到Gentle代数的整体维数等于它的箭图上的极大非平凡Forbidden路的长度.

It is showed that for Gentle algebra,the simple module and projective module can be characterized by matrix model,and a matrix representation of projective resolution of simple module is provided.Thus the global dimension of a Gentle algebra can be characterized by a special submatrix sequence induced by its matrix model.Furthermore,by showing that above special submatrix sequences correspond to maximal non-trivial Forbidden paths on the quiver of Gentle algebra,the global dimension of Gentle algebra equals the length of the maximal nontrivial Forbidden path is obtained.