自入射代数平凡扩张的复杂度

On the Complexity of the Trivial Extension of Self-injective Algebra

【目的】设Λ是一个连通的有限表示型的有限维自入射代数,T(Λ)是其平凡扩张代数。本研究主要目的是找出Λ的复杂度与T的复杂度之间的关系。【方法】首先当Λ是满足Fg假设的自入射代数时,Λ的表示维数大于等于Λ的复杂度加1,且有限表示型的表示维数等于2,所以Λ的复杂度小于等于1;又因为自入射代数Λ上的模的有无限投射维数,所以Λ的复杂度大于等于1,因而得到Λ的复杂度为1。其次,通过构造T(Λ)上单模的投射分解,具体计算T(Λ)上单模的投射分解中每一项Pt(M)的维数,得到对几乎所有的t,存在λ>0,使得dimPt(M)≤λt,利用复杂度定义即有T(Λ)的复杂度为2。【结果】因而得到T(Λ)的复杂度为Λ的复杂度加1。【结论】该结果丰富了无限表示型自入射代数与其平凡扩张代数的复杂度之间存在加1关系的结果。选取非Koszul代数的例子说明本结论成立。

[Purposes]LetΛ be a connected finite dimensional self-injective algebra of finite type,T（Λ）be its trivial extension.In this paper,we want to find out the relationship of the complexity betweenΛand its trivial extension T（Λ）.[Methods]Firstly ifΛis the self-injective algebra satisfying Fgassumption,then the representation dimension ofΛis one more than or equal to the complexity ofΛ,and the representation dimension of algebra of finite representation type is 2,so the complexity of Λ is less than or equal to 1,on the other hand,ifΛis a self-injective algebra,then theΛ-mod is infinite projective dimension,then the complexity of Λis more than or equal to 1,so the complexity ofΛis 1.Secondly by constructing the projective resolution of simple T（Λ）-mod and computing the dimension of every Pt（M）in the projective resolution of simple T（Λ）-mod,there exists λ〉0 such that dimdimPt（M）≤λt for almost all t,and by using the definition of the complexity,the complexity of T（Λ）is 2.[Findings]the complexity ofΛis one more than that of T（Λ）.[Conclusions]Which enriches the same result about the algebra of infinite representation type？Lastly an example is given in which the algebra is not Koszul to illustrate our conclusion.