Artin环上的分次模
Gradable modules over artinian rings
摘要：
设$Λ$是一个$\mathbb{Z}$ -分次Artin代数. Gordon和Green的两个经典结果是：如果$Λ$只有1/2个不可分解的分次模，直到同构，则$Λ$有有限表示型，如果$Λ$有有限表示型，则每个$Λ$ -模都是分次的.我们将这些结果推广到$\mathbb{Z}$ -分次右Artin环$R$ .关键的工具是可分次模的一个特征：f.g.。右$R$ -模是分次的当且仅当它的“上拉”是纯投射的。利用这一点，我们表明，如果有一个界的梯度长度的f.g.不可分解分次$R$ -模，则每个f.g. $R$ -模块是可分级的。作为另一个结果，我们看到，如果分次Artin代数有不可分次模，则它有一个非有限型的Prüfer模，因此它有一个Ringel工作的泛型模
英文摘要：
Let $Λ$ be a $\mathbb{Z}$-graded artin algebra. Two classical results of Gordon and Green state that if $Λ$ has only finitely many indecomposable gradable modules, up to isomorphism, then $Λ$ has finite representation type, and if $Λ$ has finite representation type then every $Λ$-module is gradable. We generalize these results to $\mathbb{Z}$-graded right artinian rings $R$. The key tool is a characterization of gradable modules: a f.g. right $R$-module is gradable if and only if its "pull-up" is pure-projective. Using this we show that if there is a bound on the graded-lengths of f.g. indecomposable graded $R$-modules, then every f.g. $R$-module is gradable. As another consequence, we see that if a graded artin algebra has an ungradable module, then it has a Prüfer module which is not of finite type, and hence it has a generic module by work of Ringel
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