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arXiv 1602.05037math.RTmath.RA
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$K[x]/(x^n)$的Auslander代数上$τ$ -倾斜模的分类

Classifying $τ$-tilting modules over the Auslander algebra of $K[x]/(x^n)$

Osamu Iyama, Xiaojin Zhang

发表信息：
Published in Journal of the Mathematical Society of Japan 2020
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摘要：

我们建立了$K[x]/(x^n)$的Auslander代数$Λ$上的基本支撑$τ$ -倾斜模的同构类的集合$\sttiltΛ$与对称群$\mathfrak{S}_{n+1}$之间的双射，它是偏序集关于$\sttiltΛ$上的生成阶和$\mathfrak{S}_{n+1}$上的左阶的反同构.这限制了基本倾斜$Λ$ -模的同构类的集合$\tiltΛ$和由于Brüstle，Hille，Ringel和Röhrle而形成的对称群$\mathfrak{S}_n$之间的双射。将Dynkin型$A_n$的预投射代数$Γ$作为$Λ$的一个因子代数，证明了张量函子$-\otimes_ΛΓ$诱导$\sttiltΛ\to\sttiltΓ$之间的一个双射.这恢复了Mizuno对类型$A_n$的双射$\mathfrak{S}_{n+1}\to\sttiltΓ$。

英文摘要：

We build a bijection between the set $\sttiltΛ$ of isomorphism classes of basic support $τ$-tilting modules over the Auslander algebra $Λ$ of $K[x]/(x^n)$ and the symmetric group $\mathfrak{S}_{n+1}$, which is an anti-isomorphism of partially ordered sets with respect to the generation order on $\sttiltΛ$ and the left order on $\mathfrak{S}_{n+1}$. This restricts to the bijection between the set $\tiltΛ$ of isomorphism classes of basic tilting $Λ$-modules and the symmetric group $\mathfrak{S}_n$ due to Brüstle, Hille, Ringel and Röhrle. Regarding the preprojective algebra $Γ$ of Dynkin type $A_n$ as a factor algebra of $Λ$, we show that the tensor functor $-\otimes_ΛΓ$ induces a bijection between $\sttiltΛ\to\sttiltΓ$. This recover Mizuno's bijection $\mathfrak{S}_{n+1}\to\sttiltΓ$ for type $A_n$.

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