论文标题
Gorenstein-Poxtive Quiver表示的反射等效性
A reflection equivalence for Gorenstein-projective quiver representations
论文作者
论文摘要
对于$λ$，一个自我注释的代数和$ q $没有定向周期的有限颤抖，代数$λq$是Gorenstein代数和类别$ {\ rm gproj}λq$ gorenstein-Prodightive $ -MODULES的Gorenstein-Prodightive $ -Modules类别。对于$ q $的水槽$ v $，我们定义一个函子$ f（v）：\下划线{\ rm gproj}λq\ to \ to \下划线{\ rm gproj}λq（v）λq（v）λq（v）$在稳定的类别之间，其中$ q（v）$ q（v）$ q（v）是从$ q $中获得$ q $ endern $ en drord n norry norder n rord nord nord n nordr of nord n rearov of nord n rof v。函子是通过对物体和同构级的显式结构给出的。我们的主要结果指出，$ f（v）$等于类别。如果$ q $的基础图是一棵树的情况，我们推断出稳定的类别$ \ useverline {\ rm gproj}λq$不取决于$ q $的方向。此外，如果$ q $是类型$ \ mathbb a_3 $和$λ= k [t]/（t^n）$ the有界的多项式代数，我们使用八面体公理中的八面体的对称性来验证三十个反射的构图会产生对象的身份。
For $Λ$ a selfinjective algebra, and $Q$ a finite quiver without oriented cycles, the algebra $ΛQ$ is a Gorenstein algebra and the category ${\rm Gproj}ΛQ$ of Gorenstein-projective $ΛQ$-modules is a Frobenius category. For a sink $v$ of $Q$, we define a functor $F(v) : \underline{\rm Gproj}ΛQ\to \underline{\rm Gproj}ΛQ(v)$ between the stable categories modulo projectives, where $Q(v)$ is obtained from $Q$ by changing the direction of each arrow ending in $v$. The functor is given by an explicit construction on the level of objects and homomorphisms. Our main result states that $F(v)$ is an equivalence of categories. In the case where the underlying graph of $Q$ is a tree, we deduce that the stable category $\underline{\rm Gproj}ΛQ$ does not depend on the orientation of $Q$. Moreover, if $Q$ is a quiver of type $\mathbb A_3$ and $Λ=k[T]/(T^n)$ the bounded polynomial algebra, we use the symmetry of the octahedron in the octahedral axiom to verify that the composition of twelve reflections yields the identity on objects.
