论文标题
涵盖理论，（单）形态类别和稳定的奥斯兰德代数
Covering theory, (mono)morphism categories and stable Auslander algebras
论文作者
论文摘要
令$ \ MATHCAL {a} $为本地有限的$ k $ - 类别，$ g $无扭矩的$ k $ - $ k $ - 线性自动形态为$ \ mathcal {a} $ fre frefe frefe fre free of ocy of $ \ \ m nathcal {a}，$ f：a}，$ f：\ natarcal a} Galois函子。我们自然地将下推函数$f_λ$扩展到函子$ \ rm {h} \ rm {f}_λ：\ rm {h}（\ rm {mod} \ rm {mod} \ mbox { - } - } - } { - } \ Mathcal {B}）$，resp。 $ \ MATHCAL {s} \ rm {f}_λ：\ Mathcal {s}（\ rm {mod} \ mbox { - } \ Mathcal {a}）类别，分别单态类别，$ \ rm {mod} \ mbox { - } \ mathcal {a} $和$ \ rm {mod} \ mbox { - } \ mathcal {b} $。在某些其他条件下，我们表明$ \ rm {h}（\ rm {mod} \ mbox { - } \ mathcal {a}）$，resp。 $ \ Mathcal {s}（\ rm {mod} \ mbox { - } \ Mathcal {a}）$在且仅当且仅当$ \ rm {h}（\ rm {mod} \ rm {mod} \ mbox { - } \ mbox { - } \ Mathcal {bbox { - mathcal {b}）中。 $ \ Mathcal {s}（\ rm {mod} \ mbox { - } \ Mathcal {b}）$是有限表示类型。作为一个应用程序，我们表明，当且仅当$λ$属于dynkin type $ \ mathbb {a} _ {a} _ {n} $带有$ n \ n \ leqslant 4 $的情况下，且仅当$λ$属于dynkin type $ \ mathbb {a} $。
Let $\mathcal{A}$ be a locally bounded $k$-category and $G$ a torsion-free group of $k$-linear automorphisms of $\mathcal{A}$ acting freely on the objects of $\mathcal{A},$ and $F:\mathcal{A}\rightarrow \mathcal{B}$ is a Galois functor. We extend naturally the push-down functor $F_λ$ to the functor $\rm{H}\rm{F}_λ:\rm{H}(\rm{mod}\mbox{-} \mathcal{A})\rightarrow \rm{H}(\rm{mod}\mbox{-} \mathcal{B})$, resp. $\mathcal{S} \rm{F}_λ:\mathcal{S}(\rm{mod}\mbox{-} \mathcal{A})\rightarrow \mathcal{S}(\rm{mod}\mbox{-} \mathcal{B})$, between the corresponding morphism categories, resp. monomorphism categories, of $\rm{mod}\mbox{-} \mathcal{A}$ and $\rm{mod}\mbox{-} \mathcal{B}$. Under some additional conditions, we show that $\rm{H}(\rm{mod}\mbox{-}\mathcal{A})$, resp. $\mathcal{S}( \rm{mod}\mbox{-}\mathcal{A})$, is locally bounded if and only if $\rm{H}(\rm{mod}\mbox{-} \mathcal{B})$, resp. $\mathcal{S}(\rm{mod}\mbox{-}\mathcal{B})$, is of finite representation type. As an application, we show that the stable Auslander algebra of a representation-finite selfinjective algebra $Λ$ is again representation-finite if and only if $Λ$ is of Dynkin type $\mathbb{A}_{n}$ with $n\leqslant 4$.
