论文标题
功能有限的子类别的根本
The radical of functorially finite subcategories
论文作者
论文摘要
令$λ$为Artin代数，$ \ Mathcal {C} $为Mod $λ$的功能有限子类别，其中包含$λ$或$dλ$。我们使用$ \ MATHCAL {C} $的无限激进的概念，并显示$ \ Mathcal {C} $具有添加剂生成器，并且仅当Rad $^\ Infty _ {\ Mathcal {C}}} $ nistation时才仅具有加法生成器。在这种情况下，我们用$ \ Mathcal {C} $的激进的词性描述了其不可医学的形态。此外，在一个温和的假设下，我们证明$ \ MATHCAL {C} $是有限表示类型，并且仅当$ \ Mathcal {C}中的不可兼容对象之间的任何单态性（表达式）（表达式）是Noyetherian（conoetherian）。此外，通过使用注射信封，投影覆盖物，剩下的$ \ Mathcal {C} $ - 近似值和正确的$ \ Mathcal {C} $ - 简单$λ$ - 模块的近似值，我们提供了其他标准来描述$ \ Mathcal {c} $是有限表示的类型。此外，我们给出了$ \ MATHCAL {C} $的激进索引，该指数独立于$ \ Mathcal {C} $中的不可兼容$λ$ -Modules的最大长度。
Let $Λ$ be an artin algebra and $\mathcal{C}$ be a functorially finite subcategory of mod$Λ$ which contains $Λ$ or $DΛ$. We use the concept of the infinite radical of $\mathcal{C}$ and show that $\mathcal{C}$ has an additive generator if and only if rad$^\infty_{\mathcal{C}}$ vanishes. In this case we describe the morphisms in powers of the radical of $\mathcal{C}$ in terms of its irreducible morphisms. Moreover, under a mild assumption, we prove that $\mathcal{C}$ is of finite representation type if and only if any family of monomorphisms (epimorphisms) between indecomposable objects in $\mathcal{C}$ is noetherian (conoetherian). Also, by using injective envelopes, projective covers, left $\mathcal{C}$-approximations and right $\mathcal{C}$-approximations of simple $Λ$-modules, we give other criteria to describe whether $\mathcal{C}$ is of finite representation type. In addition, we give a nilpotency index of the radical of $\mathcal{C}$ which is independent from the maximal length of indecomposable $Λ$-modules in $\mathcal{C}$.
