论文标题
由较高同源代数中其组成因子确定的模块
Modules determined by their composition factors in higher homological algebra
论文作者
论文摘要
抽象的。令$φ$为有限的尺寸$ k $ -algebra，让$ \ mathscr {c} = \ textrm {mod} \：φ$是有限生成的右$φ$ -MODULES的ABELIAN类别。在1985年的论文``由其组成因子决定的模块''中，Auslander和Reiten表明，在某些条件下，$ \ \ \ \ textrm {mod} \：φ$中的模块是由其组成因子确定的，并显示了与Auslander-Reiten-Reiten-Reiten转换有关的重要公式。 令$ \ mathscr {t} $为$ d $ -cluster倾斜子类别的$ \ mathscr {c} $，从定义上讲也是$ d $ -Abelian。在本文中，我们将为$ d $ -Abelian类别定义Grothendieck Group，并表明$ \ Mathscr {C} $和$ \ Mathscr {T} $的Grothendieck组是同构。我们还表明，在某些条件下，$ \ mathscr {t} $的不可分解的对象通过$ \ mathscr {c} $中的组成因子确定为同构。最后，我们从Auslander和Reiten概括了涉及较高维度的Auslander-Reiten翻译的公式。
ABSTRACT. Let $Φ$ be a finite dimensional $K$-algebra and let $\mathscr{C} = \textrm{mod}\: Φ$ be the abelian category of finitely generated right $Φ$-modules. In their 1985 paper ``Modules determined by their composition factors'', Auslander and Reiten showed that under certain conditions modules in $\textrm{mod}\: Φ$ are determined by their composition factors, and show an important formula related to the Auslander-Reiten translation. Let $\mathscr{T}$ be a $d$-cluster tilting subcategory of $\mathscr{C}$, which by definition is also $d$-abelian. In this paper we will define the Grothendieck group for a $d$-abelian category, and show that the Grothendieck groups of $\mathscr{C}$ and $\mathscr{T}$ are isomorphic. We show also that under certain conditions, the indecomposable objects of $\mathscr{T}$ are determined up to isomorphism by their composition factors in $\mathscr{C}$. Finally, we generalise the formula from Auslander and Reiten involving the higher dimensional Auslander-Reiten translation.
