论文标题
$ d $ - 阿贝尔类别的完成
The completion of $d$-abelian categories
论文作者
论文摘要
令$ a $为有限维代数，而$ \ mathfrak {m} $为$ d $ - 集群倾斜子类别mod $ a $。从较高同源代数的角度来看，一个自然的问题是$ \ mathfrak {m} $何时在mod $ a $中诱导$ d $ - 集群倾斜子类别。在本文中，我们以更通用的形式调查了这个问题。令$ \ MATHCAL {M} $为ABELIAN类别的小$ D $ -D $ -ABELIAN类别$ \ Mathcal {a} $。 $ \ Mathcal {M} $的完成，用Ind $（\ Mathcal {M}）$表示，定义为相对于被过滤的Colimits的$ \ Mathcal {M} $的通用完成。我们探索Ind $（\ Mathcal {M}）$，并演示其与Mod $ \ Mathcal {M {M} $的完整子类别$ \ MathCal {l} _D（\ Mathcal {M}）$，包含左$ D $ D $ -Exact Fightors的等价。值得注意的是，虽然Ind $（\ Mathcal {M}）$作为$ \ frac {mod \ Mathcal {m}}} {frac {mathcal {m}} {fef（\ nathcal {m}）} $，满足$ d $ - 分类subcategory的所有属性，而不是$ d $ d $ d $ d $ - falls surpers a $ d $ d $ d。对于$ d $ - 集群的倾斜子类别$ \ mathfrak {m} $ a $ a $，$ \ coperightArrow {\ mathfrak {\ mathfrak {m}} $，由$ \ mathfrak {m Mathfrak {m} $的所有过滤对象组成，是一个生成的cogenering-cogenerate-cogenerational funcogentimentimentimenter-cogenerational funptimentimed mod $ mod $ $ a $ a $ a $ a $ a $ a。 $ \ mathfrak {m} $是否为$ d $ - rigid子类别的问题仍然没有解决。但是，如果确实是$ d $ rigid，则可以作为$ d $ - 集群倾斜子类别的资格。在使用CotorSion理论的情况下，我们确定$ \ oferrightArrow {\ mathfrak {m}} $是$ 2 $ - 集群倾斜子类别，并且仅当$ \ m athfrak {m} $是有限类型的。因此，关于$ \ oferrightArrow {\ mathfrak {m}} $是否是一个$ d $ - 集体倾斜子类别的mod $ a $的倾斜子类别似乎等同于iyama关于$ \ mathfrak {m mathfrak {m} $的有限的Qestion。
Let $A$ be a finite-dimensional algebra, and $\mathfrak{M}$ be a $d$-cluster tilting subcategory of mod$A$. From the viewpoint of higher homological algebra, a natural question to ask is when $\mathfrak{M}$ induces a $d$-cluster tilting subcategory in Mod$A$. In this paper, we investigate this question in a more general form. Let $\mathcal{M}$ be a small $d$-abelian category of an abelian category $\mathcal{A}$. The completion of $\mathcal{M}$, denoted by Ind$(\mathcal{M})$, is defined as the universal completion of $\mathcal{M}$ with respect to filtered colimits. We explore Ind$(\mathcal{M})$ and demonstrate its equivalence to the full subcategory $\mathcal{L}_d(\mathcal{M})$ of Mod$\mathcal{M}$, comprising left $d$-exact functors. Notably, while Ind$(\mathcal{M})$ as a subcategory of $\frac{Mod\mathcal{M}}{Eff(\mathcal{M})}$, satisfies all properties of a $d$-cluster tilting subcategory except $d$-rigidity, it falls short of being a $d$-cluster tilting category. For a $d$-cluster tilting subcategory $\mathfrak{M}$ of mod$A$, $\overrightarrow{\mathfrak{M}}$, consists of all filtered colimits of objects from $\mathfrak{M}$, is a generating-cogenerating, functorially finite subcategory of Mod$A$. The question of whether $\mathfrak{M}$ is a $d$-rigid subcategory remains unanswered. However, if it is indeed $d$-rigid, it qualifies as a $d$-cluster tilting subcategory. In the case $d=2$, employing cotorsion theory, we establish that $\overrightarrow{\mathfrak{M}}$ is a $2$-cluster tilting subcategory if and only if $\mathfrak{M}$ is of finite type. Thus, the question regarding whether $\overrightarrow{\mathfrak{M}}$ is a $d$-cluster tilting subcategory of Mod$ A$ appears to be equivalent to the Iyama's qestion about the finiteness of $\mathfrak{M}$.
