论文标题
由简单智力系统引起的三角形类别的亚伯式子类别
Abelian subcategories of triangulated categories induced by simple minded systems
论文作者
论文摘要
如果$ k $是一个字段，$ a $ a $ a有限尺寸$ k $ -algebra，那么简单的$ a $ a $ modules在派生的类别$ \ operatorname {d}^b（\ operatatorNameAme {mod} a）$中形成一个简单的思想收藏。他们的扩展关闭是$ \ operatatorName {mod} a $;特别是，它是阿贝里安。在合适的三角类别$ \ mathcal {c} $中，由一般简单的思想集合$ \ MATHCAL {s} $模拟了这种情况。特别是，扩展封闭$ \ langle \ mathcal {s} \ rangle $是Abelian，并且对于$ \ Mathcal {C} $的Abelian子类别有一个倾斜理论。这些语句从$ \ langle \ mathcal {s} \ rangle $是有限的$ t $结构的核心。 这是简单志趣相投收藏的定义特征，其负面的自我扩展在各个程度上都消失了。将其放松到以$ \ {-w+1，\ ldots，-1 \} $中消失的放松，其中$ w $是一个积极的整数，导致了$ w $简化的概念的丰富，平行的概念，这些概念是最近成为有力兴趣的主题。 如果$ \ MATHCAL {S} $是$ W $ -SIMPLE MAINDED SYSTEM，对于某些$ W \ geqslant 2 $，则$ \ langle \ Mathcal {s} \ rangle $通常不是$ t $结构的核心。但是，使用不同的方法，我们将证明$ \ langle \ Mathcal {s} \ rangle $是Abelian，并且对于此类Abelian子类别有一个倾斜理论。我们的理论基于Quillen的确切类别的概念，特别是Dyer的定理，它提供了三角形类别的确切子类别。 简单的智力系统理论可以看作是“负集群倾斜理论”。特别是，$ \ langle \ Mathcal {s} \ rangle $的结果是一个Abelian子类别，与（较高的）正聚类倾斜理论的结果是负面的，即如果$ \ Mathcal {t} $是群集的倾斜子类别，则是$（\ nathcal），则$（\ Mathcal {\ nathcal {t}} \ Mathcal {T}] $是Abelian商类别。
If $k$ is a field, $A$ a finite dimensional $k$-algebra, then the simple $A$-modules form a simple minded collection in the derived category $\operatorname{D}^b( \operatorname{mod} A )$. Their extension closure is $\operatorname{mod} A$; in particular, it is abelian. This situation is emulated by a general simple minded collection $\mathcal{S}$ in a suitable triangulated category $\mathcal{C}$. In particular, the extension closure $\langle \mathcal{S} \rangle$ is abelian, and there is a tilting theory for such abelian subcategories of $\mathcal{C}$. These statements follow from $\langle \mathcal{S} \rangle$ being the heart of a bounded $t$-structure. It is a defining characteristic of simple minded collections that their negative self extensions vanish in every degree. Relaxing this to vanishing in degrees $\{ -w+1, \ldots, -1 \}$ where $w$ is a positive integer leads to the rich, parallel notion of $w$-simple minded systems, which have recently been the subject of vigorous interest. If $\mathcal{S}$ is a $w$-simple minded system for some $w \geqslant 2$, then $\langle \mathcal{S} \rangle$ is typically not the heart of a $t$-structure. Nevertheless, using different methods, we will prove that $\langle \mathcal{S} \rangle$ is abelian and that there is a tilting theory for such abelian subcategories. Our theory is based on Quillen's notion of exact categories, in particular a theorem by Dyer which provides exact subcategories of triangulated categories. The theory of simple minded systems can be viewed as "negative cluster tilting theory". In particular, the result that $\langle \mathcal{S} \rangle$ is an abelian subcategory is a negative counterpart to the result from (higher) positive cluster tilting theory that if $\mathcal{T}$ is a cluster tilting subcategory, then $( \mathcal{T} * Σ\mathcal{T} )/[ \mathcal{T} ]$ is an abelian quotient category.
