发表机构
Kyushu University（九州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究dg范畴的截断，证明其导出范畴是Auslander范畴的三角类比，并引入连通消解概念，涵盖非交换消解与簇倾斜对象，给出存在性充分条件及显式描述。
AI 中文摘要
我们研究dg范畴的截断$(-)^{\leq0}$。首先我们证明，给定一个具有平移的dg范畴$\mathscr{C}$，例如预三角dg范畴，典范函子$\mathscr{C}^{\leq0}\to\mathscr{C}$是一个局部化，其核由第0上同调紧致生成。接下来我们证明，截断的导出范畴可作为阿贝尔范畴上相干函子的Auslander范畴的三角类比。我们利用截断的逆对偶双模给出Auslander-Reiten-Serre对偶的描述。基于截断，我们引入dg范畴的连通消解的概念，定义为来自连通dg范畴的局部化函子。这一概念涵盖了模范畴中代数的非交换消解、三角范畴中的簇倾斜对象，以及作为万有连通消解的截断本身。我们给出三角范畴中一个对象为其增强给出连通消解的充分条件，该条件以消解维数的有限性表述。进一步，我们证明连通dg代数上的每个真dg模都是给出连通消解的dg模的直和项。因此，对于每个真连通dg代数，其有界dg导出范畴具有连通消解。最后，我们对某些范畴的截断给出显式描述，包括Dynkin箭图的导出范畴和簇范畴，以及有限维代数的Yoneda范畴。
英文摘要
We study the truncation $(-)^{\leq0}$ of dg categories. We first show that given a dg category $\mathscr{C}$ with shifts, for example a pretriangulated dg category, the canonical functor $\mathscr{C}^{\leq0}\to\mathscr{C}$ is a localization whose kernel is compactly generated by the $0$-th cohomology. Next we demonstrate that the derived category of the truncation serves as a triangulated analogue of the Auslander's category of coherent functors over abelian categories. We give a description of the Auslander-Reiten-Serre duality in terms of the inverse dualizing bimodule of the truncation. Building on truncations, we introduce the notion of connective resolutions of dg categories, defined as a localization functor from a connective dg category. This notion encompasses non-commutative resolutions of algebras in module categories, cluster tilting objects in triangulated categories, and also the truncation as the universal connective resolution. We give a sufficient condition for an object in a triangulated category to give a connective resolution of its enhancement in terms of finiteness of resolution dimension. Furthermore, we show that every proper dg module over a connective dg algebra is a direct summand of a dg module giving a connective resolution. Consequently, for every proper connective dg algebra, its bounded dg derived category has a connective resolution. Finally, we give some explicit description of the truncation for some categories including the derived and cluster categories of a Dynkin quiver, and the Yoneda category of a finite dimensional algebra.
Comments34 pages
