再余裂的提升与部分倾斜集的粘合
Lifting of recollements and gluing of partial silting sets
AI总结：
本文研究三角范畴中再余裂的提升与倾斜集的粘合，提出提升判据并给出Artin代数情形下粘合经典倾斜对象的显式构造。
摘要：
本文关注三角范畴中的再余裂与倾斜理论，包含两个主要部分。在第一部分中，证明了三角子范畴的再余裂提升为具有余积的环绕三角范畴的挠挠自由三元组（TTF三元组）的一个判据。作为推论，对于重复代数的稳定范畴或自内射有限长度代数的稳定范畴的再余裂，以及分离诺特概形的有界导出范畴的再余裂，TTF三元组的提升是可能的。此外，当再余裂中的外部子范畴是小线性范畴的导出范畴时，判据中的条件足以将再余裂提升为环绕三角范畴的再余裂，直至等价。在第二部分中，我们利用这些结果研究在再余裂的中心范畴中构造倾斜集的问题，该倾斜集生成由外部范畴中的倾斜t-结构粘合而成的t-结构。对于Artin代数的有界导出范畴的再余裂情形，我们提供了粘合经典倾斜对象的显式构造。
英文摘要：
This paper focuses on recollements and silting theory in triangulated categories. It consists of two main parts. In the first part a criterion for a recollement of triangulated subcategories to lift to a torsion torsion-free triple (TTF triple) of ambient triangulated categories with coproducts is proved. As a consequence, lifting of TTF triples is possible for recollements of stable categories of repetitive algebras or self-injective finite length algebras and recollements of bounded derived categories of separated Noetherian schemes. When, in addition, the outer subcategories in the recollement are derived categories of small linear categories the conditions from the criterion are sufficient to lift the recollement to a recollement of ambient triangulated categories up to equivalence. In the second part we use these results to study the problem of constructing silting sets in the central category of a recollement generating the t-structure glued from the silting t-structures in the outer categories. In the case of a recollement of bounded derived categories of Artin algebras we provide an explicit construction for gluing classical silting objects.
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