Silting theory in triangulated categories with coproducts
Silting theory in triangulated categories with coproducts
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DOI：
10.1016/j.jpaa.2018.07.016
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发表时间：
2015-12
影响因子：
0.8
通讯作者：
Pedro Nicolás;M. Saorín;A. Zvonareva
中科院分区：
文献类型：
--
作者：
Pedro Nicolás;M. Saorín;A. Zvonareva
我们在任意(集合索引)余积的三角范畴D中引入了非紧(部分)淤积和(部分)倾斜集合和对象的概念。我们证明了部分淤积集的等价类与其余心生成的t-结构是双射的，并且在D是紧生成的情况下，这限制为：i)自小部分淤积对象的等价类与D中的左非退化t-结构之间的双射，这些t-结构的心是模范畴，并且其相应的上同调函数保积；ii)经典淤积对象的等价类与非退化粉碎和余碎t-结构之间的双射。我们将与部分淤积集T相关的t-结构过道中的对象描述为Sum(T)[n]中具有连续锥的态射序列的Milnor(或同伦)同限体。利用这一事实，我们在非常一般的AB3阿贝尔范畴中发展了一个倾斜对象的理论，一个集合及其对偶，其中我们证明了模的倾斜和余倾理论的几个著名结果的有效性。最后，我们证明了如果T是紧生成的代数三角剖分范畴D中的有界倾斜集，并且H是相关t-结构的中心，则包含H↪D扩展到限制到有界水平的三角等价D(H)⟶∼D。
We introduce the notion of noncompact (partial) silting and (partial) tilting sets and objects in any triangulated category D with arbitrary (set-indexed) coproducts. We show that equivalence classes of partial silting sets are in bijection with t-structures generated by their co-heart whose heart has a generator, and in case D is compactly generated, this restricts to: i) a bijection between equivalence classes of self-small partial silting objects and left nondegenerate t-structures in D whose heart is a module category and whose associated cohomological functor preserves products; ii) a bijection between equivalence classes of classical silting objects and nondegenerate smashing and co-smashing t-structures whose heart is a module category. We describe the objects in the aisle of the t-structure associated to a partial silting set T as Milnor (or homotopy) colimits of sequences of morphisms with successive cones in Sum (T)[n]. We use this fact to develop a theory of tilting objects in very general AB3 abelian categories, a setting and its dual in which we show the validity of several well-known results of tilting and cotilting theory of modules. Finally, we show that if T is a bounded tilting set in a compactly generated algebraic triangulated category D and H is the heart of the associated t-structure, then the inclusion H↪ D extends to a triangulated equivalence D (H)⟶∼ D which restricts to bounded levels.