Auslander–Buchweitz Context and Co-t-structures

• Published in: Applied categorical structures 2013-10, Vol.21 (5), p.417-440
• Main Authors: Mendoza Hernández, Octavio, Sáenz Valadez, Edith Corina, Santiago Vargas, Valente, Souto Salorio, María José
• Format: Article
• Language: English
• Subjects: Convex and Discrete Geometry; Geometry; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Theory of Computation
• ISSN: 0927-2852; 1572-9095
• DOI: 10.1007/s10485-011-9271-2

We show that the relative Auslander–Buchweitz context on a triangulated category coincides with the notion of co- t -structure on certain triangulated subcategory of (see Theorem 3.8). In the Krull–Schmidt case, we establish a bijective correspondence between co- t -structures and cosuspended, precovering subcategories (see Theorem 3.11). We also give a characterization of bounded co- t -structures in terms of relative homological algebra. The relationship between silting classes and co- t -structures is also studied. We prove that a silting class ω induces a bounded non-degenerated co- t -structure on the smallest thick triangulated subcategory of containing ω . We also give a description of the bounded co- t -structures on (see Theorem 5.10). Finally, as an application to the particular case of the bounded derived category where is an abelian hereditary category which is Hom-finite, Ext-finite and has a tilting object (see Happel and Reiten, Math Z 232:559–588, 1999 ), we give a bijective correspondence between finite silting generator sets ω  =  add ( ω ) and bounded co- t -structures (see Theorem 6.7).