Infinity-tilting theory

Abstract

We define the notion of an infinitely generated tilting object of infinite homological dimension in an abelian category. A one-to-one correspondence between $\infty$-tilting objects in complete, cocomplete abelian categories with an injective cogenerator and $\infty$-cotilting objects in complete, cocomplete abelian categories with a projective generator is constructed. We also introduce $\infty$-tilting pairs, consisting of an $\infty$-tilting object and its $\infty$-tilting class, and obtain a bijective correspondence between $\infty$-tilting and $\infty$-cotilting pairs. Finally, we discuss the related derived equivalences and t-structures.

LaTeX 2e with pb-diagram and xy-pic, 34 pages, 4 figures; v.3: minor corrections, references updated

References in corpus (5)

• On purity and applications to coderived and singularity categories
• Classifying exact categories via Wakamatsu tilting
• Flat morphisms of finite presentation are very flat
• Abelian right perpendicular subcategories in module categories
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Cited by in corpus (10)

• Semi-infinite highest weight categories
• Contramodules over pro-perfect topological rings
• Derived, coderived, and contraderived categories of locally presentable abelian categories
• Abelian right perpendicular subcategories in module categories
• Covers and direct limits: a contramodule-based approach
• Differential graded Koszul duality: an introductory survey
• Tilting complexes and codimension functions over commutative noetherian rings
• Pseudo-dualizing complexes of bicomodules and pairs of t-structures
• Roos axiom holds for quasi-coherent sheaves
• Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors