Derived equivalences induced by big cotilting modules

Abstract

We prove that given a Grothendieck category G with a tilting object of finite projective dimension, the induced triangle equivalence sends an injective cogenerator of G to a big cotilting module. Moreover, every big cotilting module can be constructed like that in an essentially unique way. We also prove that the triangle equivalence is at the base of an equivalence of derivators, which in turn is induced by a Quillen equivalence with respect to suitable abelian model structures on the corresponding categories of complexes.

33 pages; version 2: more details added in Remark 1.11 and in the proofs of Lemmas 1.17 and 4.4, discussion of well-known or inessential results has been reduced (e.g. basics on model structures or the relation to co-t-structures), references have been updated

References in corpus (2)

• On exact categories and applications to triangulated adjoints and model structures
• Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves

Cited by in corpus (23)

• Torsion pairs in silting theory
• Silting objects
• The tilting-cotilting correspondence
• $t$-Structures on stable derivators and Grothendieck hearts
• Cotilting Sheaves on Noetherian Schemes
• $t$-Structures with Grothendieck hearts via functor categories
• Infinity-tilting theory
• Telescope conjecture for homotopically smashing t-structures over commutative noetherian rings
• Gorenstein homological algebra and universal coefficient theorems
• Tilting classes over commutative rings
• A functorial approach to monomorphism categories for species I
• The t-structure induced by an n-tilting module
• Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity
• Morita theory for stable derivators
• An explicit self-dual construction of complete cotorsion pairs in the relative context
• Preresolving categories and derived equivalences
• Recollements from Cotorsion Pairs
• Quasi-coherent torsion sheaves, the semiderived category, and the semitensor product: Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes
• Singular equivalences to locally coherent hearts of commutative noetherian rings
• Classifying subcategories, local algebra and the Rosenberg spectrum of a locally noetherian category
• Pseudo-dualizing complexes of bicomodules and pairs of t-structures
• Tilting modules and tilting torsion pairs - filtrations induced by tilting modules
• Roos axiom holds for quasi-coherent sheaves