Coherent rings, fp-injective modules, dualizing complexes, and covariant Serre-Grothendieck duality

Abstract

For a left coherent ring A with every left ideal having a countable set of generators, we show that the coderived category of left A-modules is compactly generated by the bounded derived category of finitely presented left A-modules (reproducing a particular case of a recent result of Stovicek with our methods). Furthermore, we present the definition of a dualizing complex of fp-injective modules over a pair of noncommutative coherent rings A and B, and construct an equivalence between the coderived category of A-modules and the contraderived category of B-modules. Finally, we define the notion of a relative dualizing complex of bimodules for a pair of noncommutative ring homomorphisms A \to R and B \to S, and obtain an equivalence between the R/A-semicoderived category of R-modules and the S/B-semicontraderived category of S-modules. For a homomorphism of commutative rings A\to R, we also construct a tensor structure on the R/A-semicoderived category of R-modules. A vision of semi-infinite algebraic geometry is discussed in the introduction.

LaTeX 2e with pb-diagram and xy-pic, 30 pages, 2 commutative diagrams; v.3: several misprints corrected, expositional improvement in Section 2; v.4: examples added in the introduction and in Sections 3 and 5; v.5: more misprints corrected, new Section 6 added; v.6: exposition improved in Section 6 -- this is intended as the final version

References in corpus (3)

• Acyclicity versus total acyclicity for complexes over noetherian rings
• On purity and applications to coderived and singularity categories
• Dedualizing complexes and MGM duality

Cited by in corpus (12)

• Dedualizing complexes and MGM duality
• Dedualizing complexes of bicomodules and MGM duality over coalgebras
• Infinity-tilting theory
• Contramodules
• Fp-projective periodicity
• Relative nonhomogeneous Koszul duality
• An explicit self-dual construction of complete cotorsion pairs in the relative context
• Koszul duality for compactly generated derived categories of second kind
• Quasi-coherent torsion sheaves, the semiderived category, and the semitensor product: Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes
• Pseudo-dualizing complexes of bicomodules and pairs of t-structures
• Resolutions as directed colimits
• Contraderived categories of CDG-modules