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

TL;DR

This paper explores duality theories for modules over noncommutative coherent rings, establishing equivalences between various derived categories and introducing new concepts like dualizing complexes of fp-injective modules.

Contribution

It defines dualizing complexes of fp-injective modules over noncommutative rings and constructs equivalences between coderived and contraderived categories, extending duality theories.

Findings

Coderived category of modules is compactly generated by finitely presented modules.

Introduces dualizing complexes for fp-injective modules over noncommutative rings.

Establishes equivalences between different derived categories and defines tensor structures.

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.