Coherent rings, fp-injective modules, dualizing complexes, and covariant Serre–Grothendieck duality
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 Št’ovíček 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 (Formula presented.) and (Formula presented.), 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 (Formula presented.), 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.