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For a left coherent ring A with every left ideal having a countable set ofgenerators, we show that the coderived category of left A-modules is compactlygenerated by the bounded derived category of finitely presented left A-modules(reproducing a particular case of a recent result of Stovicek with ourmethods). Furthermore, we present the definition of a dualizing complex offp-injective modules over a pair of noncommutative coherent rings A and B, andconstruct an equivalence between the coderived category of A-modules and thecontraderived category of B-modules. Finally, we define the notion of arelative dualizing complex of bimodules for a pair of noncommutative ringhomomorphisms A \to R and B \to S, and obtain an equivalence between theR/A-semicoderived category of R-modules and the S/B-semicontraderived categoryof S-modules. For a homomorphism of commutative rings A\to R, we also constructa tensor structure on the R/A-semicoderived category of R-modules. A vision ofsemi-infinite algebraic geometry is discussed in the introduction.

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