Coherent rings, fp-injective modules, dualizing complexes, and covariant Serre-Grothendieck duality. (English) Zbl 1394.16005
Summary: 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 J. Št’ovíček [“On purity and applications to coderived and singularity categories”, Preprint, arXiv:1412.1615] 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 \(A\longrightarrow R\) and \(B\longrightarrow S\), 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 \(A\longrightarrow R\), 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.
MSC:
| 16E35 | Derived categories and associative algebras |
| 16D90 | Module categories in associative algebras |
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 13D09 | Derived categories and commutative rings |
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