Residue complexes over noncommutative rings.
Grothendieck's concept of a residue complex is generalized to noncommutative rings, where it becomes a dualizing complex of injective bimodules (in the sense of the first author [\textit{A. Yekutieli}, J. Algebra 153, No. 1, 41-84 (1992; Zbl 0790.18005)]) which is rigid in the sense of \textit{M. Van den Bergh} [J. Algebra 195, No. 2, 662-679 (1997; Zbl 0894.16020)] and satisfies the Auslander conditions and related purity conditions. The authors prove uniqueness, existence, and functoriality of residue complexes under appropriate hypotheses. (As they warn, however, algebras such as \(U({\mathfrak{sl}}_2(\mathbb{C}))\) which are ``too noncommutative'' may not have residue complexes.) In particular, if \(A\) is an affine Noetherian PI algebra (over a field) admitting a Noetherian connected filtration, they prove that \(A\) has a residue complex \({\mathcal K}_A\), necessarily unique up to unique isomorphism in \(\text{D}(\text{Mod\,}A\otimes A^{\text{op}})\), and they describe its structure in detail. For instance, \({\mathcal K}_A^{-q}\) is a direct sum of indecomposable bimodules \(\Gamma_Z{\mathcal K}_A^{-q}\) supported on those cliques \(Z\subseteq\text{Spec\,}A\) with \(j_{{\mathcal K}_A}(A/{\mathfrak p})=-q\) for \({\mathfrak p}\in Z\), and as a left module each \(\Gamma_Z{\mathcal K}_A^{-q}\) is the direct sum over \({\mathfrak p}\in Z\) of \(\text{Goldie-rank}(A/{\mathfrak p})\) copies of the unique indecomposable injective with associated prime \(\mathfrak p\).
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