Residues and duality for singularity categories of isolated Gorenstein singularities (Q2873599)

In the very clearly written paper under review, the author considers a local Gorenstein ring \(R\) of Krull dimension \(d\) with an isolated singularity. The main result is an explicit formula for the Auslander duality pairing [\textit{M. Auslander}, in: Represent. Theory of Algebras, Proc. Phila. Conf., Lect. Notes pure appl. Math. 37, 1--244 (1978; Zbl 0383.16015)] in the singularity category \(\mathbf{D}_{sg}(R) = \mathbf{D}^b(R)/\mathrm{Perf}(R)\). The singularity category is of considerable interest due to connections with string theory, homological mirror symmetry, singularity theory, and representation theory.NEWLINENEWLINEOne model for \(\mathbf{D}_{sg}(R)\) is the homotopy category of acyclic complexes of finite free \(R\)-modules. The explicit formula is expressed in terms of the differential of such a complex, a certain collection of homotopies related to the action of \(R\) on the complex, and the generalized fraction formalism. The main technical tools in the paper are the notions of complete projective and complete injective resolutions of an \(R\)-module \(M\) and an explicit comparison morphism between \(M\) and a complete injective resolution of \(M\) constructed from a complete projective resolution.NEWLINENEWLINEThe case of an analytic hypersurface ring is an important special case of the theory. In this case, the category of singularities can be identified with a homotopy category of matrix factorizations. Via a string theoretic analysis, \textit{A. Kapustin} and \textit{Y. Li} [Adv. Theor. Math. Phys. 7, No. 4, 727--749 (2003; Zbl 1058.81061)] conjectured a formula for Serre duality in homotopy categories of matrix factorizations. There has been considerable interest in whether or not their formula defined a non-degenerate pairing and thus implemented Serre duality. This paper settles the matter by specializing the author's general explicit formula for Auslander duality to the Kapustin-Li pairing in the case of a hypersurface.