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A local Grothendieck duality for Cohen-Macaulay ideals. (English) Zbl 1276.32002

In an important paper [Ann. Sci. Éc. Norm. Supér. (4) 40, No. 6, 985–1007 (2007; Zbl 1143.32003)] M. Andersson and E. Wulcan constructed for any coherent sheaf of ideals \(\mathcal{J}\) locally a vector-valued residue current \(R\) whose annihilator is precisely \(\mathcal{J}\). If \(\mathcal{J}\) is a complete intersection, then \(R\) is just the classical Coleff-Herrera product and the statement comes down to the classical Grothendieck duality theorem (see also [A. Dickenstein and C. Sessa, Invent. Math. 80, 417–434 (1985; Zbl 0556.32005)] and [M. Passare, Math. Scand. 62, No. 1, 75–152 (1988; Zbl 0633.32005)].
In the paper under review, Lundqvist gives a new proof of the result of Andersson and Wulcan in case \(\mathcal{J}\) is Cohen-Macaulay. This proof has the advantage that it does not require Hironaka’s resolution of singularities and provides a semi-explicit realization of the residue annihilating \(\mathcal{J}\).

MSC:

32A10 Holomorphic functions of several complex variables
32A27 Residues for several complex variables
32C30 Integration on analytic sets and spaces, currents
32C37 Duality theorems for analytic spaces
32A60 Zero sets of holomorphic functions of several complex variables
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