zbMATH — the first resource for mathematics

Computational aspects of Grothendieck local residues. (English) Zbl 1089.32002
Brasselet, Jean-Paul (ed.) et al., Franco-Japanese singularities. Proceedings of the 2nd Franco-Japanese singularity conference, CIRM, Marseille-Luminy, France, September 9–13, 2002. Paris: Société Mathématique de France (ISBN 2-85629-166-X/pbk). Séminaires et Congrès 10, 287-305 (2005).
The authors derive an algorithm for the computation of Grothendieck’s residue, by studying the local residues with methods of algebraic analysis.
More precisely, let \(X\) be \(\mathbb{C}^n\), \(\mathcal{F}\) a regular sequence given by the holomorphic functions \(f_1, \ldots, f_n\) in \(X\), \(\mathcal{I}\) the ideal generated by the \(f_j\), \(V(J) = Z\) the zero-dimensional variety defined by the ideal \(\mathcal{I}, v : \text{Ext}^n_{\mathcal{O}_X} (\mathcal{O}_X/ \mathcal{I}, \Omega_X^n) \to \mathcal{H}_{[Z]}^n (\Omega_X^n)\) the canonical mapping, where \(\mathcal{H}_{[Z]}^n(\Omega_X^n)\) is the local cohomology with support on \(Z\), \(\omega_{\mathcal{F}} = [\frac{dz}{f_1 \ldots f_n}]\) the image by \(v\) of the Grothendieck symbol \([{{dz}\atop {f_1 \ldots f_n}}]\) and \(\omega_{\mathcal{F}, \beta}\) the germ at \(p \in Z\) of the algebraic cohomology class \(\omega_{\mathcal{F}} \). If Res \(\beta\) denotes the local residue map, the map \(\mathcal{H}_{\{\beta\}}(\Omega_{\overline X}^n) \times \mathcal{O}_{X, p} \to \chi_{\{\beta\}}(\Omega_X^n)\) composed with the local residue map, defines a natural pairing between the topological vector spaces \(\mathcal{H}_{\{\beta\}}^n(\Omega_X^n)\) and \(\mathcal{O}_{X, \beta}\) and thus \(\omega_{\mathcal{F}, \beta}\) induces a linear functional \(\text{Res}_\beta(\omega_{\mathcal{F}})\) (defined to be \(\text{Res}_\beta(\varphi(z) \omega_{\mathcal{F}, \beta})\) for \(\varphi(z) \in \mathcal{O}_{X, \beta}\), \(\beta \in Z\).
The authors give a description of the kernel of this functional, in terms of partial differential operators, using result of M. Kashiwara [Publ. Res. Inst. Math. Sci., Kyoto Univ. 10, 563–579 (1975; Zbl 0313.58019)] and M. Kashiwara and T. Kawai [Publ. Res. Inst. Math. Sci. 17, 813–979 (1981; Zbl 0505.58033)]. In particular, the holonomic system \(\mathcal{D}_X/ \text{Ann}_{\mathcal{D}_X}(\omega_{\mathcal{F}}\)) completely characterizes the algebraic local cohomology class \(\omega_{\mathcal{F}}\) as its solution. Some examples are given.
In §3 the authors devise a method for computing the local residues, by using a decomposition of \(\omega_{\mathcal{F}}\) into a direct sum \(\omega_{\mathcal{F}} = \omega_1 + \ldots + \omega_l\) and using the linear functional Res\((\omega_{\mathcal{F}})\). In §4 the authors give an algorithm for computing residues with first order differential operators, using a result proved in a yet unpublished preprint.
Finally they give an algorithm \(R\) for the computation of the Grothendieck local residue which, uses the fact that the first order annulator \(\text{Ann}_{A_n}^{(1)}(\omega_\lambda)\) is equal with \(\text{Ann}_{A_n}(\omega_\lambda)\), (\(A_n\) denotes the Weyl algebra), and an algorithm \(A\) (for the construction of first order annihilators).
For the entire collection see [Zbl 1061.14001].

32A27 Residues for several complex variables
32C36 Local cohomology of analytic spaces
32C38 Sheaves of differential operators and their modules, \(D\)-modules
Full Text: Link