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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].

MSC:
 32A27 Residues for several complex variables 32C36 Local cohomology of analytic spaces 32C38 Sheaves of differential operators and their modules, $$D$$-modules
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