The simple \({\mathcal D}\)-module associated to the intersection homology complex for a class of plane curves
Let Y be a non singular affine algebraic variety over an algebraically closed field k and let \(X\subset Y\) be a hypersurface defined by an equation \(f\in {\mathcal O}(Y)\). One can consider \({\mathcal O}(Y)_ f={\mathcal O}(Y-X)/{\mathcal O}(X)\) as a left module over the ring \({\mathcal D}\) of algebraic linear differential operators on Y. This \({\mathcal D}\)-module has a finite length (Bernstein-Kashiwara) and contains a unique simple submodule L(Y,X) [\textit{J. L. Brylinski}, in Systèmes différentielles et singularités, Colloq. Luminy 1983, Astérisque 130, 260-271 (1985; Zbl 0568.14010)]. This \({\mathcal D}\)-module corresponds by the Riemann- Hilbert correspondence to the intersection cohomology of Goresky and MacPherson. In the paper under review the author proves that \(L(Y,X)={\mathcal O}(Y)_ f\) or equivalently that \({\mathcal O}(Y)_ f\) is a simple \({\mathcal D}\)-module, when \(Y=A\) 2 and X is an irreducible curve which is at each of its points analytically locally irreducible. This result was already known, for \(k={\mathbb{C}}\), as it can be proved by using the Riemann-Hilbert correspondence. However this paper gives a new proof which is purely algebraic. The author considers the ring \(A=R/f\) nR where \(R=k[x,y]\). The ring of differential operators over A can be defined, and one can prove the inclusion \(A\subset C\subset Fract(A)\), where \(C={\mathcal O}(\tilde X)\times k[z]/(z\) n), \(\tilde X\to^{\pi}X\) normalisation of X. The hypothesis means that \(\pi\) is 1-1 and one can deduce from this that \({\mathcal D}(A)\) and \({\mathcal D}(C)\) are equivalent in the sense of Morita and then that \({\mathcal D}(A)\) is simple and that A is a simple \({\mathcal D}(A)\)- module. Then the author proves that A is isomorphic as a \({\mathcal D}(A)\)- module to \(f^{-n}R/R\), and finally by a limit process gets the fact that \({\mathcal O}(A\) \(2)_ f\) is simple. The proofs are given with all details except for some references to another paper by the author and \textit{J. T. Stafford} [Proc. Lond. Math. Soc., III. Ser. 56, No.2, 229-259 (1988)].
scientific article; zbMATH DE number 4067082