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Higher relative primitive ideals


Authors: Guangfeng Jiang and Aron Simis
Journal: Proc. Amer. Math. Soc. 129 (2001), 647-655
MSC (2000): Primary 13N15, 14B05; Secondary 13N10, 13P99, 16S32, 32S05
DOI: https://doi.org/10.1090/S0002-9939-00-05597-0
Published electronically: September 19, 2000
MathSciNet review: 1707149
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Abstract: The main object of this note is to introduce a higher order analog of the so-called primitive ideal of $Y$ relative to $X$ introduced by Jiang-Pellikaan-Siersma, where $X\supset Y$ are germs of analytic subspaces of $(\mathbb{C}^n,0)$. Our treatment of the problem is ideal-theoretic throughout, using the notion of iterated higher differential operators. Some examples from singularity theory are worked out. We establish the connection between higher primitive ideals and (relative) symbolic powers of an ideal and give an effective algorithm to compute both.


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  • 1. A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361 (French). MR 0238860
  • 2. G. Jiang, Functions with non-isolated singularities on singular spaces, Thesis, Universiteit Utrecht, 1997.
  • 3. G. Jiang, D. Siersma, Local embeddings of lines in singular hypersurfaces, Ann. Inst. Fourier, Grenoble 49, 4 (1999), 1129-1147.
  • 4. J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987. With the cooperation of L. W. Small; A Wiley-Interscience Publication. MR 934572
  • 5. Yoshikazu Nakai, High order derivations. I, Osaka J. Math. 7 (1970), 1–27. MR 0263804
  • 6. R. Pellikaan, Hypersurface singularities and resolutions of Jacobi modules, Thesis, Rijkuniversiteit Utrecht, 1985.
  • 7. Ruud Pellikaan, Finite determinacy of functions with nonisolated singularities, Proc. London Math. Soc. (3) 57 (1988), no. 2, 357–382. MR 950595, https://doi.org/10.1112/plms/s3-57.2.357
  • 8. Kyoji Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 265–291. MR 586450
  • 9. Peter Seibt, Differential filtrations and symbolic powers of regular primes, Math. Z. 166 (1979), no. 2, 159–164. MR 525620, https://doi.org/10.1007/BF01214042
  • 10. Wolmer V. Vasconcelos, Computational methods in commutative algebra and algebraic geometry, Algorithms and Computation in Mathematics, vol. 2, Springer-Verlag, Berlin, 1998. With chapters by David Eisenbud, Daniel R. Grayson, Jürgen Herzog and Michael Stillman. MR 1484973

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Additional Information

Guangfeng Jiang
Affiliation: Department of Mathematics, Jinzhou Normal University, Jinzhou City, Liaoning 121000, People’s Republic of China

Aron Simis
Affiliation: Departamento de Matemática, CCEN, Universidade Federal de Pernambuco, Av. Prof. Luis Freire, 50740-540 Recife, PE, Brazil
Email: aron@dmat.ufpe.br

DOI: https://doi.org/10.1090/S0002-9939-00-05597-0
Keywords: Primitive ideal, symbolic power, differential operator, Jacobian matrix, surface singularity
Received by editor(s): March 18, 1999
Received by editor(s) in revised form: May 12, 1999
Published electronically: September 19, 2000
Additional Notes: The first author was supported by JSPS: P98028, and the second author was partially supported by CNPq, Brazil.
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2000 American Mathematical Society