×

Direct methods for primary decomposition. (English) Zbl 0770.13018

Let \(I\) be an ideal in a polynomial ring \(S=k[x_ 1,\dots,x_ n]\) over a field \(k\). This paper gives new methods for computing the equidimensional parts of \(I\); the radical of \(I\), the localization of \(I\) at an ideal \(J\), and the primary decomposition of \(I\). These methods are based on ideas of modern commutative algebra and avoid the use of generic projections used by Hermann (1926) and all others. These techniques extend to arbitrary ideal operations previously only possible for principal ideals. Most of these results are stated for modules and it is usually assumed that \(k\) is a perfect field. Explicit algorithms are given for solving these problems computationally and they have been implemented in the computer algebra system Macaulay of Bayer and Stillman.

MSC:

13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13A05 Divisibility and factorizations in commutative rings
13A15 Ideals and multiplicative ideal theory in commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13A10 Radical theory on commutative rings (MSC2000)

Software:

Macaulay2
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Avramov, L.: Homology of local flat extensions and complete intersection defects. Math. Ann.228, 27-37 (1977) · Zbl 0345.13018 · doi:10.1007/BF01360771
[2] Bayer, D.: The division algorithm and the Hilbert scheme. Thesis, Harvard University, 1982. Order number 82-22588, Univ. Microfilms Intl., Ann Arbor Michigan (1982)
[3] Bayer, D., Galligo, A., Stillman, M.: Computing primary decompositions (in preparation)
[4] Bayer, D., Stillman, M.: Macaulay: A system for computation in algebraic geometry and commutative algebra. Source and object code available for Unix and Macintosh computers. Contact the authors, or download from zariski. harvard. edu via anonymous ftp. (login: anonymous, password: any, cd Macaulay) (1982-1990)
[5] Bayer, D., Stillman, M.: A criterion for detectingm-regularity. Invent. Math.87, 1-11 (1987) · Zbl 0625.13003 · doi:10.1007/BF01389151
[6] Bayer, D., Stillman, M.: Computation of Hilbert functions. J. Symb. Comput.14, 31-50 (1992) · Zbl 0763.13007 · doi:10.1016/0747-7171(92)90024-X
[7] Bayer, D., Mumford, D.: What can be computed in algebraic geometry? In: Eisenbud, D., Robbiano, L. (eds.), Proceedings of the Cortona conference on computational algebraic geometry Cambridge: Cambridge University Press 1993 · Zbl 0846.13017
[8] Bertram, A., Ein L., Lazarsfeld R.: Vanishing theorems, a theorem of Severi, and the equations defining projective varieties, (preprint) · Zbl 0762.14012
[9] Brennan, J.P., Vasconcelos, W.: Effective computation of the integral closure of a morphism. J. Pure Appl. Alg. (to appear) · Zbl 0830.13018
[10] Buchsbaum, D.A., Eisenbud, D.: What makes a complex exact? J. Algebra25, 259-268 (1973) · Zbl 0264.13007 · doi:10.1016/0021-8693(73)90044-6
[11] Buchsbaum, D.A., Eisenbud, D.: Some structure theorems for finite free resolutions. Adv. Math.12, 84-139 (1974) · Zbl 0297.13014 · doi:10.1016/S0001-8708(74)80019-8
[12] Buchsbaum, D.A., Eisenbud, D.: What annihilates a module. J. Algebra47, 231-243 (1977) · Zbl 0372.13002 · doi:10.1016/0021-8693(77)90223-X
[13] Cox, D., Little, J., O’Shea, D.: Ideals, varieties and algorithms. Berlin Heidelberg New York: Springer 1992
[14] Eisenbud, D.: Commutative algebra with a view toward algebraic geometry. (Brandeis Lect. Notes. 1989) · Zbl 0819.13001
[15] Eisenbud, D., Levine, H.: An algebraic formula for the degree of aC ? map germ. Ann. Math.106, 19-44 (1977) · Zbl 0398.57020 · doi:10.2307/1971156
[16] Eisenbud, D., Stillman, M.: Methods in comp algebraic geometry and commutative algebra (in preparation)
[17] Eisenbud, D., Sturmfels, B.: Finding sparse systems of parameters. (in preparation) · Zbl 0807.13012
[18] Gianni, P., Trager, B., Zacharias, G.: Gröbner bases and primary decomposition of polynomials ideals. J. Symb. Comput6, 149-167 (1988) · Zbl 0667.13008 · doi:10.1016/S0747-7171(88)80040-3
[19] Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique IV. Publ. Math., Inst. Hautes Étud. Sci.32 (1967)
[20] Gruson, L., Lazarsfeld, R., Peskine, C.: On a theorem of Castelnuovo and the equations defining space curves. Invent Math.72, 491-506 (1983) · Zbl 0565.14014 · doi:10.1007/BF01398398
[21] Hartshorne, R.: Algebraic geometry. Berlin Heidelberg New York: Springer 1977 · Zbl 0367.14001
[22] Hermann, G.: Die Frage der endlich vielen Schritte in der Theorie der Polynomideale. Math. Ann.95, 736-788 (1926) · JFM 52.0127.01 · doi:10.1007/BF01206635
[23] Hilbert, D.: Über die Theorie der algebraischer Formen. Math. Ann.36, 473-534 (1890) · JFM 22.0133.01 · doi:10.1007/BF01208503
[24] Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic 0. Ann. Math.79, 205-326 (1964) · Zbl 0122.38603 · doi:10.2307/1970547
[25] Hochster, M.: Symbolic powers in Noetherian domains. Ill. J. Math.15, 9-27 (1971) · Zbl 0211.06701
[26] Kaplansky, I.: Commutative Rings. Boston: Allyn and Bacon 1970 · Zbl 0203.34601
[27] Knuth, D.: The art of computer programming vol. 2: Seminumerical algorithms. Reading: Addison-Wesley 1971 · Zbl 0191.18001
[28] Krick, T., Logar, A.: An algorithm for the computation of the radical of an ideal in the ring of polynomials. In: Mattson, H.F. et al. (eds.) Proceedings 9th AAEEC. (Lect. Notes Comput. Sci., vol. 539, pp. 195-205) Berlin Heidelberg New York: Springer 1991 · Zbl 0823.13018
[29] Kunz, E.: Kähler Differentials. Wiesbaden: Viehweg 1986 · Zbl 0587.13014
[30] Lazard, D.: Ideal bases and primary decomposition: case of two variables. J. Symb. Comput. 261-270 (1985) · Zbl 0616.68036
[31] Lazard, D.: Commutative algebra and computer algebra. (Lect. Notes Comput. Sci., vol. 144, pp. 40-48) Berlin Heidelberg New York: Springer 1982 · Zbl 0552.68047
[32] Lazard, D.: Solving zero-dimensional algebraic systems. J. Symb. Comput. (to appear) · Zbl 0753.13012
[33] Lazarsfeld, R.: A sharp Castelnuovo bound for smooth surfaces. Duke Math. J.55, 423-429 (1987) · Zbl 0646.14005 · doi:10.1215/S0012-7094-87-05523-2
[34] Matsumura, H.: Commutative algebra. New York: Benjamin 1970 · Zbl 0211.06501
[35] Matsumura, H.: Commutative ring theory. Cambridge: Cambridge University Press 1986 · Zbl 0603.13001
[36] Mumford, D.: Varieties defined by quadratic equations. In: Proceedings, of the conference at the Centro Int. Mat. Estivo (CIME). Varenna 1969. Rome: Cremonese 1970 · Zbl 0169.23301
[37] Nagata, M.: Local rings. New York: Interscience 1962 · Zbl 0123.03402
[38] Northcott, D.G.: A homological investigation of a certain residual ideal. Math. Ann.150, 99-110 (1963) · Zbl 0112.02904 · doi:10.1007/BF01396585
[39] Peskine, C., Szpiro, L.: Liaison des variétés algébriques I. Invent. Math.26, 271-302 (1974) · Zbl 0298.14022 · doi:10.1007/BF01425554
[40] Vasconcelos, W.: Computing the integral closure of an affine domain. Proc. Am. Math. Soc.113, 633-638 (1991) · Zbl 0739.13014 · doi:10.1090/S0002-9939-1991-1055780-6
[41] Scheja, G., Storch, U.: Über Spurfunktionen bei vollständigen Durchschnitten, J. Reine Angew. Math.278, 157-170 (1975) · Zbl 0316.13003
[42] Seidenberg, A.: On the Lasker-Noether decomposition theorem. Am. J. Math.106, 611-638 (1984) · Zbl 0567.13006 · doi:10.2307/2374287
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.