Linkage and sums of ideals

Johnson, Mark

doi:10.1090/S0002-9947-98-01976-X

Linkage and sums of ideals
HTML articles powered by AMS MathViewer

by Mark R. Johnson PDF

Trans. Amer. Math. Soc. 350 (1998), 1913-1930 Request permission

Abstract:

It is shown (under mild conditions) that the sum of transversal ideals in a regular local ring cannot lie in the linkage class of a complete intersection. For a sum of geometrically linked Cohen-Macaulay ideals, we compute the depths of the conormal module and the first Koszul homology. As applications, we construct general examples of ideals which are strongly Cohen-Macaulay, strongly nonobstructed but not in the linkage class of a complete intersection, and Gorenstein ideals which are strongly nonobstructed but not syzygetic.

References

P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
Maurice Auslander and Idun Reiten, Modules determined by their composition factors, Illinois J. Math. 29 (1985), no. 2, 280–301. MR 784524
R.-O. Buchweitz, Contributions à la thèorie des singularités, Thesis, l’Université de Paris, 1981.
R.-O. Buchweitz and B. Ulrich, Homological properties which are invariant under linkage, preprint.
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Jürgen Herzog, Deformationen von Cohen-Macaulay Algebren, J. Reine Angew. Math. 318 (1980), 83–105 (German). MR 579384, DOI 10.1515/crll.1980.318.83
Silvio Greco and Giuseppe Valla (eds.), Commutative algebra, Heidelberger Taschenbücher [Heidelberg Paperbacks], vol. 80, Marcel Dekker, Inc., New York, 1983. MR 686936
Craig Huneke, Linkage and the Koszul homology of ideals, Amer. J. Math. 104 (1982), no. 5, 1043–1062. MR 675309, DOI 10.2307/2374083
Craig Huneke, Strongly Cohen-Macaulay schemes and residual intersections, Trans. Amer. Math. Soc. 277 (1983), no. 2, 739–763. MR 694386, DOI 10.1090/S0002-9947-1983-0694386-5
Craig Huneke and Bernd Ulrich, Divisor class groups and deformations, Amer. J. Math. 107 (1985), no. 6, 1265–1303 (1986). MR 815763, DOI 10.2307/2374407
Craig Huneke and Bernd Ulrich, The structure of linkage, Ann. of Math. (2) 126 (1987), no. 2, 277–334. MR 908149, DOI 10.2307/1971402
Craig Huneke and Bernd Ulrich, Algebraic linkage, Duke Math. J. 56 (1988), no. 3, 415–429. MR 948528, DOI 10.1215/S0012-7094-88-05618-9
Craig Huneke and Bernd Ulrich, Residual intersections, J. Reine Angew. Math. 390 (1988), 1–20. MR 953673, DOI 10.1515/crll.1988.390.1
C. Huneke and B. Ulrich, Powers of licci ideals, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 339–346. MR 1015526, DOI 10.1007/978-1-4612-3660-3_{1}7
Craig Huneke and Bernd Ulrich, Local properties of licci ideals, Math. Z. 211 (1992), no. 1, 129–154. MR 1179785, DOI 10.1007/BF02571423
Mark Johnson and Bernd Ulrich, Artin-Nagata properties and Cohen-Macaulay associated graded rings, Compositio Math. 103 (1996), no. 1, 7–29. MR 1404996
Andrew R. Kustin and Matthew Miller, A general resolution for grade four Gorenstein ideals, Manuscripta Math. 35 (1981), no. 1-2, 221–269. MR 627935, DOI 10.1007/BF01168458
Andrew R. Kustin and Matthew Miller, Deformation and linkage of Gorenstein algebras, Trans. Amer. Math. Soc. 284 (1984), no. 2, 501–534. MR 743730, DOI 10.1090/S0002-9947-1984-0743730-X
Stephen Lichtenbaum, On the vanishing of $\textrm {Tor}$ in regular local rings, Illinois J. Math. 10 (1966), 220–226. MR 188249
E. Lopez, Licci Gorenstein ideals of deviation two, Ph.D. thesis, Michigan State University (1988).
C. Peskine and L. Szpiro, Liaison des variétés algébriques. I, Invent. Math. 26 (1974), 271–302 (French). MR 364271, DOI 10.1007/BF01425554
A. Simis and W. V. Vasconcelos, The syzygies of the conormal module, Amer. J. Math. 103 (1981), no. 2, 203–224. MR 610474, DOI 10.2307/2374214
Aron Simis, Wolmer V. Vasconcelos, and Rafael H. Villarreal, On the ideal theory of graphs, J. Algebra 167 (1994), no. 2, 389–416. MR 1283294, DOI 10.1006/jabr.1994.1192
Hema Srinivasan, A grade five Gorenstein algebra with no minimal algebra resolutions, J. Algebra 179 (1996), no. 2, 362–379. MR 1367854, DOI 10.1006/jabr.1996.0016
M. Nagata and H. Matsumura (eds.), Commutative algebra and combinatorics, Advanced Studies in Pure Mathematics, vol. 11, North-Holland Publishing Co., Amsterdam; Kinokuniya Company Ltd., Tokyo, 1987. Papers from the U.S.-Japan Joint Seminar held in Kyoto, August 25–31, 1985. MR 951190, DOI 10.2969/aspm/01110000
Bernd Ulrich, Sums of linked ideals, Trans. Amer. Math. Soc. 318 (1990), no. 1, 1–42. MR 964902, DOI 10.1090/S0002-9947-1990-0964902-8
Bernd Ulrich, Artin-Nagata properties and reductions of ideals, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 373–400. MR 1266194, DOI 10.1090/conm/159/01519
B. Ulrich, Parafactoriality and small divisor class groups, in preparation.
Wolmer V. Vasconcelos, Koszul homology and the structure of low codimension Cohen-Macaulay ideals, Trans. Amer. Math. Soc. 301 (1987), no. 2, 591–613. MR 882705, DOI 10.1090/S0002-9947-1987-0882705-X
Wolmer V. Vasconcelos, Arithmetic of blowup algebras, London Mathematical Society Lecture Note Series, vol. 195, Cambridge University Press, Cambridge, 1994. MR 1275840, DOI 10.1017/CBO9780511574726
W. V. Vasconcelos and R. Villarreal, On Gorenstein ideals of codimension four, Proc. Amer. Math. Soc. 98 (1986), no. 2, 205–210. MR 854019, DOI 10.1090/S0002-9939-1986-0854019-X
Rafael H. Villarreal, Cohen-Macaulay graphs, Manuscripta Math. 66 (1990), no. 3, 277–293. MR 1031197, DOI 10.1007/BF02568497
Junzo Watanabe, A note on Gorenstein rings of embedding codimension three, Nagoya Math. J. 50 (1973), 227–232. MR 319985