Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Gorenstein algebras, symmetric matrices,
self-linked ideals, and symbolic powers


Authors: Steven Kleiman and Bernd Ulrich
Journal: Trans. Amer. Math. Soc. 349 (1997), 4973-5000
MSC (1991): Primary 13C40, 13H10, 13A30, 14E05
DOI: https://doi.org/10.1090/S0002-9947-97-01960-0
MathSciNet review: 1422609
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Inspired by recent work in the theory of central projections onto hypersurfaces, we characterize self-linked perfect ideals of grade $2$ as those with a Hilbert-Burch matrix that has a maximal symmetric subblock. We also prove that every Gorenstein perfect algebra of grade $1$ can be presented, as a module, by a symmetric matrix. Both results are derived from the same elementary lemma about symmetrizing a matrix that has, modulo a nonzerodivisor, a symmetric syzygy matrix. In addition, we establish a correspondence, roughly speaking, between Gorenstein perfect algebras of grade $1$ that are birational onto their image, on the one hand, and self-linked perfect ideals of grade $2$ that have one of the self-linking elements contained in the second symbolic power, on the other hand. Finally, we provide another characterization of these ideals in terms of their symbolic Rees algebras, and we prove a criterion for these algebras to be normal.


References [Enhancements On Off] (What's this?)

  • 1. M. Artin and M. Nagata, Residual intersections in Cohen-Macaulay rings, J. Math. Kyoto Univ. 12 1972 307-23. MR 46:166
  • 2. M. Brodmann, The asymptotic nature of analytic spread, Math. Proc. Camb. Phil. Soc. 86 1979 35-39. MR 81e:13033
  • 3. W. Bruns, The Eisenbud-Evans generalized principal ideal theorem and determinantal ideals, Proc. Amer. Math. Soc. 83 1981 19-24. MR 82k:13010
  • 4. D. Buchsbaum and D. Eisenbud, What annihilates a module? J. Algebra 47 1977 231-43. MR 57:16293
  • 5. D. Buchsbaum and D. Eisenbud, Algebraic structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math. 99 1977 447-85. MR 56:11983
  • 6. F. Catanese, Commutative algebra methods and equations of regular surfaces, Algebraic Geometry, Bucharest 1982, L. Badescu and D. Popescu (eds.), Lecture Notes in Math. 1056, Springer-Verlag, 1984, pp. 68-111. MR 86c:14027
  • 7. T. de Jong and D. van Straten, Deformations of the normalization of hypersurfaces, Math. Ann. 288 1990 527-47. MR 92d:32050
  • 8. J. Eagon and D. Northcott, Ideals defined by matrices and a certain complex associated to them, Proc. Royal Soc. A269 1962 188-204. MR 26:161
  • 9. D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 1980 35-64. MR 82d:13013
  • 10. D. Eisenbud and B. Mazur, Symbolic squares, Fitting ideals, and evolutions, Preprint.
  • 11. H. Flenner, Die Sätze von Bertini für lokale Ringe, Math. Ann. 229 1977 97-111. MR 57:311
  • 12. S. Goto and K. Nishida, The Cohen-Macaulay and Gorenstein Rees algebras associated to filtrations, Mem. Amer. Math. Soc., vol. 110, 1994, no. 526. MR 95b:13001
  • 13. S. Goto, K. Nishida, and Y. Shimoda, The Gorensteinness of symbolic Rees algebras for space curves, J. Math. Soc. Japan 43 1991 465-81. MR 92h:13019
  • 14. S. Goto, K. Nishida, and K. Watanabe, Non-Cohen-Macaulay symbolic blow-ups for space monomial curves and counterexamples to Cowsik's question, Proc. Amer. Math. Soc. 120 1994 383-92. MR 94d:13005
  • 15. M. Grassi, Koszul modules and Gorenstein algebras, J. Algebra 180 1996 918-53. MR 96b:13021
  • 16. A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, (SGA II), Exposé XIV, Advanced Studies in Math, Vol. II, North Holland Publishing Co., Amsterdam, 1968. MR 57:16294
  • 17. J. Herzog and M. Kühl, Maximal Cohen-Macaulay modules over Gorenstein rings and Bourbaki sequences, in ``Commutative Algebra and Combinatorics,'' M. Nagata and H. Matsumura (eds.), Advanced Studies in Pure Math 11, North-Holland, Amsterdam (1987), 65-92. MR 89h:13029
  • 18. J. Herzog and B. Ulrich, Self-linked curve singularities, Nagoya Math J. 120 1990 129-53. MR 92c:13010
  • 19. T. Jozéfiak, Ideals generated by minors of a symmetric matrix, Comment. Math. Helv. 53 1978 594-607. MR 80g:14041
  • 20. T. Jozéfiak and P. Pragacz, Ideals generated by Pfaffians, J. Algebra 61 1979 189-98. MR 81e:13005
  • 21. S. Kleiman, J. Lipman, and B. Ulrich, The source double-point cycle of a finite map of codimension one, in ``Complex Projective Varieties,'' G. Ellingsrud, C. Peskine, G. Sacchiero, and S. A. Stromme (eds.), London Math. Soc. Lecture Note Series 179 1992 199-212. MR 94a:14003
  • 22. S. Kleiman, J. Lipman, and B. Ulrich, The multiple-point schemes of a finite curvilinear map of codimension one, Ark. Mat. 34 1996 285-326. CMP 97:03
  • 23. E. Kunz, Kähler differentials, Advandced Lectures in Math., Vieweg, 1986. MR 88e:14025
  • 24. R. Kutz, Cohen-Macaulay rings and ideal theory in rings of invariants of algebraic groups, Trans. Amer. Math. Soc. 194 1974 115-29. MR 50:4570
  • 25. H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Math 8, 1986. MR 88h:13001
  • 26. D. Mond and R. Pellikaan, Fitting ideals and multiple points of analytic mappings, Algebraic geometry and complex analysis, E. Ramirez de Arellano (ed.), Proc. Conf., Pátzcuaro 1987, Lecture Notes in Math. 1414, Springer-Verlag, 1989, pp. 107-161. MR 91e:32035
  • 27. I. Reiten, The converse of a theorem of Sharp on Gorenstein modules, Proc. Amer. Math. Soc. 40 1972 417-20. MR 45:5128
  • 28. P. Roberts, A prime ideal in a polynomial ring whose symbolic blow-up is not Noetherian, Proc. Amer. Math. Soc. 94 1985 589-92. MR 86k:13017
  • 29. P. Roberts, An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert's fourteenth problem, J. Algebra 132 1990 461-73. MR 91j:13006
  • 30. J. Shamash, The Poincaré series of a local ring, J. Algebra 12 1969 453-70. MR 39:2751
  • 31. P. Valabrega and G. Valla, Form rings and regular sequences, Nagoya Math J. 72 1978 93-101. MR 80d:14010
  • 32. G. Valla, On set-theoretic complete intersections, Complete Intersections, S. Greco and R. Strano (eds.), Proc. Conf., Arcireale 1983 Lecture Notes in Math. 1092, Springer-Verlag, 1984, pp. 85-101. MR 86f:14033

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 13C40, 13H10, 13A30, 14E05

Retrieve articles in all journals with MSC (1991): 13C40, 13H10, 13A30, 14E05


Additional Information

Steven Kleiman
Affiliation: Department of Mathematics, Room 2-278, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
Email: Kleiman@math.MIT.edu

Bernd Ulrich
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
Email: Ulrich@math.MSU.edu

DOI: https://doi.org/10.1090/S0002-9947-97-01960-0
Received by editor(s): June 2, 1996
Additional Notes: The first author was supported in part by NSF grant 9400918-DMS. It is a pleasure for this author to thank the Mathematical Institute of the University of Copenhagen for its hospitality during the summer of 1995 when this work was completed
The second author was supported in part by NSF grant DMS-9305832
Dedicated: To David Eisenbud on his fiftieth birthday
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society