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

Kleiman, Steven; Ulrich, Bernd

doi:10.1090/S0002-9947-97-01960-0

Gorenstein algebras, symmetric matrices, self-linked ideals, and symbolic powers
HTML articles powered by AMS MathViewer

by Steven Kleiman and Bernd Ulrich

Trans. Amer. Math. Soc. 349 (1997), 4973-5000

DOI: https://doi.org/10.1090/S0002-9947-97-01960-0

PDF | Request permission

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

M. Artin and M. Nagata, Residual intersections in Cohen-Macaulay rings, J. Math. Kyoto Univ. 12 (1972), 307–323. MR 301006, DOI 10.1215/kjm/1250523522
I. M. Sheffer, Some properties of polynomial sets of type zero, Duke Math. J. 5 (1939), 590–622. MR 81, DOI 10.1215/S0012-7094-39-00549-1
Winfried Bruns, The Eisenbud-Evans generalized principal ideal theorem and determinantal ideals, Proc. Amer. Math. Soc. 83 (1981), no. 1, 19–24. MR 619972, DOI 10.1090/S0002-9939-1981-0619972-3
David A. Buchsbaum and David Eisenbud, What annihilates a module?, J. Algebra 47 (1977), no. 2, 231–243. MR 476736, DOI 10.1016/0021-8693(77)90223-X
David A. Buchsbaum and David Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$, Amer. J. Math. 99 (1977), no. 3, 447–485. MR 453723, DOI 10.2307/2373926
Fabrizio Catanese, Commutative algebra methods and equations of regular surfaces, Algebraic geometry, Bucharest 1982 (Bucharest, 1982) Lecture Notes in Math., vol. 1056, Springer, Berlin, 1984, pp. 68–111. MR 749939, DOI 10.1007/BFb0071770
T. de Jong and D. van Straten, Deformations of the normalization of hypersurfaces, Math. Ann. 288 (1990), no. 3, 527–547. MR 1079877, DOI 10.1007/BF01444547
J. A. Eagon and D. G. Northcott, Ideals defined by matrices and a certain complex associated with them, Proc. Roy. Soc. London Ser. A 269 (1962), 188–204. MR 142592, DOI 10.1098/rspa.1962.0170
David Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), no. 1, 35–64. MR 570778, DOI 10.1090/S0002-9947-1980-0570778-7
D. Eisenbud and B. Mazur, Symbolic squares, Fitting ideals, and evolutions, Preprint.
Hubert Flenner, Die Sätze von Bertini für lokale Ringe, Math. Ann. 229 (1977), no. 2, 97–111 (German). MR 460317, DOI 10.1007/BF01351596
Shiro Goto and Koji Nishida, The Cohen-Macaulay and Gorenstein Rees algebras associated to filtrations, American Mathematical Society, Providence, RI, 1994. Mem. Amer. Math. Soc. 110 (1994), no. 526. MR 1287443
Shiro Goto, Koji Nishida, and Yasuhiro Shimoda, The Gorensteinness of symbolic Rees algebras for space curves, J. Math. Soc. Japan 43 (1991), no. 3, 465–481. MR 1111598, DOI 10.2969/jmsj/04330465
Shiro Goto, Koji Nishida, and Keiichi Watanabe, Non-Cohen-Macaulay symbolic blow-ups for space monomial curves and counterexamples to Cowsik’s question, Proc. Amer. Math. Soc. 120 (1994), no. 2, 383–392. MR 1163334, DOI 10.1090/S0002-9939-1994-1163334-9
Hermann Kober, Transformationen von algebraischem Typ, Ann. of Math. (2) 40 (1939), 549–559 (German). MR 96, DOI 10.2307/1968939
Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux $(SGA$ $2)$, Advanced Studies in Pure Mathematics, Vol. 2, North-Holland Publishing Co., Amsterdam; Masson & Cie, Editeur, Paris, 1968 (French). Augmenté d’un exposé par Michèle Raynaud; Séminaire de Géométrie Algébrique du Bois-Marie, 1962. MR 0476737
Jürgen Herzog and Michael Kühl, Maximal Cohen-Macaulay modules over Gorenstein rings and Bourbaki-sequences, Commutative algebra and combinatorics (Kyoto, 1985) Adv. Stud. Pure Math., vol. 11, North-Holland, Amsterdam, 1987, pp. 65–92. MR 951197, DOI 10.2969/aspm/01110065
Jürgen Herzog and Bernd Ulrich, Self-linked curve singularities, Nagoya Math. J. 120 (1990), 129–153. MR 1086575, DOI 10.1017/S0027763000003305
Tadeusz Józefiak, Ideals generated by minors of a symmetric matrix, Comment. Math. Helv. 53 (1978), no. 4, 595–607. MR 511849, DOI 10.1007/BF02566100
Tadeusz Józefiak and Piotr Pragacz, Ideals generated by Pfaffians, J. Algebra 61 (1979), no. 1, 189–198. MR 554859, DOI 10.1016/0021-8693(79)90313-2
Steven Kleiman, Joseph Lipman, and Bernd Ulrich, The source double-point cycle of a finite map of codimension one, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 199–212. MR 1201384, DOI 10.1017/CBO9780511662652.015
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.
Ernst Kunz, Kähler differentials, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1986. MR 864975, DOI 10.1007/978-3-663-14074-0
Ronald E. Kutz, Cohen-Macaulay rings and ideal theory in rings of invariants of algebraic groups, Trans. Amer. Math. Soc. 194 (1974), 115–129. MR 352082, DOI 10.1090/S0002-9947-1974-0352082-2
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
David Mond and Ruud Pellikaan, Fitting ideals and multiple points of analytic mappings, Algebraic geometry and complex analysis (Pátzcuaro, 1987) Lecture Notes in Math., vol. 1414, Springer, Berlin, 1989, pp. 107–161. MR 1042359, DOI 10.1007/BFb0090254
Idun Reiten, The converse to a theorem of Sharp on Gorenstein modules, Proc. Amer. Math. Soc. 32 (1972), 417–420. MR 296067, DOI 10.1090/S0002-9939-1972-0296067-7
Paul C. Roberts, A prime ideal in a polynomial ring whose symbolic blow-up is not Noetherian, Proc. Amer. Math. Soc. 94 (1985), no. 4, 589–592. MR 792266, DOI 10.1090/S0002-9939-1985-0792266-5
Paul 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), no. 2, 461–473. MR 1061491, DOI 10.1016/0021-8693(90)90141-A
Jack Shamash, The Poincaré series of a local ring, J. Algebra 12 (1969), 453–470. MR 241411, DOI 10.1016/0021-8693(69)90023-4
Paolo Valabrega and Giuseppe Valla, Form rings and regular sequences, Nagoya Math. J. 72 (1978), 93–101. MR 514892, DOI 10.1017/S0027763000018225
Giuseppe Valla, On set-theoretic complete intersections, Complete intersections (Acireale, 1983) Lecture Notes in Math., vol. 1092, Springer, Berlin, 1984, pp. 85–101. MR 775878, DOI 10.1007/BFb0099358