A differential criterion for complete intersections

• Autores: Bart de Smit
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 48, Fasc. 1-2, 1997, págs. 85-96
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • Let $A$ be a noetherian ring whose maximal spectrum has dimension at most 1. For instance, $A$ can be a noetherian local ring or an order in a number field. Let $B$ be a finite projective $A$-algebra that becomes étale over the total ring of quotients of $A$. In this note it is shown that $B$ is of the form $A[X_1,\dots , X_n]/(f_1,\dots , f_n)$ if and only if the Fitting ideal Fit$_B(\Omega_{B/A})$ of the module of differentials of $B$ over $A$ is free of rank 1 as a $B$-module. In particular, the ring of integers in a number field $K$ is of the form $\mathbb{Z}[X_1,\dots , X_n]/(f_1,\dots , f_n)$ if and only if the different of $K$ over $\mathbb{Q}$ is a principal ideal.

• Referencias bibliográficas
  • M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading, Mass, 1969.
  • M. Auslander and D. A. Buchsbaum, Homological dimension in local rings, Trans. Amer. Math. Soc. 85 (1957), 390–405.
  • S. de Smit and H. W. Lenstra, Jr., Finite complete intersection algebras and the completeness radical, in preparation.
  • D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Math. 150, Springer-Verlag, New York, 1994.
  • R. Hartshorne, Algebraic geometry, Graduate Texts in Math. 52, Springer-Verlag, New York, 1977.
  • E. Kunz, Kähler differentials, F. Vieweg, Braunschweig, 1986.
  • N. Mohan Kumar, Complete intersections, J. Math. Kyoto Univ. 17 (1977), 533–538.
  • S. Lang, Algebra, 3rd ed., Addison-Wesley, Reading, Mass, 1993.
  • H. W. Lenstra, Jr., Complete intersections and Gorenstein rings, pp. 99–109 in: J. Coates, S. T. Yau (eds.), Elliptic curves, modular forms,...
  • J. Lipman, On the Jacobian ideal of the module of differentials, Proc. Amer. Math. Soc. 21 (1969), 422–426.
  • H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1986.
  • J.-P. Serre, Local fields, Graduate Texts in Math. 67, Springer-Verlag, New York, 1979.
  • J.-P. Serre, Modules projectifs et espaces fibrés à fibre vectorielle, Seminaire Dubreil-Pisot 1957/58 no. 23; pp. 531–543 in: Collected Papers,...
  • W. V. Vasconcelos, Ideals generated by $R$-sequences, J. Algebra 6 (1967), 309–316.