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)



Induced formal deformations and the Cohen-Macaulay property

Author: Phillip Griffith
Journal: Trans. Amer. Math. Soc. 353 (2001), 77-93
MSC (2000): Primary 13B10, 13B15, 13D10, 13F40; Secondary 13H10, 13N05, 14B07
Published electronically: June 13, 2000
MathSciNet review: 1675194
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The main result states: if $A/B$ is a module finite extension of excellent local normal domains which is unramified in codimension two and if $S/\varkappa S \simeq \hat B$ represents a deformation of the completion of $B$, then there is a corresponding $S$-algebra deformation $T/\varkappa T \simeq \hat A$ such that the ring homomorphism $S \hookrightarrow T$ represents a deformation of $\hat B \hookrightarrow \hat A$. The main application is to the ascent of the arithmetic Cohen-Macaulay property for an étale map $f : X \to Y$ of smooth projective varieties over an algebraically closed field.${}^*$

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

  • 1. M. Auslander and D. Buchsbaum, On ramification theory in Noetherian rings, Amer. J. Math. 81 (1959), 749-765. MR 21:5659
  • 2. M. Auslander, S. Ding and $\emptyset $. Solberg, Liftings and weak liftings of modules, J. of Alg. 156 (1993), 273-317. MR 94d:16007
  • 3. A. Beauville, Complex Algebraic Surfaces, London Math. Soc. Lecture Notes Series, Vol. 68, Cambridge Univ. Press, Cambridge, 1983. MR 85a:14024
  • 4. J. Bingener and U. Storch, Zur Berechnung der Divisorenklassengruppen kompleter lokaler Ringe, Nova Acta Leopoldina N.F. 52 Nr. 7 63 (1981), 7-63. MR 83m:13017
  • 5. A. Borek and P. Griffith, Weak purity for Gorenstein rings is determined in codimension four, J. Algebraic Geometry 5 (1996), 415-437. MR 97b:13030
  • 6. W. L. Chow, On unmixedness theorems, Amer. J. Math 86 (1964), 799-822.MR 30:2031
  • 7. S. D. Cutkosky, Purity of the branch locus and Lefschetz theorems, Compositio Mathematica 96 (1995), 173-195. MR 96h:13023
  • 8. D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Springer, Berlin-Heidelberg-New York, 1994. MR 97a:13001
  • 9. R. Fossum, The Divisor Class Group of a Krull Domain, in Ergebnisse der Mathematik und iher Grenzgebiete, vol. 74, Springer, Berlin-Heidelberg-New York, 1973. MR 52:3139
  • 10. R. Fossum, H. Foxby, P. Griffith and I. Reiten, Minimal injective resolutions with applications to dualizing modules and Gorenstein modules, Inst. Hautes Études Sci. Publ. Math. 45 (1976), 193-215. MR 53:392
  • 11. H. Gillet and C. Soulé, $K$-théorie et nullité des multiplicités d'intersection, C. R. Acad. Sc. Paris Série I Math. 300 (1985), 71-74. MR 86k:13027
  • 12. P. Griffith, Normal extensions of regular local rings, J. of Alg. 106 (1987), 465-475. MR 88c:13020
  • 13. -, Some results in local rings on ramification in low codimension, J. of Alg. 137 (1991), 473-490. MR 92c:13017
  • 14. P. Griffith and D. Weston, Restrictions of torsion divisor classes to hypersurfaces, J. of Alg. 167 (1994), 473-487. MR 95c:13008
  • 15. A. Grothendieck, Cohomologie locale des faisceaux cohérents et théormes de Lefschetz locaux et globaux, Séminaire de Géométrie Algébrique du Bois-Marie, 1962 (SGA2), fasc. 1, Inst. Hautes Études Sci., Paris, 1962; 3rd ed., 1965; reprint, Adv. Stud. Pure Math., vol. 2, North-Holland, Amsterdam, and Masson, Paris, 1968. MR 35:1604; MR 57:16294
  • 16. -, Élements de géométrie algébrique (EGA), Part IV: Étude locale des schémes et des morphismes de schémes. IV, Inst. Hautes Études Sci. Publ. Math. No. 32 (1967). MR 39:220
  • 17. R. Hartshorne, Algebraic Geometry, Springer, Berlin-Heidelberg-New York, 1977. MR 57:3116
  • 18. R. Hartshorne and A. Ogus, On the factoriality of local rings of small embedding codimension, Communications in Algebra 1 (1974), 415-437.MR 50:322
  • 19. J. Herzog, Deformationen von Cohen-Macaulay Algebren, J. Reine Angew. Math. 318 (1980), 83-105. MR 81m:13012
  • 20. M. Hochster and J. Eagon, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020-1058. MR 46:1787
  • 21. M. Hochster and J. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Advances in Math. 13 (1974), 115-175. MR 50:311
  • 22. C. Huneke, A remark concerning multiplicities, Proc. AMS 85 (1982), 331-332. MR 83m:13016
  • 23. H. Matsumura, Commutative Algebra, Cambridge Studies in Advanced Mathematics no. 8, Cambridge Univ. Press, Cambridge, 1989. MR 42:8213; MR 82i:13003 (earlier eds.)
  • 24. M. Nagata, On the purity of branch locus in regular local rings, Ill. Jour. of Math. 3 (1959), 328-333. MR 21:5660
  • 25. M. Raynaud, Anneaux Locaux Henséliens, in Lecture Notes in Mathematics, no. 169, Springer, Berlin- Heidelberg-New York, 1970. MR 43:3252
  • 26. P. Roberts, Multiplicities and Chern Classes in Local Algebra, in Cambridge Tracts in Mathematics, no. 133, Cambridge Univ. Press, Cambridge (1998). CMP 99:13
  • 27. D. Smith, Ph.D. Thesis, University of Illinois (1998).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13B10, 13B15, 13D10, 13F40, 13H10, 13N05, 14B07

Retrieve articles in all journals with MSC (2000): 13B10, 13B15, 13D10, 13F40, 13H10, 13N05, 14B07

Additional Information

Phillip Griffith
Affiliation: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801

Keywords: Cohen-Macaulay local rings, normal domains, ramification, deformations, Segre products.
Received by editor(s): August 15, 1998
Published electronically: June 13, 2000
Additional Notes: The author would like to thank the referee for several corrections and helpful suggestions.
$^{*}$ See Added in Proof for correction
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society