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)

 
 

 

Differentials on quotients of algebraic varieties


Author: Carol M. Knighten
Journal: Trans. Amer. Math. Soc. 177 (1973), 65-89
MSC: Primary 14F10
DOI: https://doi.org/10.1090/S0002-9947-1973-0323788-5
MathSciNet review: 0323788
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The relations between differentials invariant with respect to a finite group acting on a variety and the differentials on the quotient variety are studied. If the quotient map is unramified in codimension 1 we have an isomorphism for Zariski differentials, but not in general for Käahler differentials. Necessary and sufficient conditions for isomorphism of the Zariski differentials are given when the finite group acts linearly. Examples illustrate the scope of the theorems and some open problems.


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

  • [1] R. E. Briney, Intersection theory on quotients of algebraic varieties, Amer. J. Math. 84 (1962), 217-238. MR 25 #3935. MR 0140515 (25:3935)
  • [2] A. Grothendieck, Revêtements étales et groupe fondamental (Séminaire de géométrie algébrique du Bois Marie 1960/61), Lecture Notes in Math., vol. 224, Springer-Verlag, Berlin and New York, 1971. MR 0354651 (50:7129)
  • [3] -, Éléments de géométrie algébrique. IV, Inst. Hautes Études Sci. Publ. Math. No. 32 (1967), 361 pp. MR 39 #220.
  • [4] J. Lipman, Free derivation modules on algebraic varieties, Amer. J. Math. 87 (1965), 874-898. MR 32 #4130. MR 0186672 (32:4130)
  • [5] A. Mattuck, Symmetric products and Jacobians, Amer. J. Math. 83 (1961), 189-206. MR 26 #122. MR 0142553 (26:122)
  • [6] -, On symmetric products of curves, Proc. Amer. Math. Soc. 13 (1962), 82-87. MR 25 #76. MR 0136608 (25:76)
  • [7] A. Mattuck and A. Mayer, The Riemann-Roch theorem for algebraic curves, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 223-237. MR 29 #102. MR 0162798 (29:102)
  • [8] P. Samuel, Anneaux gradués factoriels et modules réflexifs, Bull. Soc. Math. France 92 (1964), 237-249. MR 32 #4160. MR 0186702 (32:4160)
  • [9] J. P. Serre, Groupes algébriques et corps de classes, Actualités Sci. Indust., no. 1264, Hermann, Paris, 1959. MR 21 #1973; errata, MR 30 p. 1200. MR 0103191 (21:1973)
  • [10] S. Suzuki, On torsion of the module of differentials of a locality which is a complete intersection, J. Math. Kyoto Univ. 4 (1964/65), 471-475. MR 31 #4809. MR 0180575 (31:4809)
  • [11] O. Zariski, An introduction to the theory of algebraic surfaces, Lecture Notes in Math., no. 83, Springer-Verlag, Berlin and New York, 1969. MR 41 #8418. MR 0323385 (48:1742)
  • [12] O. Zariski and P. Samuel, Commutative algebra. Vol. I, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1957. MR 19, 833. MR 0090581 (19:833e)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 14F10

Retrieve articles in all journals with MSC: 14F10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0323788-5
Keywords: Quotient of algebraic variety, Zariski differentials, Kähler differentials, invariant differentials, ramified, action of finite group
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society