Differentials on quotients of algebraic varieties

Knighten, Carol M.

doi:10.1090/S0002-9947-1973-0323788-5

Differentials on quotients of algebraic varieties
HTML articles powered by AMS MathViewer

by Carol M. Knighten

Trans. Amer. Math. Soc. 177 (1973), 65-89

DOI: https://doi.org/10.1090/S0002-9947-1973-0323788-5

PDF | Request permission

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

R. E. Briney, Intersection theory on quotients of algebraic varieties, Amer. J. Math. 84 (1962), 217–238. MR 140515, DOI 10.2307/2372760
Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1); Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud. MR 0354651

Éléments de géométrie algébrique

39

Joseph Lipman, Free derivation modules on algebraic varieties, Amer. J. Math. 87 (1965), 874–898. MR 186672, DOI 10.2307/2373252
Arthur Mattuck, Symmetric products and Jacobians, Amer. J. Math. 83 (1961), 189–206. MR 142553, DOI 10.2307/2372727
Arthur Mattuck, On symmetric products of curves, Proc. Amer. Math. Soc. 13 (1962), 82–87. MR 136608, DOI 10.1090/S0002-9939-1962-0136608-7
A. Mattuck and A. Mayer, The Riemann-Roch theorem for algebraic curves, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 223–237. MR 162798
Pierre Samuel, Anneaux gradués factoriels et modules réflexifs, Bull. Soc. Math. France 92 (1964), 237–249 (French). MR 186702, DOI 10.24033/bsmf.1608
Jean-Pierre Serre, Groupes algébriques et corps de classes, Publications de l’Institut de Mathématique de l’Université de Nancago, VII, Hermann, Paris, 1959 (French). MR 0103191
Satoshi 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 180575, DOI 10.1215/kjm/1250524601
R. E. Kalman, Introduction to the algebraic theory of linear dynamical systems, Mathematical systems theory and economics, I, II (Proc. Internat. Summer School, Varenna, 1967) Lecture Notes in Operations Research and Mathematical Economics, Vols. 11, vol. 12, Springer, Berlin, 1969, pp. 41–65. MR 0323385
Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581