Hostname: page-component-7bb8b95d7b-dvmhs Total loading time: 0 Render date: 2024-09-12T12:18:29.995Z Has data issue: false hasContentIssue false
  • English
  • Français

On F-integrable actions of the restricted Lie algebra of a formal group F in characteristic p > 0

Published online by Cambridge University Press:  22 January 2016

Andrzej Tyc
Andrzej Tyc*
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, ul. Chopina 12/18, 87-100 Toruń, Poland
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let k be an integral domain, let F = (F1X, Y),…, Fn(X, Y)), X = (X1,…, Xn), Y = (Y1,…, Yn), be an n-dimensional formal group over k, and let L(F) be the Lie algebra of all F-invariant k-derivations of the ring of formal power series k[X] (cf. § 2). If A is a (commutative) k-algebra and Derk (A) denotes the Lie algebra of all k-derivations d: A → A, then by an action of L(F) on A we mean a morphism of Lie algebras φ: L(F) → Derk (A) such that φ(dp) = φ(d)p, provided char (k) = p > 0. An action of the formal group F on A is a morphism of k-algebras D: A-→A[X] such that D(a)≡a mod (X) for a ∊ A, and FA ∘ D = DY ∘ D, where FA: A[X] → A[X, Y], DF: A[X] → A[X, Y] are morphisms of k-algebras given by FA(g(X)) = g(F), DY(Σa aaXa) = Σa D(aa)Y; for a motivation of this notion, see [15]. Let D: A → A[X] be such an action.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 115 , September 1989 , pp. 125 - 137
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

References

[1] Carfi, V., Integrable derivations in rings of analytic type over a DVR, Atti Sem. Mat. Fis. Università di Modena, 32 (1983), No. 1, 1–10.Google Scholar
[2] Fossum, R. M., Invariants and Formal Group Law Actions, in Contemporary Mathematics, vol. 43, 1985.Google Scholar
[3] Fogarty, J., Kähler differentials and Hilbert’s fourteenth problem for finite groups, Amer. J. Math., 102 (1980), 1159–1174.Google Scholar
[4] Fröhlich, A., Formal Groups, in Lecture Notes in Math., 74, Springer-Verlag, New York/Berlin, 1968.Google Scholar
[5] Furuya, M., Note on local rings with integrable derivations, TRU Math., 17 (1981), No. 1, 39–45.Google Scholar
[6] Hazewinkel, M., Formal Groups and Applications, Academic Press, New York/San Francisco, 1978.Google Scholar
[7] Heerema, N., Derivations and embeddings of a field in its power series ring, II, Michigan Math. J., 8 (1961), 129–134.Google Scholar
[8] Hochschild, G., Simple algebras with purely inseparable splitting fields of exponent 1, Trans. Amer. Math. Soc, 79 (1955), 477–489.Google Scholar
[9] Matsumura, H., Commutative Algebra, 2nd. ed., Benjamin, 1980.Google Scholar
[10] Matsumura, H., Integrable derivations, Nagoya Math. J., 87 (1982), 227–245.Google Scholar
[11] Nowicki, A., Stiff derivations of commutative rings, Colloquium Math., 47 (1984), Fasc. 1, 7–16.Google Scholar
[12] Restuccia, G., Matsumura, H., Integrable derivations in rings of unequal characteristic, Nagoya Math. J., 93 (1984), 173–178.Google Scholar
[13] Rudakov, A., Shafarevich, I., Inseparable morphisms of algebraic surfaces, Izv. AN SSSR, Ser. mat., 40 (1976), 1269–1307 in Russian, Engl, transl.: Math. USSR Izv., 10 (1976), 1205–1237.Google Scholar
[14] Sheshadri, C. S., L’operation de Cartier. Application, in Chevalley, Seminaire C., 3e, année, 1958/59.Google Scholar
[15] Tyc, A., Invariants of linearly reductive formal group actions, J. Algebra, 101 (1986), no. 1, 166–187.Google Scholar
[16] Tyc, A., Differential basis, p-basis, and smoothness in characteristic p > 0, to appear in Proc. Amer. Math. Soc.+0,+to+appear+in+Proc.+Amer.+Math.+Soc.>Google Scholar