no. 7
Hasse–Schmidt derivations, divided powers and differential smoothness
Narváez Macarro, Luis1
1Universidad de Sevilla Facultad de Matemáticas Instituto de Matemáticas (IMUS) Departamento de Álgebra P.O. Box 1160 41080 Sevilla (Spain)
Annales de l'Institut Fourier, Volume 59 (2009) no. 7, pp. 2979-3014

Let k be a commutative ring, A a commutative k-algebra and D the filtered ring of k-linear differential operators of A. We prove that: (1) The graded ring gr D admits a canonical embedding θ into the graded dual of the symmetric algebra of the module Ω A/k of differentials of A over k, which has a canonical divided power structure. (2) There is a canonical morphism ϑ from the divided power algebra of the module of k-linear Hasse–Schmidt integrable derivations of A to gr D. (3) Morphisms θ and ϑ fit into a canonical commutative diagram.

Soit k un anneau commutatif, A une k-algèbre commutative et D l’anneau filtré des opérateurs différentiels k-linéaires de A. Nous montrons que  : (1) l’anneau gradué gr D admet un plongement canonique θ dans le dual gradué de l’algèbre symétrique du module Ω A/k des différentielles de A sur k, qui a une structure canonique de puissances divisées. (2) Il existe un morphisme canonique ϑ de l’algèbre des puissances divisées du module des dérivations k-linéaires et intégrables dans le sens de Hasse-Schmidt de A vers gr D. (3) Les morphismes θ et ϑ forment partie d’un diagramme commutatif canonique.

DOI: 10.5802/aif.2513
Classification: 13N15, 13N10
Keywords: Derivation, integrable derivation, differential operator, divided powers structure
Mots-clés : dérivation, dérivation intégrable, opérateur différentiel, structure de puissances divisées

Narváez Macarro, Luis  1

1 Universidad de Sevilla Facultad de Matemáticas Instituto de Matemáticas (IMUS) Departamento de Álgebra P.O. Box 1160 41080 Sevilla (Spain) 
Narváez Macarro, Luis. Hasse–Schmidt derivations, divided powers and differential smoothness. Annales de l'Institut Fourier, Volume 59 (2009) no. 7, pp. 2979-3014. doi: 10.5802/aif.2513
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[1] Becker, J. Higher derivations and integral closure, Amer. J. Math., Volume 100 (1978) no. 3, pp. 495-521 | DOI | Zbl | MR

[2] Berthelot, P.; Ogus, A. Notes on crystalline cohomology, Mathematical Notes, 21, Princeton Univ. Press, Princeton, N.J., 1978 | Zbl | MR

[3] Eisenbud, D. Commutative Algebra with a view toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer Verlag, New York, 1995 | Zbl | MR

[4] Fernández-Lebrón, M.; Narváez-Macarro, L. Hasse-Schmidt derivations and coefficient fields in positive characteristics, J. Algebra, Volume 265 (2003) no. 1, pp. 200-210 | DOI | Zbl | MR

[5] Gerstenhaber, M. The fundamental form of an inseparable extension, Trans. Amer. Math. Soc., Volume 227 (1977), pp. 165-184 | DOI | Zbl | MR

[6] Grothendieck, A.; Dieudonné, J. Éléments de Géométrie Algébrique IV: Étude locale des schémas et de morphismes de schémas (Quatrième Partie), Inst. Hautes Études Sci. Publ. Math., 32, Press Univ. de France, Paris, 1967 | Zbl | Numdam

[7] Hasse, H.; Schmidt, F. K. Noch eine Begründung der Theorie der höheren Differrentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten, J. Reine U. Angew. Math., Volume 177 (1937), pp. 223-239 | Zbl

[8] Laksov, D. Divided powers (2006) (Unpublished notes)

[9] Matsumura, H. Integrable derivations, Nagoya Math. J., Volume 87 (1982), pp. 227-245 | Zbl | MR

[10] Matsumura, H. Commutative Ring Theory, Cambridge studies in advanced mathematics, 8, Cambridge Univ. Press, Cambidge, 1986 | Zbl | MR

[11] Roby, N. Lois polynomes et lois formelles en théorie des modules, Ann. Sci. École Norm. Sup., Volume 80 (1963) no. 3, pp. 213-348 | Zbl | MR | Numdam

[12] Roby, N. Les algèbres à puissances divisées, Bull. Sci. Math., Volume 89 (1965) no. 2, pp. 75-91 | Zbl | MR

[13] Traves, W. N. Differential Operators and Nakai’s Conjecture, University of Toronto, Department of Mathematics (1998) (Ph. D. Thesis)

[14] Vojta, P. Jets via Hasse–Schmidt derivations, Diophantine geometry (CRM Series), Volume 4, Ed. Norm., Pisa, 2007, pp. 335-361 | MR

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