Let be a commutative ring, a commutative -algebra and the filtered ring of -linear differential operators of . We prove that: (1) The graded ring admits a canonical embedding into the graded dual of the symmetric algebra of the module of differentials of over , which has a canonical divided power structure. (2) There is a canonical morphism from the divided power algebra of the module of -linear Hasse–Schmidt integrable derivations of to . (3) Morphisms and fit into a canonical commutative diagram.
Soit un anneau commutatif, une -algèbre commutative et l’anneau filtré des opérateurs différentiels -linéaires de . Nous montrons que : (1) l’anneau gradué admet un plongement canonique dans le dual gradué de l’algèbre symétrique du module des différentielles de sur , qui a une structure canonique de puissances divisées. (2) Il existe un morphisme canonique de l’algèbre des puissances divisées du module des dérivations -linéaires et intégrables dans le sens de Hasse-Schmidt de vers . (3) Les morphismes et forment partie d’un diagramme commutatif canonique.
Accepted:
DOI: 10.5802/aif.2513
Classification: 13N15, 13N10
Keywords: Derivation, integrable derivation, differential operator, divided powers structure
@article{AIF_2009__59_7_2979_0, author = {Narv\'aez Macarro, Luis}, title = {Hasse{\textendash}Schmidt derivations, divided powers and differential smoothness}, journal = {Annales de l'Institut Fourier}, pages = {2979--3014}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {7}, year = {2009}, doi = {10.5802/aif.2513}, zbl = {1184.13076}, mrnumber = {2649344}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2513/} }
TY - JOUR TI - Hasse–Schmidt derivations, divided powers and differential smoothness JO - Annales de l'Institut Fourier PY - 2009 DA - 2009/// SP - 2979 EP - 3014 VL - 59 IS - 7 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2513/ UR - https://zbmath.org/?q=an%3A1184.13076 UR - https://www.ams.org/mathscinet-getitem?mr=2649344 UR - https://doi.org/10.5802/aif.2513 DO - 10.5802/aif.2513 LA - en ID - AIF_2009__59_7_2979_0 ER -
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. https://aif.centre-mersenne.org/articles/10.5802/aif.2513/
[1] Higher derivations and integral closure, Amer. J. Math., Tome 100 (1978) no. 3, pp. 495-521 | Article | MR: 501221 | Zbl: 0386.13008
[2] Notes on crystalline cohomology, Mathematical Notes, Tome 21, Princeton Univ. Press, Princeton, N.J., 1978 | MR: 491705 | Zbl: 0383.14010
[3] Commutative Algebra with a view toward Algebraic Geometry, Graduate Texts in Mathematics, Tome 150, Springer Verlag, New York, 1995 | MR: 1322960 | Zbl: 0819.13001
[4] Hasse-Schmidt derivations and coefficient fields in positive characteristics, J. Algebra, Tome 265 (2003) no. 1, pp. 200-210 | Article | MR: 1984906 | Zbl: 1099.13518
[5] The fundamental form of an inseparable extension, Trans. Amer. Math. Soc., Tome 227 (1977), pp. 165-184 | Article | MR: 429861 | Zbl: 0354.12023
[6] Éléments de Géométrie Algébrique IV: Étude locale des schémas et de morphismes de schémas (Quatrième Partie), Inst. Hautes Études Sci. Publ. Math., Tome 32, Press Univ. de France, Paris, 1967 | Numdam | Zbl: 0153.22301
[7] Noch eine Begründung der Theorie der höheren Differrentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten, J. Reine U. Angew. Math., Tome 177 (1937), pp. 223-239 | Zbl: 0017.10101
[8] Divided powers (2006) (Unpublished notes)
[9] Integrable derivations, Nagoya Math. J., Tome 87 (1982), pp. 227-245 | MR: 676593 | Zbl: 0458.13002
[10] Commutative Ring Theory, Cambridge studies in advanced mathematics, Tome 8, Cambridge Univ. Press, Cambidge, 1986 | MR: 879273 | Zbl: 0603.13001
[11] Lois polynomes et lois formelles en théorie des modules, Ann. Sci. École Norm. Sup., Tome 80 (1963) no. 3, pp. 213-348 | Numdam | MR: 161887 | Zbl: 0117.02302
[12] Les algèbres à puissances divisées, Bull. Sci. Math., Tome 89 (1965) no. 2, pp. 75-91 | MR: 193127 | Zbl: 0145.04503
[13] Differential Operators and Nakai’s Conjecture (1998) (Ph. D. Thesis)
[14] Jets via Hasse–Schmidt derivations, Diophantine geometry (CRM Series) Tome 4, Ed. Norm., Pisa, 2007, pp. 335-361 | MR: 2349665 | Zbl: pre05263295
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