Hasse–Schmidt derivations, divided powers and differential smoothness
Annales de l'Institut Fourier, Volume 59 (2009) no. 7, p. 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 : https://doi.org/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--Schmidt derivations, divided powers and differential smoothness},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {7},
     year = {2009},
     pages = {2979-3014},
     doi = {10.5802/aif.2513},
     mrnumber = {2649344},
     zbl = {1184.13076},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2009__59_7_2979_0}
}
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. http://www.numdam.org/item/AIF_2009__59_7_2979_0/

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