no. 7
Hasse–Schmidt derivations, divided powers and differential smoothness
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.

Received:
Accepted:
DOI: 10.5802/aif.2513
Classification: 13N15,  13N10
Keywords: Derivation, integrable derivation, differential operator, divided powers structure 
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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/

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