The flatness of the $𝒪$-module of smooth functions and integral representation

Andersson, Mats

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 32 (2023) no. 1, pp. 1-14

We give a proof of the well-known fact that the $𝒪$-module $ℰ$ of smooth functions is flat by means of residue theory and integral formulas. A variant of the proof gives a similar result for classes of functions of lower regularity. We also prove a Briançon–Skoda type theorem for ideals of the form $ℰa$, where $a$ is an ideal in $𝒪$.

Nous donnons une preuve du fait bien connu que le $𝒪$-module $ℰ$ des fonctions lisses est plat au moyen de la théorie des résidus et des formules intègrales. Une variante de la preuve donne une résultat similaire pour classes de fonctions de moindre régularité. Nous prouvons également un théorème de type Briançon-Skoda pour des idéaux de la forme $ℰa$, où $a$ est un idéal dans $𝒪$.

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