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In this paper we prove that the projective dimension of Mn = R 4 =hAni is 2n ? 1, where R is the ring of polynomials in 4n variables with complex coeecients, and hAni is the module generated by the columns of a 44n matrix which arises as the Fourier transform of the matrix of diierential operators associated with the regularity condition for a function of n quater-nionic variables. As a corollary we show that the sheaf R of regular functions has abby dimension 2n?1, and we prove a cohomology vanishing theorem for open sets in the space H n of quaternions. We also show that Ext j (Mn; R) = 0; for j = 1; :: : ;2n?2 and Ext 2n?1 (Mn; R) 6 = 0; and we use this result to show the removability of certain singularities of the Cauchy{Fueter system. R esum e. Soit R l'anneau des polynomes de 4n variables. Soit An la transformation de Fourier de la matrice d'op erateurs dii erentiels associ ee a la condition de r egularit e impos ee a une fonction de n variables quaterniones. Soit aussi hAni le module d eeni par les colonnes de An. Dans cet article nous prouvons que la dimension projective du module Mn = R 4 =hAni est 2n ? 1. Nous prouvons ensuite, dans un corollaire, que la dimension asque du fais-ceau R des fonctions r eguli eres est 2n ? 1, et nous prouvons que certains groups de cohomologie sont z ero pour les ouverts de l'espace H n de quater-nions. Nous prouvons que Ext j (Mn; R) = 0; pour j = 1; :: : ;2n ? 2 et que Ext 2n?1 (Mn; R) 6 = 0, et nous utilisons ce r esultat pour prouver que certaines singularit es du system de Cauchy-Fueter peuvent ^ etre elimin ees.

Hartog's phenomenon for polyregular functions and projective dimension of related modules over a polynomial ring