no. 2
Hartog's phenomenon for polyregular functions and projective dimension of related modules over a polynomial ring
Annales de l'Institut Fourier, Tome 47 (1997) no. 2, pp. 623-640.

Soit R l’anneau des polynômes de 4n variables. Soit A n la transformation de Fourier de la matrice d’opérateurs différentiels associée à la condition de régularité imposée à une fonction de n variables quaternioniques, et 〈A n 〉 le module défini par les colonnes de A n . Dans cet article nous prouvons que la dimension projective du module ℳ n =R 4 /〈A n 〉 est 2n-1. Nous prouvons ensuite, comme corollaire, que la dimension flasque du faisceau ℛ des fonctions régulières est 2n-1, et que certains groupes de cohomologie sont nuls pour les ouverts de l’espace ℍ n de quaternions. Nous démontrons que Ext j (ℳ n ,R)=0, pour j=1,⋯,2n-2 et que Ext 2n-1 (ℳ n ,R)≠0, et nous utilisons ce résultat pour prouver que certaines singularités du système de Cauchy-Fueter peuvent être éliminées.

In this paper we prove that the projective dimension of ℳ n =R 4 /〈A n 〉 is 2n-1, where R is the ring of polynomials in 4n variables with complex coefficients, and 〈A n 〉 is the module generated by the columns of a 4×4n matrix which arises as the Fourier transform of the matrix of differential operators associated with the regularity condition for a function of n quaternionic variables. As a corollary we show that the sheaf ℛ of regular functions has flabby 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 (ℳ n ,R)=0, for j=1,⋯,2n-2 and Ext 2n-1 (ℳ n ,R)≠0, and we use this result to show the removability of certain singularities of the Cauchy–Fueter system.

@article{AIF_1997__47_2_623_0,
     author = {Adams, William W. and Loustaunau, Philippe and Palamodov, Victor P. and Struppa, Daniele C.},
     title = {Hartog's phenomenon for polyregular functions and projective dimension of related modules over a polynomial ring},
     journal = {Annales de l'Institut Fourier},
     pages = {623--640},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {2},
     year = {1997},
     doi = {10.5802/aif.1576},
     zbl = {0974.32005},
     mrnumber = {98f:32013},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1576/}
}
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Adams, William W.; Loustaunau, Philippe; Palamodov, Victor P.; Struppa, Daniele C. Hartog's phenomenon for polyregular functions and projective dimension of related modules over a polynomial ring. Annales de l'Institut Fourier, Tome 47 (1997) no. 2, pp. 623-640. doi : 10.5802/aif.1576. https://aif.centre-mersenne.org/articles/10.5802/aif.1576/

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