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 n =R 4 /A n est 2n-1. Nous prouvons ensuite, comme corollaire, que la dimension flasque du faisceau des fonctions régulières est 2n-1, et que certains groupes de cohomologie sont nuls pour les ouverts de l’espace n de quaternions. Nous démontrons que Ext j ( n ,R)=0, pour j=1,,2n-2 et que Ext 2n-1 ( 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 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 of regular functions has flabby dimension 2n-1, and we prove a cohomology vanishing theorem for open sets in the space n of quaternions. We also show that Ext j ( n ,R)=0, for j=1,,2n-2 and Ext 2n-1 ( 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},
     mrnumber = {98f:32013},
     zbl = {0974.32005},
     language = {en},
     url = {http://archive.numdam.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. http://archive.numdam.org/articles/10.5802/aif.1576/

[1] W.W. Adams, C.A. Berenstein, P. Loustaunau, I. Sabadini, and D.C. Struppa, Regular Functions of Several Quaternionic Variables and the Cauchy-Fueter Complex, to appear in J. of Complex Variables. | Zbl

[2] W.W. Adams and P. Loustaunau, An Introduction to Gröbner Bases, Graduate Studies in Mathematics, Vol. 3, American Mathematical Society, Providence, (RI), 1994. | MR | Zbl

[3] J. Eagon and D.G. Northcott, Ideals defined by matrices, and a certain complex associated to them, Proc. Royal Soc., 269 (1962), 188-204. | Zbl

[4] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer Verlag, New York (NY), 1994. | Zbl

[5] A. Fabiano, G. Gentili, D.C. Struppa, Sheaves of quaternionic hyperfunctions and microfunctions, Compl. Var. Theory and Appl., 24 (1994), 161-184. | MR | Zbl

[6] H. Komatsu, Relative Cohomology of Sheaves of Solutions of Differential Equations, Springer LNM, 287 (1973), 192-261. | MR | Zbl

[7] B. Malgrange, Faisceaux sur des variétés analytiques réelles, Bull. Soc. Math. France, 85 (1957), 231-237. | Numdam | MR | Zbl

[8] V.P. Palamodov, Linear Differential Operators with Constant Coefficients, Springer Verlag, New York, 1970. (English translation of Russian original, Moscow, 1967.) | Zbl

[9] J.J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979. | MR | Zbl

[10] M. Sato, T. Kawai, and M. Kashiwara, Microfunctions and Pseudo-Differential Equations, Springer LNM, 287 (1973), 265-529. | MR | Zbl

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