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

Adams, William W.; Loustaunau, Philippe; Palamodov, Victor P.; Struppa, Daniele C.

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 $\langle A_n\rangle$ le module défini par les colonnes de $A_n$. Dans cet article nous prouvons que la dimension projective du module ${ℳ}_n=R^4/\langle A_n\rangle$ 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 ${\mathrm{Ext}}^j({ℳ}_n,R)=0,$ pour $j=1,\cdots ,2n-2$ et que ${\mathrm{Ext}}^{2n-1}({ℳ}_n,R)\ne 0$, 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 ${ℳ}_n=R^4/\langle A_n\rangle$ is $2n-1$, where $R$ is the ring of polynomials in $4n$ variables with complex coefficients, and $\langle A_n\rangle$ is the module generated by the columns of a $4\times 4n$ 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 ${\mathrm{Ext}}^j({ℳ}_n,R)=0$, for $j=1,\cdots ,2n-2$ and ${\mathrm{Ext}}^{2n-1}({ℳ}_n,R)\ne 0,$ and we use this result to show the removability of certain singularities of the Cauchy–Fueter system.

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