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), p. 623-640 / Harvested from Numdam

Résumé

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 ${\cal M}_n = R^4/\langle A_n\rangle $ est $2n-1$. Nous prouvons ensuite, comme corollaire, que la dimension flasque du faisceau ${\cal R}$ des fonctions régulières est $2n-1$, et que certains groupes de cohomologie sont nuls pour les ouverts de l’espace ${\Bbb H}^n$ de quaternions. Nous démontrons que ${\rm Ext}^j({\cal M}_n,R) = 0,$ pour $j=1,\dots ,2n-2$ et que ${\rm Ext}^{2n-1}({\cal M}_n,R) \ne 0$, et nous utilisons ce résultat pour prouver que certaines singularités du système de Cauchy-Fueter peuvent être éliminées.

Abstract

In this paper we prove that the projective dimension of ${\cal 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 ${\cal R}$ of regular functions has flabby dimension $2n-1$, and we prove a cohomology vanishing theorem for open sets in the space ${\Bbb H}^n$ of quaternions. We also show that ${\rm Ext}^j({\cal M}_n,R)=0$, for $j=1, \dots ,2n-2$ and ${\rm Ext}^{2n-1}({\cal M}_n,R) \ne 0,$ and we use this result to show the removability of certain singularities of the Cauchy–Fueter system.

Détail

Publié le : 1997-01-01

MR 98f:32013 | Zbl 0974.32005

DOI : https://doi.org/10.5802/aif.1576

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