| | |
Summary: HARTOG'S PHENOMENON FOR POLYREGULAR FUNCTIONS
AND PROJECTIVE DIMENSION OF RELATED MODULES
OVER A POLYNOMIAL RING
W.W. ADAMS, P. LOUSTAUNAU, V.P. PALAMODOV, D.C. STRUPPA
Abstract. In this paper we prove that the projective dimension of Mn =
R 4 =hAn i is 2n \Gamma 1, where R is the ring of polynomials in 4n variables with
complex coefficients, and hAn i is the module generated by the columns of a
4 \Theta 4n matrix which arises as the Fourier transform of the matrix of differential
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 flabby dimension 2n \Gamma 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 \Gamma 2 and Ext 2n\Gamma1 (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 trans
formation de Fourier de la matrice d'op'erateurs diff'erentiels associ'ee `a la con
dition de r'egularit'e impos'ee ` a une fonction de n variables quaterniones. Soit
aussi hAn i le module d'efini par les colonnes de An . Dans cet article nous
prouvons que la dimension projective du module Mn = R 4 =hAn i est 2n \Gamma 1.
Nous prouvons ensuite, dans un corollaire, que la dimension flasque du fais
|
