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 4x4n 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 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.