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

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 3 x 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 Ext(j) (M-n, R) = 0, for j = 1, ..., 2n - 2 and Ext(2n-1)(M-n, R) not equal 0, and we use this result to show the removability of certain singularities of the Cauchy-Fueter system.