In this paper we prove that the projective dimension of Mn = R^4/(An) is 2n -1, where R is the ring of polynomials in 4n variables with complex coefficients and (An) 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 R of regular functions has flabby dimension 2n -1, and we prove a cohomology vanishing theorem for open sets in the space Hn of quaternions. We also show that Ext3(Mn, R) = 0, for j = 1,...., 2n - 2, and Ext ^(2n -1) (Mn, R) =/= 0, and we use this result to show the removability of certain singularities of the Cauchy Fueter system.
Adams, W.W., Loustaunau, P., Palamodov, VP., & Struppa, D.C. (1997). Hartog's phenomenon for polyregular functions and projective dimension of related modules over a polynomial ring. Annales de l'Institut Fourier, 47(2), 623-640. Retrieved from http://aif.cedram.org/aif-bin/item?id=AIF_1997__47_2_623_0