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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 $4\times 4n$ 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 $ℛ$ of regular functions has flabby dimension $2n-1$, and we prove a cohomology vanishing theorem for open...