### Hartog's phenomenon for polyregular functions and projective dimension of related modules over a polynomial ring

In this paper we prove that the projective dimension of ${\mathcal{M}}_{n}={R}^{4}/\u27e8{A}_{n}\u27e9$ is $2n-1$, where $R$ is the ring of polynomials in $4n$ variables with complex coefficients, and $\u27e8{A}_{n}\u27e9$ 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 $\mathcal{R}$ of regular functions has flabby dimension $2n-1$, and we prove a cohomology vanishing theorem for open...