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Hartog’s phenomenon for polyregular functions and projective dimension of related modules over a polynomial ring. (English) Zbl 0974.32005
Summary: We prove that the projective dimension of \({\mathcal M}_n=R^4/\langle A_n\rangle\) is \(2n-1\), where \(R\) is the ring of polynomials in \(4n\) variables with complex coefficients, and \(\langle A_n\rangle\) 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 sets in the space \({\mathbb H}^n\) of quaternions. We also show that \(\text{Ext}^j({\mathcal M}_n,R)=0\), for \(j=1, \dots,2n-2\) and \(\text{Ext}^{2n-1}({\mathcal M}_n,R) \neq 0,\) and we use this result to show the removability of certain singularities of the Cauchy-Fueter system.

MSC:
32D20 Removable singularities in several complex variables
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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