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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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##### References:
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