Adams, William W.; Loustaunau, Philippe; Palamodov, Victor P.; Struppa, Daniele C. Hartog’s phenomenon for polyregular functions and projective dimension of related modules over a polynomial ring. (English) Zbl 0974.32005 Ann. Inst. Fourier 47, No. 2, 623-640 (1997). 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. Cited in 1 ReviewCited in 26 Documents MSC: 32D20 Removable singularities in several complex variables 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Keywords:Hartog’s phenomenon; regular functions; projective dimension; Cauchy-Fueter; system quaternions PDFBibTeX XMLCite \textit{W. W. Adams} et al., Ann. Inst. Fourier 47, No. 2, 623--640 (1997; Zbl 0974.32005) Full Text: DOI Numdam EuDML References: [1] [1] , , , , and , Regular Functions of Several Quaternionic Variables and the Cauchy-Fueter Complex, to appear in J. of Complex Variables. · Zbl 0966.35088 [2] [2] and , An Introduction to Gröbner Bases, Graduate Studies in Mathematics, Vol. 3, American Mathematical Society, Providence, (RI), 1994. · Zbl 0803.13015 [3] [3] and , Ideals defined by matrices, and a certain complex associated to them, Proc. Royal Soc., 269 (1962), 188-204. · Zbl 0106.25603 [4] [4] , Commutative Algebra with a View Toward Algebraic Geometry, Springer Verlag, New York (NY), 1994. · Zbl 0819.13001 [5] [5] , , , Sheaves of quaternionic hyperfunctions and microfunctions, Compl. Var. Theory and Appl., 24 (1994), 161-184. · Zbl 0819.30030 [6] [6] , Relative Cohomology of Sheaves of Solutions of Differential Equations, Springer LNM, 287 (1973), 192-261. · Zbl 0278.58010 [7] [7] , Faisceaux sur des variétés analytiques réelles, Bull. Soc. Math. France, 85 (1957), 231-237. · Zbl 0079.39201 [8] [8] , Linear Differential Operators with Constant Coefficients, Springer Verlag, New York, 1970. (English translation of Russian original, Moscow, 1967.) · Zbl 0191.43401 [9] [9] , An Introduction to Homological Algebra, Academic Press, New York, 1979. · Zbl 0441.18018 [10] [10] , , and , Microfunctions and Pseudo-Differential Equations, Springer LNM, 287 (1973), 265-529. · Zbl 0277.46039 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.