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

William W. Adams; Philippe Loustaunau; Victor P. Palamodov; Daniele C. Struppa

Annales de l'institut Fourier (1997)

- Volume: 47, Issue: 2, page 623-640
- ISSN: 0373-0956

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topAdams, William W., et al. "Hartog's phenomenon for polyregular functions and projective dimension of related modules over a polynomial ring." Annales de l'institut Fourier 47.2 (1997): 623-640. <http://eudml.org/doc/75240>.

@article{Adams1997,

abstract = {In this paper we prove that the projective dimension of $\{\cal 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 $\{\cal R\}$ of regular functions has flabby dimension $2n-1$, and we prove a cohomology vanishing theorem for open sets in the space $\{\Bbb H\}^n$ of quaternions. We also show that $\{\rm Ext\}^j(\{\cal M\}_n,R)=0$, for $j=1, \dots ,2n-2$ and $\{\rm Ext\}^\{2n-1\}(\{\cal M\}_n,R) \ne 0,$ and we use this result to show the removability of certain singularities of the Cauchy–Fueter system.},

author = {Adams, William W., Loustaunau, Philippe, Palamodov, Victor P., Struppa, Daniele C.},

journal = {Annales de l'institut Fourier},

keywords = {Hartog's phenomenon; regular functions; projective dimension; Cauchy-Fueter; system quaternions},

language = {eng},

number = {2},

pages = {623-640},

publisher = {Association des Annales de l'Institut Fourier},

title = {Hartog's phenomenon for polyregular functions and projective dimension of related modules over a polynomial ring},

url = {http://eudml.org/doc/75240},

volume = {47},

year = {1997},

}

TY - JOUR

AU - Adams, William W.

AU - Loustaunau, Philippe

AU - Palamodov, Victor P.

AU - Struppa, Daniele C.

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

JO - Annales de l'institut Fourier

PY - 1997

PB - Association des Annales de l'Institut Fourier

VL - 47

IS - 2

SP - 623

EP - 640

AB - In this paper we prove that the projective dimension of ${\cal 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 ${\cal R}$ of regular functions has flabby dimension $2n-1$, and we prove a cohomology vanishing theorem for open sets in the space ${\Bbb H}^n$ of quaternions. We also show that ${\rm Ext}^j({\cal M}_n,R)=0$, for $j=1, \dots ,2n-2$ and ${\rm Ext}^{2n-1}({\cal M}_n,R) \ne 0,$ and we use this result to show the removability of certain singularities of the Cauchy–Fueter system.

LA - eng

KW - Hartog's phenomenon; regular functions; projective dimension; Cauchy-Fueter; system quaternions

UR - http://eudml.org/doc/75240

ER -

## References

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- [2] W.W. ADAMS and P. LOUSTAUNAU, An Introduction to Gröbner Bases, Graduate Studies in Mathematics, Vol. 3, American Mathematical Society, Providence, (RI), 1994. Zbl0803.13015MR95g:13025
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- [5] A. FABIANO, G. GENTILI, D.C. STRUPPA, Sheaves of quaternionic hyperfunctions and microfunctions, Compl. Var. Theory and Appl., 24 (1994), 161-184. Zbl0819.30030MR95d:30091
- [6] H. KOMATSU, Relative Cohomology of Sheaves of Solutions of Differential Equations, Springer LNM, 287 (1973), 192-261. Zbl0278.58010MR52 #14681
- [7] B. MALGRANGE, Faisceaux sur des variétés analytiques réelles, Bull. Soc. Math. France, 85 (1957), 231-237. Zbl0079.39201MR20 #1340
- [8] V.P. PALAMODOV, Linear Differential Operators with Constant Coefficients, Springer Verlag, New York, 1970. (English translation of Russian original, Moscow, 1967.) Zbl0191.43401
- [9] J.J. ROTMAN, An Introduction to Homological Algebra, Academic Press, New York, 1979. Zbl0441.18018MR80k:18001
- [10] M. SATO, T. KAWAI, and M. KASHIWARA, Microfunctions and Pseudo-Differential Equations, Springer LNM, 287 (1973), 265-529. Zbl0277.46039MR54 #8747

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