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A free resolution of the module of derivations for generic arrangements. (English) Zbl 0732.13009

For a generic arrangement \({\mathcal A}\) of n hyperplanes \(H_ 1,...,H_ n\) in the \(\ell\)-dimensional K-vectorspace V with dual \(V^*\) and corresponding symmetric algebra \(S=Sym(V^*)\) let \(Der(S)=Der_ K(S,S)\) be the S-module of derivations. Fix \(\alpha_ i\in V^*\), \(i=1,...,n\), such that \(H_ i=Ker(\alpha_ i)\). Then the module of logarithmic derivations along \({\mathcal A}\), \(D=D({\mathcal A})\), is defined as D(\({\mathcal A})=\{\theta \in Der(S)| \theta (\alpha_ i)\in S\alpha_ i\), \(i=1,...,n\}\). A (new) construction of a minimal free resolution of D is described and the homological dimension \(hd_ SD\) is proved to be equal to \(\ell -2 (n>\ell)\). To this end one writes \(D=D_ 0\oplus_ SS\theta_ 0\), where \(D_ 0=D_ 0({\mathcal A})=\{\theta \in D| \theta (\alpha_ n)=0\}\) and \(\theta_ 0\) is the Euler derivation defined by \(\theta_ 0(\alpha)=\alpha\) for all \(\alpha \in V^*\). Thus the construction of a resolution of D is reduced to one for \(D_ 0\). This is accomplished by explicit construction of suitable exact complexes involving the intersection lattice of \({\mathcal A}\). As a final step one constructs an exact complex \(F_*=(F_ k,\delta_ k)\), \(\delta_ k: F_ k\to F_{k-1}\), \(k=0,...,\ell -1\), with \(rank_ S(F_ k)=(^{n- 1}_{\ell -1-k})(_{k-1}^{n-\ell +k-2})\). The sequence \(F_*\) gives a minimal resolution of \(D_ 0\) (at least when \(n>\ell >2\) and K is algebraically closed).

MSC:

13N10 Commutative rings of differential operators and their modules
05B30 Other designs, configurations
13D25 Complexes (MSC2000)
13B10 Morphisms of commutative rings
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References:

[1] Northcott, D., Finite Free Resolutions (1976), Cambridge Univ. Press: Cambridge Univ. Press London/New York · Zbl 0328.13010
[2] L. Rose and H. Terao, A free resolution of the module of logarithmic forms of a generic arrangement, preprint.; L. Rose and H. Terao, A free resolution of the module of logarithmic forms of a generic arrangement, preprint. · Zbl 0732.13010
[3] Stückrad, J.; Vogel, W., Buchsbaum Rings and Applications (1986), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0606.13018
[4] Terao, H., Free arrangements of hyperplanes over an arbitrary field, (Proc. Japan. Acad., 59 (1987)), 301-304 · Zbl 0634.05019
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