On Generators of the Module of Logarithmic 1-Forms with Poles Along an Arrangement
Sergey Yuzvinsky
University of Oregon Eugene OR 97403
DOI: 10.1023/A:1022480128534
Abstract
For each element X of codimension two of the intersection lattice of a hyperplane arrangement we define a differential logarithmic 1-forms 
X with poles along the arrangement. Then we describe the class of arrangements for which forms X generate the whole module of the logarithmic 1-forms with poles along the arrangement. The description is done in terms of linear relations among the functionals defining the hyperplanes. We construct a minimal free resolution of the module generated by X that in particular defines the projective dimension of this module. In order to study relations among X we construct free resolutions of certain ideals of a polynomial ring generated by products of linear forms. We give examples and discuss possible generalizations of the results.
X with poles along the arrangement. Then we describe the class of arrangements for which forms X generate the whole module of the logarithmic 1-forms with poles along the arrangement. The description is done in terms of linear relations among the functionals defining the hyperplanes. We construct a minimal free resolution of the module generated by X that in particular defines the projective dimension of this module. In order to study relations among X we construct free resolutions of certain ideals of a polynomial ring generated by products of linear forms. We give examples and discuss possible generalizations of the results.Pages: 253–269
Keywords: hyperplane arrangement; logarithmic form; module; free resolution; ideal
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References
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2. P. Deligne, "Theorie de Hodge," Publ. Math. IHES 40 (1972), 5-57.
3. T. Jozefiak and B. Sagan, "Basic derivations for subarrangements of Coxeter arrangements," J. Alg. Combin. 2 (1993), 291-320.
4. E. Kunz, Introduction to commutative algebra and algebraic geometry, Birkhauser, Boston, 1985.
5. K.S. Lee, "On logarithmic forms and arrangements of hyperplanes," Thesis, University of Wisconsin, Madison, 1994.
6. D. Northcott, Finite free resolutions, Cambridge Univ. Press, London, 1976.
7. D. Quillen, "Homotopy properties of the poset of non-trivial p-subgroups of a group," Advances in Math. 28 (1978), 101-128.
8. L. Rose and H. Terao, "A free resolution of the module of logarithmic forms of a generic arrangement," J. of Algebra 136 (1991), 376-400.
9. K. Saito, 'Theory of logarithmic differential forms and logarithmic vector fields," J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 27 (1981), 265-291.
10. L. Solomon and H. Terao, "A formula for the characteristic polynomial of an arrangement," Advances in Math. 64 (1987), 305-325.
11. J. StUckard and W. Vogel, Buchsbaum Rings and Applications, Springer-Verlag, Berlin/New York, 1986.
12. S. Yuzvinsky, "A free resolution of the module of derivations for generic arrangements," J. of Algebra 136 (1991), 432-438.
13. S. Yuzvinsky, "The first two obstructions to the freeness of arrangements," Trans. AMS 335 (1993), 231-244.
14. G.M. Ziegler, "Combinatorial constructions of logarithmic differential forms," Advances in Math. 76 (1989), 116-154.