×

Cohomology of local sheaves on arrangement lattices. (English) Zbl 0758.32014

Summary: We apply cohomology of sheaves to arrangements of hyperplanes. In particular we prove an inequality for the depth of cohomology modules of local sheaves on the intersection lattice of an arrangement. This generalizes a result of Solomon-Terao about the commulative property of local functors. We also prove a characterization of free arrangements by certain properties of the cohomology of a sheaf of derivation modules. This gives a condition on the Möbius function of the intersection lattice of a free arrangement. Using this condition we prove that certain geometric lattices cannot afford free arrangements although their Poincaré polynomials factor.

MSC:

32S20 Global theory of complex singularities; cohomological properties
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
05B35 Combinatorial aspects of matroids and geometric lattices
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kenneth Bacławski, Whitney numbers of geometric lattices, Advances in Math. 16 (1975), 125 – 138. · Zbl 0326.05027 · doi:10.1016/0001-8708(75)90145-0
[2] Jon Folkman, The homology groups of a lattice, J. Math. Mech. 15 (1966), 631 – 636. · Zbl 0146.01602
[3] Roger Godement, Topologie algébrique et théorie des faisceaux, Actualit’es Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13, Hermann, Paris, 1958 (French). · Zbl 0080.16201
[4] Robin Hartshorne, Local cohomology, A seminar given by A. Grothendieck, Harvard University, Fall, vol. 1961, Springer-Verlag, Berlin-New York, 1967. · Zbl 0237.14008
[5] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001
[6] Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. · Zbl 0441.13001
[7] D. G. Northcott, Finite free resolutions, Cambridge University Press, Cambridge-New York-Melbourne, 1976. Cambridge Tracts in Mathematics, No. 71. · Zbl 0328.13010
[8] P. Orlik, Introduction to arrangements, Amer. Math. Soc., Providence, R. I., 1989. · Zbl 0722.51003
[9] Lauren L. Rose and Hiroaki Terao, A free resolution of the module of logarithmic forms of a generic arrangement, J. Algebra 136 (1991), no. 2, 376 – 400. · Zbl 0732.13010 · doi:10.1016/0021-8693(91)90052-A
[10] L. Solomon and H. Terao, A formula for the characteristic polynomial of an arrangement, Adv. in Math. 64 (1987), no. 3, 305 – 325. · Zbl 0625.05001 · doi:10.1016/0001-8708(87)90011-9
[11] R. P. Stanley, Supersolvable lattices, Algebra Universalis 2 (1972), 197 – 217. · Zbl 0256.06002 · doi:10.1007/BF02945028
[12] Hiroaki Terao, Arrangements of hyperplanes and their freeness. I, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 293 – 312. Hiroaki Terao, Arrangements of hyperplanes and their freeness. II. The Coxeter equality, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 313 – 320. · Zbl 0509.14006
[13] Hiroaki Terao, Free arrangements of hyperplanes and unitary reflection groups, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), no. 8, 389 – 392. Hiroaki Terao, Generalized exponents of a free arrangement of hyperplanes and Shepherd-Todd-Brieskorn formula, Invent. Math. 63 (1981), no. 1, 159 – 179. · Zbl 0476.14016 · doi:10.1007/BF01389197
[14] Hiroaki Terao, Free arrangements of hyperplanes over an arbitrary field, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 7, 301 – 303. · Zbl 0634.05019
[15] -, On the homological dimensions of arrangements, in preparation.
[16] Sergey Yuzvinsky, Cohen-Macaulay seminormalizations of unions of linear subspaces, J. Algebra 132 (1990), no. 2, 431 – 445. · Zbl 0705.13011 · doi:10.1016/0021-8693(90)90139-F
[17] Sergey Yuzvinsky, A free resolution of the module of derivations for generic arrangements, J. Algebra 136 (1991), no. 2, 432 – 438. · Zbl 0732.13009 · doi:10.1016/0021-8693(91)90054-C
[18] Günter M. Ziegler, Matroid representations and free arrangements, Trans. Amer. Math. Soc. 320 (1990), no. 2, 525 – 541. · Zbl 0727.05019
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.