Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Cohomology of local sheaves on arrangement lattices

Author: Sergey Yuzvinsky
Journal: Proc. Amer. Math. Soc. 112 (1991), 1207-1217
MSC: Primary 52B30; Secondary 05B35, 13D40, 32S20
MathSciNet review: 1062840
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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 cummulative property of local functors. We also prove a characterization of free arrangements by certain properties of the cohomlogy 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.

References [Enhancements On Off] (What's this?)

  • [B] K. Baclawski, Whitney numbers of geometric lattices, Adv. in Math. 16 (1975), 125-138. MR 0387086 (52:7933)
  • [F] J. Folkman, The homology group of a lattice, J. Math. Mech. 15 (1966), 631-636. MR 0188116 (32:5557)
  • [Go] R. Godement, Topologie algébrique et théorie des faisceaux, Hermann, Paris, 1958. MR 0102797 (21:1583)
  • [Gr] A. Grothendieck, Local cohomology, Lecture Notes in Math., vol. 41, 1967. MR 0224620 (37:219)
  • [H] R. Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
  • [M] H. Matsumura, Commutative algebra, Benjamin, New York, 1970. MR 0266911 (42:1813)
  • [N] D. Northcott, Finite free resolutions, Cambridge Univ. Press, London, 1976. MR 0460383 (57:377)
  • [O] P. Orlik, Introduction to arrangements, Amer. Math. Soc., Providence, R. I., 1989.
  • [RT] L. Rose and H. Terao, A free resolution of the module of logarithmic forms of a generic arrangement, J. Algebra (to appear). MR 1089305 (93h:32048)
  • [ST] L. Solomon and H. Terao, A formula for the characteristic polynomial of an arrangement, Adv. Math. 64 (1987), 305-325. MR 888631 (88m:32022)
  • [S] R. Stanley, Supersolvable lattices, Algebra Universalis 2 (1972), 197-217. MR 0309815 (46:8920)
  • [T1] H. Terao, Arrangements of hyperplanes and their freeness. I, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. (2) 27 (1980), 293-312. MR 586451 (84i:32016a)
  • [T2] -, Generalized exponents of a free arrangements of hyperplanes and Shephard-Todd-Brieskorn formula, Invent. Math. 63 (1981), 159-179. MR 608532 (82e:32018b)
  • [T3] -, Free arrangements of hyperplanes over an arbitrary field, Proc. Japan Acad. Ser. A Math. Sci. 59 (1987), 301-304. MR 726186 (85f:32017)
  • [T4] -, On the homological dimensions of arrangements, in preparation.
  • [Y1] S. Yuzvinsky, Cohen-Macaulay seminormalizations of unions of linear subspaces, J. Algebra 132 (1990), 431-445. MR 1061489 (91g:14046)
  • [Y2] -, A free resolution of the module of derivations for generic arrangements, J. Algebra (to appear). MR 1089307 (92b:52026)
  • [Z] G. Ziegler, Matroid representations and free arrangements, Trans. Amer. Math. Soc. (to appear). MR 986703 (90k:32045)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 52B30, 05B35, 13D40, 32S20

Retrieve articles in all journals with MSC: 52B30, 05B35, 13D40, 32S20

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society