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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$\beta \,\mathbf {nbc}$-bases for cohomology of local systems
on hyperplane complements


Authors: Michael Falk and Hiroaki Terao
Journal: Trans. Amer. Math. Soc. 349 (1997), 189-202
MSC (1991): Primary 52B30
MathSciNet review: 1401770
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Abstract: We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes ${\mathcal A} $. The cohomology plays an important role for the theory of generalized hypergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for the only non-vanishing cohomology group, under some nonresonance conditions on the local system, for any arrangement ${\mathcal A} $. The bases are determined by a linear ordering of the hyperplanes, and are indexed by certain ``no-broken-circuits" bases of ${\mathcal A} $. The basic forms depend on the local system, but any two bases constructed in this way are related by a matrix of integer constants which depend only on the linear orders and not on the local system. In certain special cases we show the existence of bases of monomial logarithmic forms.


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Additional Information

Michael Falk
Affiliation: Department of Mathematics, University of Wisconsin - Madison, Madison, Wisconsin 53704
Email: mjf@odin.math.nau.edu

Hiroaki Terao
Affiliation: Department of Mathematics, University of Wisconsin - Madison, Madison, Wisconsin 53704
Email: terao@math.wisc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01844-8
PII: S 0002-9947(97)01844-8
Received by editor(s): April 30, 1995
Additional Notes: The first author was partially supported by a Northern Arizona University Organized Research Grant
Dedicated: In memory of Michitake Kita
Article copyright: © Copyright 1997 American Mathematical Society