On the algebra associated with a geometric lattice

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23 Citations (Scopus)

Abstract

Let L be a geometric lattice. Following P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. math. 56 (1980), 167-189, we associate with L a graded commutative algebra A(L). In this paper we introduce a new invariant ψ of the algebra A(L) which suffices to distinguish algebras for which all other known invariants coincide. This result is applied to the study of arrangements of complex hyperplanes, with L being the intersection lattice. In this case A(L) is isomorphic to the cohomology algebra of the associated hyperplane complement. The goal is to find examples of arrangements with non-isomorphic lattices but homotopy equivalent complements. The invariant introduced here effectively narrows the list of candidates. Nevertheless, we exhibit combinatorially inequivalent arrangements for which all known invariants, including ψ, coincide.

Original languageEnglish (US)
Pages (from-to)152-163
Number of pages12
JournalAdvances in Mathematics
Volume80
Issue number2
DOIs
StatePublished - 1990

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Geometric Lattice
Hyperplane
Arrangement
Algebra
Invariant
Complement
Graded Algebra
Commutative Algebra
Combinatorics
Homotopy
Cohomology
Isomorphic
Intersection
Topology

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

On the algebra associated with a geometric lattice. / Falk, Michael J.

In: Advances in Mathematics, Vol. 80, No. 2, 1990, p. 152-163.

Research output: Contribution to journalArticle

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