### 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 language | English (US) |
---|---|

Pages (from-to) | 152-163 |

Number of pages | 12 |

Journal | Advances in Mathematics |

Volume | 80 |

Issue number | 2 |

DOIs | |

State | Published - 1990 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

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

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 80, no. 2, pp. 152-163. https://doi.org/10.1016/0001-8708(90)90024-H

}

TY - JOUR

T1 - On the algebra associated with a geometric lattice

AU - Falk, Michael J

PY - 1990

Y1 - 1990

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=38249019721&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38249019721&partnerID=8YFLogxK

U2 - 10.1016/0001-8708(90)90024-H

DO - 10.1016/0001-8708(90)90024-H

M3 - Article

AN - SCOPUS:38249019721

VL - 80

SP - 152

EP - 163

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 2

ER -