### Abstract

If A. is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G = π _{1}(X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A = H*(X,k), viewed as a module over the exterior algebra E on A: θ _{k}(G) = dim _{k} Tor _{k-1} ^{E} (A,k) _{k}, for k ≥ 2, where k is a field of characteristic 0. The Chen ranks conjecture asserts that, for k sufficiently large, θ _{k}(G) = (k - 1) ∑ _{r≥1} h _{r} ( ^{r+k-1} _{k}, where h _{r} is the number of r-dimensional components of the projective resonance variety R ^{1}(A). Our earlier work on the resolution of A over E and the above equality yield a proof of the conjecture for graphic arrangements. Using results on the geometry of R ^{1}(A) and a localization argument, we establish the inequality θ _{k}(k-1) ∑ _{r>1} ^{hr} ( _{k} ^{r+k-1}), f k ≫ 0 for arbitrary A. Finally, we show that there is a polynomial P(t) of degree equal to the dimension of R ^{1}l(A), such that θ _{k}(G) = P(k), for all k ≫ 0.

Original language | English (US) |
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Pages (from-to) | 2269-2289 |

Number of pages | 21 |

Journal | Transactions of the American Mathematical Society |

Volume | 358 |

Issue number | 5 |

DOIs | |

State | Published - May 2006 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Transactions of the American Mathematical Society*,

*358*(5), 2269-2289. https://doi.org/10.1090/S0002-9947-05-03853-5