## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Arrangements – Tokyo 1998, M. Falk and H. Terao, eds. (Tokyo: Mathematical Society of Japan, 2000), 27 - 49

### On the Cohomology of Discriminantal Arrangements and Orlik–Solomon Algebras

#### Abstract

We relate the cohomology of the Orlik–Solomon algebra of a discriminantal arrangement to the local system cohomology of the complement. The Orlik–Solomon algebra of such an arrangement (viewed as a complex) is shown to be a linear approximation of a complex arising from the fundamental group of the complement, the cohomology of which is isomorphic to that of the complement with coefficients in an arbitrary complex rank one local system. We also establish the relationship between the cohomology support loci of the complement of a discriminantal arrangement and the resonant varieties of its Orlik–Solomon algebra.

#### Article information

**Source***Arrangements – Tokyo 1998*, M. Falk and H. Terao, eds. (Tokyo: Mathematical Society of Japan, 2000), 27-49

**Dates**

First available in Project Euclid:
20 August 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1534788964

**Digital Object Identifier**

doi:10.2969/aspm/02710027

**Mathematical Reviews number (MathSciNet)**

MR1796892

**Zentralblatt MATH identifier**

0978.32028

**Subjects**

Primary: 32S22: Relations with arrangements of hyperplanes [See also 52C35] 55N25: Homology with local coefficients, equivariant cohomology

Secondary: 20F36: Braid groups; Artin groups

**Keywords**

discriminantal arrangement local system Orlik–Solomon algebra

#### Citation

Cohen, Daniel C. On the Cohomology of Discriminantal Arrangements and Orlik–Solomon Algebras. Arrangements – Tokyo 1998, 27--49, Mathematical Society of Japan, Tokyo, Japan, 2000. doi:10.2969/aspm/02710027. https://projecteuclid.org/euclid.aspm/1534788964