Editing Arrangement of hyperplanes (section)

=== The Orlik–Solomon algebra ===

The intersection semilattice determines another combinatorial invariant of the arrangement, the [[Orlik–Solomon algebra]]. To define it, fix a commutative subring ''K'' of the base field and form the [[exterior algebra]] ''E'' of the vector space
:<math>\bigoplus_{H \in A} K e_H </math>
generated by the hyperplanes.
A [[chain complex]] structure is defined on ''E'' with the usual boundary operator <math>\partial</math>.
The Orlik–Solomon algebra is then the quotient of ''E'' by the [[Ideal (ring theory)|ideal]] generated by elements of the form <math>e_{H_1} \wedge \cdots \wedge e_{H_p}</math> for which <math>H_1, \dots, H_p</math> have empty intersection, and by boundaries of elements of the same form for which <math>H_1 \cap \cdots \cap H_p</math> has [[codimension]] less than ''p''.