Cohmology of Orlik-Solomon Algebras with respect to Rank 2 Modular Elements

Jianming Yu and Guangfeng Jiang
Department of mathematics and information science, Faculty of Science, Beijing university of
chemical technology, Beijing 100029, P. R. China

Abstract

Let R be a commutative ring with 1. Let E be the graded exterior algebra over R generated by 1 and the degree one elements e₁,..., e_n. Define an R-linear map , for i = 1,..., n, and

for p 2, where ^ indicates an omitted factor. Let M be a simple matroid on [n]={1,..., n} with rank l. The Orlik-Solomon algebra OS(M) of M is the quotient of E modulo ideal I generated by the circuit boundaries is a circuit >. The Orlik-Solomon algebra has both combinatorial and topological significance. Let A be the hyperplane arrangement arising from a complex realization of M. It was proved by P. Orlik and L. Solomon that OS(M) is isomorphic to the cohomology algebra of the complement of Since ideal I is graded, the Orlik-Solomon algebra OS(M) has the natural grading. For an element the left multiplication by defines a map OS^p(M) OS^p+1(M)which squares to zero. Thus we have a cochain complex (OS(M),). It is well-known that the cohomology H^p(OS(M), )=0 for all p if . It has been general interest to compute the cohomology for In this paper, we compute the cohomology of (OS(M), ) in case there is a rank two modular element X in the flat lattice L of M and is concentrated on X, i.e., _j 0 if and only if j X in L. We express the dimension of H^p(OS(M), ) by the dimension of OSⁱ(M) , i p .