The ring structure on the cohomology of coordinate subspace arrangements (Q1974152)

Every simplicial complex \(\Delta\subset 2^{[n]}\) on the vertex set \([n]= \{1,\dots, n\}\) defines a real resp. complex arrangement of coordinate subspaces in \(\mathbb{R}^n\) resp. \(\mathbb{C}^n\) via the correspondence \(\Delta\ni \sigma\mapsto \text{span}\{e_i:i\in \sigma\}\). The linear structure of the cohomology of the complement of such an arrangement is explicitly given in terms of the combinatorics of \(\Delta\) and its links by the Goresky-MacPherson formula. Here we derive, by combinatorial means, the ring structure on the integral cohomology in terms of data of \(\Delta\). We provide a non-trivial example of different cohomology rings in the real and complex case. Furthermore, we give an example of a coordinate arrangement that yields non-trivial multiplication of torsion elements.