the result, we need an essential combinatorial ingredient, which we describe next.

Denote by $E^*({{\mathcal A}})$ the exterior algebra generated, over an arbitrary unitary commutative ring $\mathbb{K}$ by elements $e_H$ of degree $1$, one for each $H \in {{\mathcal A}}$. The [*Orlik-Solomon algebra*]{} of ${{\mathcal A}},\; A_{\mathbb{K}}^*({{\mathcal A}})$ is defined as a quotient of $E^*({{\mathcal A}})$ by an ideal $I$ generated by dependency relations in $\mathcal L({{\mathcal A}})$. More precisely, one has a canonical boundary operator, $\partial \colon E^*({{\mathcal A}})\to E^{*-1}({{\mathcal A}})$, defined by $$\label{eq:32}
 \partial (e_{H_{i_1}}\cdots e_{H_{i_t}})= \sum_{j=1}^t (-1)^{j-1} e_{H_{i_1}}\cdots 
 \widehat{e_{H_{i_j}}} \cdots e_{H_{i_t}}\, ,$$ where $\widehat{\bullet}$ denotes deletion of $\bullet$. For central arrangements, $I$ is generated by boundaries of elements $e_{H_{i_1}} \dots e_{H_{i_t}}$ such that $r(\bigcap_{j=1}^{t}H_{i_{j}}) <t$. For arbitrary affine arrangements, one also assumes in the above construction that $\bigcap_{j=1}^{t}H_{i_{j}}\ne \emptyset$, and then also adds to the generating set of $I$ elements $e_{H_{i_1}} \dots e_{H_{i_t}}$ such that $ \bigcap_{j=1}^{t}H_{i_{j}}= \emptyset$. We denote the generators of $A_{\mathbb{K}}^*({{\mathcal A}})$ again by $e_{H}$, for $H \in {{\mathcal A}}$.

The simplest example of OS-algebra is the one associated to the boolean arrangement. Since there are no dependency relations among the hyperplanes, the OS-algebra is just the exterior algebra, $E^*({{\mathcal A}})$.

\[thm:OS\] There is 