Toral arrangements and hyperplane arrangements (Q1293506)

The author defines a toral arrangement to be a finite set \({\mathcal A}\) of characters of an algebraic torus \(T\). Such a set corresponds to an integral hyperplane arrangement \(d{\mathcal A}\) in the Lie algebra of the torus given by the kernels of the derivatives of the characters. Let \(D({\mathcal A})\) denote the set of derivations \(\theta\) of the coordinate ring \({\mathcal O}(T)\) of the torus such that \(\theta(y)\in y {\mathcal O}(T)\) where \(y:=\prod_{\chi\in{\mathcal A}}(\chi-1)\) is the defining equation of \({\mathcal A}\). The author proves that \(d{\mathcal A}\) is a free hyperplane arrangement if and only if the localization of \(D({\mathcal A})\) at the identity \(e\) of \(T\) is a free module over the localization of \({\mathcal O}(T)\) in \(e\). Moreover, if that is the case then the exponents of the hyperplane arrangement \(d{\mathcal A}\) can be recovered from \({\mathcal A}\).