Syzygies of unimodular Lawrence ideals (Q2717081)

Let \(L\subset\mathbb{Z}^n\) be a lattice. The Lawrence ideal \(J_L \subset R=K[X_1, \dots, X_n,Y_1,\dots,Y_n]\) is generated by all binomials \(X^a Y^b -X^bY^a\), for which \(a-b\in L\) (here \(K\) is a field). These ideals have much stronger properties than lattice ideals in general.NEWLINENEWLINENEWLINEThe authors start by showing that the lattice \(L\) is unimodular if (and only if) \(R/J_L\) is normal. Unimodularity of \(L\) is characterized by the existence of a basis that, as a matrix, has all its maximal minors equal to \(0,\pm 1\).NEWLINENEWLINENEWLINEFrom \(L\) an infinite hyperplane arrangement \({\mathcal H}\) with vertices exactly in \(L\) is constructed. The cell structure of \({\mathcal H}\) modulo the action of \(L\) supports a complex \({\mathcal F}\) of free \(R\)-modules. This complex is the main object of the paper: It is a minimal free graded resolution of \(J_L\). -- The authors furthermore construct minimal free resolutions of the so-called fiber monomial ideals and the initial monomial ideal \(\text{ini} (J_L)\) of \(J_L\) (for an arbitrary term order on \(R)\). A surprising fact: \(\text{ini} ({\mathcal F})\) resolves \(\text{ini} (J_L)\). -- The results are applied to Lawrence ideals arising from graphs, which turn out to be unimodular.NEWLINENEWLINENEWLINEIn the last section, the authors consider the diagonal embedding of a unimodular toric variety and give a suitable version of the Beilinson spectral sequence in this context.