Binomial edge ideals over an exterior algebra (Q6539988)

Let \(G\) be a finite simple graph with \(n\) vertices, \(k\) a field, \(S=k[x_1,\ldots,x_n]\) a polynomial ring and \(E=k\langle x_1,\ldots,x_n\rangle\) the exterior algebra over \(k\). The author considers the binomial edge ideal of \(G\) in the exterior algebra \(E\) and studies some of its properties. Let \(J\subset S\) and \(\tilde{J}\subset E\) be the binomial edge ideal of \(G\) in \(S\) and \(E\), respectively. The author constructs a complex supported on a poset and proves that the linear strand of the minimal free resolution of \(\tilde{J}\) over \(E\) is undelined by this poset. Moreover, the author pays attention to the particular case when \(G\) is the complete graph on \(n\) vertices. The author computes the Hilbert and the Poincaré series of \(E/\tilde{J}\) and determines formulas for some Betti numbers of \(E/\tilde{J}\).