Initial simplicial complexes of prime ideals
Let \(I\) be an ideal in a polynomial ring \(R = k[x_0, \dots, x_n]\) over a field~\(k\) and \(<\) a term order on \(R\). The initial complex of~\(I\) is a simplicial complex \(\Delta = \Delta_<(I)\) such that the Stanley-Reisner ideal \(I_\Delta\) is equal to the radical of the initial ideal \(\text{in}_< I\) of~\(I\). The initial ideal and hence the initial complex of \(I\) is useful to study the residue ring~\(R/I\). \textit{M. Kalkbrener} and \textit{B. Sturmfels} [Adv. Math. 116, No. 2, 365--376 (1995; Zbl 0877.13025)] proved that \(\Delta_<(I)\) is pure and strongly connected if \(I\) is a prime ideal. Is the converse true? What simplicial complex can be the initial complex of a prime ideal? The present article is an approach to these questions. The main theorem of this article is the following: Assume that \(k\) is algebraically closed. For any integer \(n \geq d > 0\), there are a prime ideal \(I \subset R\) and a term order~\(>\) such that \(R/I\) is complete intersection and the initial complex of~\(I\) is the \(d\)-skeleton of the \(n\)-simplex, that is, the subcomplex of the \(n\)-simplex generated by all the \(d\)-faces.
scientific article; zbMATH DE number 2159745
scientific article