The Gröbner fan of an \(A_n\)-module
Let \(I\) be a non-zero left ideal of the Weyl algebra \(A_n\) of order \(n\) defined over a field \(k\) and let \(L:\mathbb{R}^{2n} \to\mathbb{R}\) be a linear form defined by \(L(\alpha, \beta)= \sum^n_{i=1} e_i\alpha_i+ \sum^n_{i=1} f_i \beta_i\). If \(e_i+ f_i\geq 0\), then \(L\) defines a filtration \(F^L_\bullet\) on \(A_n\). Let \(\text{gr}^L(I)\) be the graded ideal associated with the filtration induced by \(F^L_\bullet\) on \(I\subset A_n\). Let \(U\) be the set of all linear forms \(L\) for which \(e_i+f_i\geq 0\) for all \(1\leq i\leq n\). The aim of this paper is to prove, by using the theory of Gröbner basis, that \(\text{gr}^L(I)\) does not vary when \(L\) moves in a subset of \(U\). Let \(h(I)\) be the homogenization of the ideal \(I\). The main result of this paper is the following: there exists a partition \({\mathcal E}\) of \(U\) into convex rational polyhedral cones, such that for all element \(\sigma \in {\mathcal E}\), \(\text{gr}^L (h(I))\) and \(\text{Ext}_{\prec_L} (h(I))\) do not depend on \(L\in\sigma\). The same is true for \(\text{gr}^L(I)\).
scientific article; zbMATH DE number 1488531