Multigraded Hilbert schemes parametrizing ideals in the Weyl algebra
The author shows how the multigraded Hilbert scheme construction of \textit{M. Haiman} and \textit{B. Sturmfels} [J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)] can be used to construct a quasi-projective scheme which parametrize left homogeneous ideals in the Weyl algebra having fixed Hilbert function. Fixing an integral domain \(k\) of characteristic zero, the Weyl algebra \(W=k \langle x_1, \dots, x_n, \partial_1, \dots, \partial_n \rangle\) has a \(k\)-basis consisting of the set \(\mathcal B = \{ x^\alpha \partial^\beta | \alpha, \beta \in \mathbb N^n \}\). If \(A\) is an abelian group, then any \(A\)-grading \(\mathbb N^n \to A\) on the polynomial ring \(S = k[x_1,\dots,x_n]\) extends to an \(A\)-grading \(\mathbb N^{2n} \to A\) on \(W\) by \(\deg (x^\alpha \partial^\beta) = \deg \alpha - \deg \beta\), which induces a decomposition \(W = \bigoplus_{a \in A} W_a\). Given a Hilbert function \(h:A \to \mathbb N\), the corresponding Hilbert functor \(H^h_W\) takes a \(k\)-algebra \(R\) to the set of homogeneous ideals \(I \subset R \otimes_k W\) such that \((R \otimes _k W_a)/I_a\) is a locally free \(R\)-module of rank \(h(a)\) for each \(a \in A\). The main theorem says that \(H^h_W\) is representable by a quasi-projective scheme over \(k\). The strategy of the proof is similar to that of Haiman and Sturmfels [loc. cit.], but there are some new behaviors regarding monomials in the Weyl algebra \(W\) not seen in the polynomial ring \(S\). An obvious difference is that a product of monomials in \(W\) need not be a monomial. Another difference is that \(W\) has infinite antichains of monomial ideals, unlike the polynomial case: see work of \textit{D. MacLagan} [Proc. Am. Math. Soc. 129, 1609--1615 (2001; Zbl 0984.13013)]. Moreover, the natural extension of Gröbner basis theory for \(S\) to \(W\) does not work well, so the author considers the initial ideal of a left ideal in the associated graded algebra \(\text{gr} W\) and uses Gröbner basis theory for \(W\) developed by \textit{M. Saito} et al. [Gröbner deformations of hypergeometric differential equations. Berlin: Springer (2000; Zbl 0946.13021)]. The arguments are well presented along with examples showing the novel points.
scientific article; zbMATH DE number 6840994