\(d\)-simple rings and simple \(\mathcal D\)-modules.

*(English)*Zbl 0935.16017Let \(K\) be a field of characteristic zero, and let \(A_n=K[x_1,\ldots,x_n;\partial/\partial x_1,\ldots,\partial/\partial x_n]\) denote the \(n\)-th Weyl algebra over \(K\). For some time it was believed that all simple \(A_n\)-modules were holonomic, that is, their Gelfand-Kirillov dimension was equal to \(n\), until J. T. Stafford [Invent. Math. 79, 619-638 (1985; Zbl 0558.17011)] provided examples to the contrary. A different approach by J. Bernstein and V. Lunts [Invent. Math. 94, No. 2, 223-243 (1988; Zbl 0658.32009)] produced many interesting families of non-holonomic simple \(A_n\)-modules. However, none of these do include Stafford’s examples; so to obtain a new family that does, is the aim of the paper under review. Furthermore, the method of construction can be extended to rings of differential operators \({\mathcal D}(X)\) on other smooth affine varieties \(X\).

The starting point for the approach is the well known fact that there exist derivations \(d\) with respect to which the polynomial ring in \(n\) variables over \(K\) is \(d\)-simple, that is, \(d(I)\nsubseteqq I\) for any ideal \(0\neq I\subset R\). For some of these, a polynomial \(f\) can be found such that \(A_n/A_n(d+f)\) is a simple non-holonomic \(A_n\)-module. Specifically, the main result of the paper is the following. For \(2\leq i\leq n\), let \(a_i\), \(b_i\), \(h_i\) be non-zero polynomials in \(K[x_1]\) such that \(a_2,\ldots,a_n\) are linearly independent over the field of rationals and \(\deg(a_i)>\max\{\deg(b_i),\deg(h_i)\}\) for \(2\leq i\leq n\). Choosing the derivation \(d=\partial_1+\sum_{i\geq 2}(x_ia_i+b_i)\partial_i\) and the polynomial \(f=\sum_{i\geq 2}h_ix_i\) produces the simple non-holonomic module \(A_n/A_n(d+f)\). Some theorems are proved which allow to reduce the construction of simple non-holonomic modules over a ring of differential operators of certain smooth varieties to the same problem over the Weyl algebra. Particular applications include the ring \({\mathcal D}(X)\), where \(X\) is a non-singular quadric surface, or any select variety.

The starting point for the approach is the well known fact that there exist derivations \(d\) with respect to which the polynomial ring in \(n\) variables over \(K\) is \(d\)-simple, that is, \(d(I)\nsubseteqq I\) for any ideal \(0\neq I\subset R\). For some of these, a polynomial \(f\) can be found such that \(A_n/A_n(d+f)\) is a simple non-holonomic \(A_n\)-module. Specifically, the main result of the paper is the following. For \(2\leq i\leq n\), let \(a_i\), \(b_i\), \(h_i\) be non-zero polynomials in \(K[x_1]\) such that \(a_2,\ldots,a_n\) are linearly independent over the field of rationals and \(\deg(a_i)>\max\{\deg(b_i),\deg(h_i)\}\) for \(2\leq i\leq n\). Choosing the derivation \(d=\partial_1+\sum_{i\geq 2}(x_ia_i+b_i)\partial_i\) and the polynomial \(f=\sum_{i\geq 2}h_ix_i\) produces the simple non-holonomic module \(A_n/A_n(d+f)\). Some theorems are proved which allow to reduce the construction of simple non-holonomic modules over a ring of differential operators of certain smooth varieties to the same problem over the Weyl algebra. Particular applications include the ring \({\mathcal D}(X)\), where \(X\) is a non-singular quadric surface, or any select variety.

Reviewer: G.Krause (Winnipeg)

##### MSC:

16S32 | Rings of differential operators (associative algebraic aspects) |

16P90 | Growth rate, Gelfand-Kirillov dimension |

16W50 | Graded rings and modules (associative rings and algebras) |

16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |

16S36 | Ordinary and skew polynomial rings and semigroup rings |

32C38 | Sheaves of differential operators and their modules, \(D\)-modules |