NONHOLONOMIC SIMPLE D-MODULES FROM SIMPLE DERIVATIONS

Abstract

Abstract.We give new examples of affine sufaces whose rings of coordinates are d-simple and use these examples to construct simple nonholonomic D-modules over these surfaces.2000 Mathematics Subject Classification. Primary: 13N15, 13N10. Secondary: 16S32. Introduction.The d-simplicity of commutative rings was the subject of several papers in the 1970s and early 1980s, at least partly because of its applications to the construction of simple noncommutative noetherian rings [12, Proposition 1.14]. Although little was published on it from the mid 1980s to the mid 1990s, the subject has known something of a revival in recent years, fuelled perhaps by the construction of new examples of derivations with respect to which the ring of polynomials is d-simple Despite these advances, some aspects of the theory have progressed very little since the 1980s. One of these is the construction of new examples of d-simple rings. The only examples known up to now were the ones already given in J. Archer's PhD thesis [1]; namely, coordinate rings of affine spaces, tori, quadrics, and products of these varieties with affine space. This is precisely the question that we tackle in this paper. As an application of the theorems proved in Section 3, we give several new examples of smooth surfaces whose coordinate rings are d-simple. Two of these lead to new families of nonholonomic simple D-modules over surfaces. One of these families is particularly interesting because all the previous examples of simple nonholonomic D-modules required the affine surface to have trivial Picard group. However, the surface of Example 4.1 is the product of an elliptic curve E with an affine line, so its Picard group, which is isomorphic to Pic(E), must be nonzero. As a bonus we construct, in Example 4.4, an irreducible nonholonomic D-module over an explicit surface of ‫ރ‬ 3 , taking a singular derivation as our starting point.

On the isotropy group of a simple derivation

“…On the other hand, there have been many recent connections of d− simplicity with commutative algebra and algebraic geometry via the theory of holomorphic foliations (see, for example [5], [7] and [16] ), D-modules (see, for example, [4] and [6]) and also with the question of algebraic independence of solutions of certain differential equations in the power series ring K[[t]] (see [3]).…”

A Class of Simple Derivations ofk[x,y]

“…If d is a simple derivation of A, then the ring A is called d-simple. The question of d-simplicity of a commutative ring and in particular of a polynomial algebra in finitely many variables over a field of characteristic zero has been studied by several authors, e.g., [1,4,10,12,15,16].…”

“…In recent years the construction of examples of simple derivations of a polynomial algebra in finitely many variables over a field of characteristic zero has been studied intensively and the subject has undergone a revival because of its application to the construction of nonholonomic irreducible modules over Weyl algebra [1]. Let A 2 be the second Weyl algebra, that is, the ring of differential operators over the ring k x y and d = x + f y be a simple k-derivation of k x y .…”

Derivações simples de álgebras afins

“…Por outro lado, são muitas as aplicações de d-simplicidade em álgebra comutativa e geometria algébrica. Entre elas destacam-se aplicações na teoria de folheações holomorfas (ver, por exemplo, (COUTINHO, 2008), (COUTINHO; LEVCOVITZ, 2014) e (OLIVEIRA, 2012)), D-módulos (ver, por exemplo, (COUTINHO, 2013) e (COUTINHO, 2007) ) e até mesmo na questão da independência algébrica de soluções de certas equações diferenciais no anel da série de potências formais K [[t]] (ver (BRUMATTI;LEQUAIN;LEVCOVITZ, 2003)).…”