An example of a simple derivation in two variables

Nowicki, Andrzej

doi:10.4064/cm113-1-3

Cited by 19 publications

(16 citation statements)

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“…A similar construction can be made by taking the simple derivation defined in [16] as a starting point. Instead of that, we give a more general example in the same vein.…”

Section: D(l)/d(l)((nmentioning

confidence: 99%

NONHOLONOMIC SIMPLE `D`-MODULES FROM SIMPLE DERIVATIONS

Coutinho¹

2007

Glasgow Math. J.

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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.

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“…A similar construction can be made by taking the simple derivation defined in [16] as a starting point. Instead of that, we give a more general example in the same vein.…”

Section: D(l)/d(l)((nmentioning

confidence: 99%

NONHOLONOMIC SIMPLE `D`-MODULES FROM SIMPLE DERIVATIONS

Coutinho¹

2007

Glasgow Math. J.

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“…For derivation (2), only certain sufficient conditions were indicated. In [5], it was proved that derivation (3) is simple. We prove the following theorem, which significantly generalizes the last result:…”

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confidence: 98%

“…Assume that n > 1 because, for n = 1, the theorem was proved in [5]. Assume that the derivation d has a σ-invariant Darboux polynomial…”

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confidence: 98%

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Simple derivations of higher degree in two variables

Havran

2009

Ukr Math J

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We present a new class of simple derivations of arbitrary degree in the ring of polynomials in two variables.Let k be a field of characteristic 0 and let R = k [x, y] be the ring of polynomials over k. A derivation δ in the ring R is defined as a k-linear mapping δ :Every derivation δ of the ring R can be uniquely represented in the formthe ring R is called simple if R does not have δ-invariant ideals other than 0 and R. Derivations of this type play an important role in numerous problems. For example, the ring of twisted polynomials R[t, δ] is simple if and only if δ is a simple derivation [1] (Theorem 8.4). Similarly, a Lie algebra defined on the space R according to the rule [a, b] = aδ(b) − δ(a)b is simple if and only if δ is simple [2]. Also recall that, in the case where the derivationis simple, there exist polynomials G such that the quotient module A 2 /A 2 (δ + G) is a simple nonholonomic A 2 -module, where A 2 is a Weyl algebra or an algebra of differential operators on a plane [3] (Theorem 2.

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“…Example 7.3. ( [10], [19], [20]). The derivation ∂ x + (xy + 1)∂ y has no essential Darboux polynomial.…”

Section: Polynomials M D In Two Variablesmentioning

confidence: 99%

Divergence-free polynomial derivations

Nowicki¹

2017

Analytic and Algebraic Geometry 2

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Abstract. In this paper we present some new and old properties of divergences and divergence-free derivations.Throughout the paper all rings are commutative with unity. Let k be a ring and let d be a k-derivation of the polynomial ringThe derivation d is said to be divergence-free if d = 0. PreliminariesLet k be a ring, and let R be a k-algebra. A k-linear mapping d : R → R is said to be a k-derivation of R iffor all a, b ∈ R. We denote by Der k (R) the set of all k-derivations of R.Thus, the set Der k (R) is an R-module which is also a Lie algebra.We denote by R d the kernel of d, that is,This set is a subring of R, called the ring of constants of R (with respect to d). If R is a field, then R d is a subfield of R.

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An example of a simple derivation in two variables

Cited by 19 publications

References 12 publications

NONHOLONOMIC SIMPLE `D`-MODULES FROM SIMPLE DERIVATIONS

NONHOLONOMIC SIMPLE `D`-MODULES FROM SIMPLE DERIVATIONS

Simple derivations of higher degree in two variables

Divergence-free polynomial derivations

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An example of a simple derivation in two variables

Cited by 19 publications

References 12 publications

NONHOLONOMIC SIMPLE D-MODULES FROM SIMPLE DERIVATIONS

NONHOLONOMIC SIMPLE D-MODULES FROM SIMPLE DERIVATIONS

Simple derivations of higher degree in two variables

Divergence-free polynomial derivations

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Product

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NONHOLONOMIC SIMPLE `D`-MODULES FROM SIMPLE DERIVATIONS

NONHOLONOMIC SIMPLE `D`-MODULES FROM SIMPLE DERIVATIONS