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Rings of differential operators over rational affine curves

1990; Société Mathématique de France; Volume: 118; Issue: 2 Linguagem: Inglês

10.24033/bsmf.2143

2102-622X

Gail Letzter, Leonid Makar-Limanov,

Polynomial and algebraic computation

RESUME.-Soit X une courbe algebrique irreductible sur C dont la normalisee est la droite affine et telle sur Ie morphisme de normalisation est injectif.Soit D(X) Panneau des operateurs differentiels sur X. Nous etudions un invariant pour 1'anneau D(X) des operateurs differentiels sur X, note codimD(X).En particulier, nous montrons que D(X) ^ D(Y) implique codimD(X) = codimD(y).Cela permet de distinguer dans certains cas les anneaux d'operateurs differentiels de courbes nonisomorphes.En outre, nous decrivons les sous-algebres ad-nilpotentes maximales de D(X).Nous montrons que si B est une sous-algebre ad-nilpotente maximales de D(X), alors B est un sous-anneau de type fini d'un C[b] ou b designe un element du corps des fractions de D(X); de plus, la cloture integrale de B est C[b\.ABSTRACT.-Let X be an irreducible algebraic curve over the complex numbers such that its normalization is the affine line, and the normalization map is injective.Let D(X) denote its ring of differential operators.We find an invariant for D(X) denoted as codimD(X).In particular, we show that D(X) ^ D(Y) implies codimD(X) = codimD(y).This allows us to distinguish certain rings of differential operators of non-isomorphic curves.We also describe the maximal ad-nilpotent subalgebras of D(X).We show that if B is a maximal ad-nilpotent subalgebra of D(X), then B is a finitely generated subring of C [b] for some element b of the quotient field of D(X) and the integral closure of B is C [b].