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Isomorphisms of rings of differential operators on curves. (English) Zbl 0752.16011

Let \(X\) be a complex affine curve. It is natural to ask whether \({\mathcal D}(X)\), the ring of differential operators on \(X\), determines \(X\). That is, does \({\mathcal D}(X)\cong{\mathcal D}(Y)\) imply that \(X\cong Y\)? The subalgebra \(N(X)\) of \({\mathcal D}(X)\) consisting of ad-nilpotent elements is obviously an invariant of \({\mathcal D}(X)\). It contains \({\mathcal O}(X)\), the ring of regular functions on \({\mathcal D}(X)\). The combined work of Makar- Limanov and Perkins shows that \(N(X)={\mathcal O}(X)\), unless there is a finite, injective birational map \(\pi:\mathbb{A}^ 1\to X\). So the answer is ‘yes’ except possible for cures of the latter type. Note that, for such a curve, the work of Smith and Stafford shows that \({\mathcal D}(X)\) is Morita equivalent to \({\mathcal D}(\mathbb{A}^ 1)\) and hence is simple. Thus the above problem is part of the important general question of finding (non-Morita) invariants of simple rings.
Remarkably, there are examples of non-isomorphic curves with isomorphic rings of differential operators. G. Letzter found the first examples and the author presents some further families of examples in his paper. The proofs are rather computational and it remains an intriguing problem to: (1) determine exactly which isomorphisms there are between rings of differential operators on curves, and (2) interpret then these isomorphisms in a geometric way. Naturally, one expects that the answer to (2) will precede the answer to (1).

MSC:

16S32 Rings of differential operators (associative algebraic aspects)
14H99 Curves in algebraic geometry
13N10 Commutative rings of differential operators and their modules
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32C38 Sheaves of differential operators and their modules, \(D\)-modules
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
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