Formal punctured ribbons and two-dimensional local fields

Kurke, Herbert; Osipov, D. V.; Zheglov, Alexander Borisovich

doi:10.1515/crelle.2009.029

Cited by 9 publications

(36 citation statements)

References 6 publications

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“…At the end of this section we prove a theorem about algebraic-geometric properties of maximal commutative subrings of partial differential operators in two variables; in particular, we show that all such rings must be Cohen-Macaulay. Some applications of constructions described in this paper to the theory of ribbons (see [16], [17]) and theory of generalized Parshin-KP's hierarchies (see [30], [41]), as well as several explicit examples of commuting operators, will appear in a separate paper (see [19]), part of which is a recent work [18] (cf. also work [42] for a comparison with Baker-Akhieser-modulesapproach).…”

Section: Introductionmentioning

confidence: 99%

On rings of commuting partial differential operators

Zheglov¹

2014

St. Petersburg Math. J.

105

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We give a natural generalization of the classification of commutative rings of ordinary differential operators, given in works of Krichever, Mumford, Mulase, and determine commutative rings of operators in a completed ring of partial differential operators in two variables (satisfying certain mild conditions) in terms of Parshin's generalized geometric data. It uses a generalization of M.Sato's theory and is constructible in both ways.

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Section: Introductionmentioning

confidence: 99%

On rings of commuting partial differential operators

Zheglov¹

2014

St. Petersburg Math. J.

105

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“…The aim of this paper, except the main aim (formulated in the beginning) to study algebrogeometric properties of commutative rings of PDOs, is to give also the answers on these questions. These answers will be important in order to apply the theory of ribbons developed in [25], [26] to study rings of PDO and their isospectral deformations.…”

Section: Introductionmentioning

confidence: 99%

Commuting differential operators and higher-dimensional algebraic varieties

Kurke

Osipov

Zheglov

2014

Sel. Math. New Ser.

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Several algebro-geometric properties of commutative rings of partial differential operators as well as several geometric constructions are investigated. In particular, we show how to associate a geometric data by a commutative ring of partial differential operators, and we investigate the properties of these geometric data. This construction is in some sense similar to the construction of a formal module of Baker-Akhieser functions. On the other hand, there is a recent generalization of Sato's theory which belongs to the third author of this paper. We compare both approaches to the commutative rings of partial differential operators in two variables. As a by-product we get several necessary conditions on geometric data describing commutative rings of partial differential operators.

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“…We introduced the notion of a torsion free sheaf on a ribbon. An importance of such sheaves followed from theorem 1 of article [10], where we proved that torsion free sheaves on some ribbons plus some geometrical data such as formal trivialization of sheaves, local parameters at smooth points of ribbons and so on are in one-to-one correspondence with generalized Fredholm subspaces of two-dimensional local field k((u))((t)) (see also section 2 of this article).…”

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confidence: 97%

“…In article [10] we investigated new geometric objectsX ∞ = (C, A) , which are ringed spaces: formal punctured ribbons with the underlying topological space C as an algebraic curve. (For simplicity we call such objects "ribbons".)…”

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confidence: 99%