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Orlov's Equivalence and Maximal Cohen-Macaulay Modules over the Cone of an Elliptic Curve

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Lennart Galinat

Algebraic Geometry · Submitted 6 Feb 2013

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Since every indecomposable Maximal Cohen-Macaulay module over the completion of a smooth cubic curve is gradable, we obtain explicit descriptions of all indecomposable rank one matrix factorisations of such potentials.

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arXiv ID1302.1383 (version 1)
Pith IDZTHBMK2F, Pith's permanent record of this paper and its review
CitationLennart Galinat. "Orlov's Equivalence and Maximal Cohen-Macaulay Modules over the Cone of an Elliptic Curve." arXiv:1302.1383 (2013). https://pith.science/paper/ZTHBMK2F
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,704 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.29 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.10 · mentions code or data availability 0.05.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We describe a method for doing computations with Orlov's equivalence between the bounded derived category of certain hypersurfaces and the stable category of graded matrix factorisations of the polynomials describing these hypersurfaces. In the case of a smooth elliptic curve over an algebraically closed field we describe the indecomposable graded matrix factorisations of rank one. Since every indecomposable Maximal Cohen-Macaulay module over the completion of a smooth cubic curve is gradable, we obtain explicit descriptions of all indecomposable rank one matrix factorisations of such potentials. Finally, we explain how to compute all indecomposable matrix factorisations of higher rank with the help of a computer algebra system.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We describe a method for doing computations with Orlov's equivalence between the bounded derived category of certain hypersurfaces and the stable category of graded matrix factorisations of the polynomials describing these hypersurfaces. contribution 0.76 0.91 0.29 asserted 1.00
s1 In the case of a smooth elliptic curve over an algebraically closed field we describe the indecomposable graded matrix factorisations of rank one. contribution 0.62 0.93 0.14 asserted 1.00
s2 Since every indecomposable Maximal Cohen-Macaulay module over the completion of a smooth cubic curve is gradable, we obtain explicit descriptions of all indecomposable rank one matrix factorisations of such potentials. result 0.79 0.52 0.28 asserted 1.00
s3 Finally, we explain how to compute all indecomposable matrix factorisations of higher rank with the help of a computer algebra system. method 0.87 0.84 0.10 asserted 0.99

What kind of paper the classifier reports this is: method or tool 0.99 · evidence from proof or derivation 0.95.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

We describe a method for doing computations with Orlov's equivalence between the bounded derived category of certain hypersurfaces and the stable category of graded matrix factorisations of the polynomials describing these hypersurfaces. In the case of a smooth elliptic curve over an algebraically closed field we describe the indecomposable graded matrix factorisations of rank one. Since every indecomposable Maximal Cohen-Macaulay module over the completion of a smooth cubic curve is gradable, we obtain explicit descriptions of all indecomposable rank one matrix factorisations of such potentials. Finally, we explain how to compute all indecomposable matrix factorisations of higher rank with the help of a computer algebra system.

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