Orlov's equivalence and maximal Cohen-Macaulay modules over the cone of an elliptic curve
type: Item
instance of: scholarly article
title: Orlov's equivalence and maximal Cohen-Macaulay modules over the cone of an elliptic curve
zbMATH Open document ID: 1329.14034
author: Lennart Galinat
published in: Mathematische Nachrichten
publication date: 2014-11-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1302.1383
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 16G50
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 6368922
zbMATH Keywords: MCM modules
zbMATH Keywords: matrix factorization
zbMATH Keywords: Orlov's equivalence
zbMATH Keywords: elliptic curve
describes a project that uses: SINGULAR
MaRDI profile type: Publication
OpenAlex ID: W2963462469
arXiv ID: 1302.1383
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DOI: 10.1002/MANA.201300106
review text: Let \(K\) be a field and \(f\in K[x_0, \ldots, x_n]\) be a homogeneous polynomial of degree \(n+1\) and \(X=\mathrm{Proj}\;K[x_0, \ldots, x_n]/f\). \textit{D. Orlov} [Prog. Math. 270, 503--531 (2009; Zbl 1200.18007)] proved that there is an equivalence between the bounded derived category of coherent sheaves on \(X\) and the homotopy category of graded matrix factorisations of \(f\). It is described how to compute this equivalence in case of a smooth elliptic curve over an algebraically closed field. The indecomposable graded matrix factorisations of rank one are described. Every indecomposable maximal Cohen-Macaulay module over the completion of a smooth curve is gradable. This gives explicit descriptions of all indecomposable rank one matrix factorisations. It is explained how to compute all indecomposable matrix factorisations of higher rank using a computer algebra system as \texttt{Singular}.
reviewed by: Gerhard Pfister
description: scientific article; zbMATH DE number 6368922