Vector bundles of degree zero over an elliptic curve.
type: Item
instance of: scholarly article
title: Vector bundles of degree zero over an elliptic curve.
zbMATH Open document ID: 1052.14034
DOI: 10.1016/S1631-073X(02)02478-0
author: Silke Lekaus
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2002-11-11T00:00:00Z
review text: To every vector bundle \(V\) of finite rank over a complete, connected, reduced scheme \(X\) over a perfect field \(k\) there exist principal \(G\)-bundles (for some group scheme \(G\)) over which the pullback of \(V\) is trivial. Let \(G_V\) denote such a group scheme of smallest dimension. On the other hand, using the Grothendieck ring \(K(X)\) of isomorphism classes of vector bundles over \(X\), to every \(V\) one can associate a \(\mathbb Q\)-algebra \(R(V) \subset K(X) \otimes \mathbb Q\). \textit{M. V. Nori} [Compos. Math. 33, 29--41 (1976; Zbl 0337.14016)] showed that for any finite vector bundle \(V\) one has \(\dim R(V) = \dim G_V = 0\). The author proves that for every vector bundle \(V\) of degree \(0\) over an elliptic curve \(X\) again \[ \dim R(V) = \dim G_V \] holds. An example shows that these dimensions may differ for curves \(X\) of genus~\(2\).
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 1809873
zbMATH Keywords: principal \(G\)-bundles
zbMATH Keywords: Grothendieck ring
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1809873