1964/10/01 by Gary Horrocks · 6 citations
Chemistry · Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Chemistry #Citation #Computer science #Information retrieval #Library science #Mathematics #Physics #Ring (chemistry) #Rings, Modules, and Algebras #Spectrum (functional analysis)
paper · doi:10.1112/plms/s3-14.4.689
openalex publication_date 1964/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/04/04
THIS paper is concerned with the problem of determining whether a vector bundle defined on an open set (in the Zariski topology) of an algebraic variety can be extended to a vector bundle on the whole variety. It deals only with the special case of this problem in which the vector bundle is defined on the complement of a simple point. When the variety is a curve and the divisor class of the point is non-zero, the extension (which always exists) is not unique; for tensoring a given extension with the line bundle corresponding to the divisor class of the point gives a different extension. When the variety has dimension at least 2 there is at most one extension. For if a rational section of a vector bundle has a pole at a simple point, then it has a pole on a divisor through that point. So the group of sections of a vector bundle over a neighbourhood of a point is completely determined by the group of sections over the punctured neighbourhood. It follows that the extension is unique. Because of this, the results obtained here could be applied to the more general problem of extending a vector bundle