### Abstract

We give a generalized definition of an elementary transformation of vector bundles on regular schemes by using Maximal Cohen-Macaulay sheaves on divisors. This definition is a natural extension of that given by Maruyama, and has a connection with that given by Sumihiro. By this elementary transformation, we can construct, up to tensoring line bundles, all vector bundles from trivial bundles on nonsingular quasi-projective varieties over an algebraically closed field. Moreover, we give an application of this theory to reflexive sheaves.

Original language | English |
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Pages (from-to) | 4285-4295 |

Number of pages | 11 |

Journal | Transactions of the American Mathematical Society |

Volume | 359 |

Issue number | 9 |

DOIs | |

Publication status | Published - Dec 1 2007 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

**The elementary transformation of vector bundles on regular schemes.** / Abe, Takuro.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 359, no. 9, pp. 4285-4295. https://doi.org/10.1090/S0002-9947-07-04161-X

}

TY - JOUR

T1 - The elementary transformation of vector bundles on regular schemes

AU - Abe, Takuro

PY - 2007/12/1

Y1 - 2007/12/1

N2 - We give a generalized definition of an elementary transformation of vector bundles on regular schemes by using Maximal Cohen-Macaulay sheaves on divisors. This definition is a natural extension of that given by Maruyama, and has a connection with that given by Sumihiro. By this elementary transformation, we can construct, up to tensoring line bundles, all vector bundles from trivial bundles on nonsingular quasi-projective varieties over an algebraically closed field. Moreover, we give an application of this theory to reflexive sheaves.

AB - We give a generalized definition of an elementary transformation of vector bundles on regular schemes by using Maximal Cohen-Macaulay sheaves on divisors. This definition is a natural extension of that given by Maruyama, and has a connection with that given by Sumihiro. By this elementary transformation, we can construct, up to tensoring line bundles, all vector bundles from trivial bundles on nonsingular quasi-projective varieties over an algebraically closed field. Moreover, we give an application of this theory to reflexive sheaves.

UR - http://www.scopus.com/inward/record.url?scp=77951033727&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951033727&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-07-04161-X

DO - 10.1090/S0002-9947-07-04161-X

M3 - Article

AN - SCOPUS:77951033727

VL - 359

SP - 4285

EP - 4295

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 9

ER -