### Abstract

The main purpose of this paper is to obtain the Hilbert-Samuel polynomial of a module via blowing up and applying intersection theory rather than employing associated graded objects. The result comes in the form of a concrete Riemann-Roch formula for the blow-up of a nonsingular affine scheme at its closed point. To achieve this goal, we note that the blow-up sits naturally between two projective spaces, one over a field and one a regular local ring, and then apply the Grothendieck-Riemann-Roch Theorem to each containment.

Original language | English (US) |
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Pages (from-to) | 3003-3025 |

Number of pages | 23 |

Journal | Journal of Algebra |

Volume | 322 |

Issue number | 9 |

DOIs | |

State | Published - Nov 1 2009 |

### Keywords

- Blow-up
- Hilbert-Samuel polynomial
- Riemann-Roch theorem

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Chan, C. Y. J., & Miller, C. (2009). A Riemann-Roch formula for the blow-up of a nonsingular affine scheme.

*Journal of Algebra*,*322*(9), 3003-3025. https://doi.org/10.1016/j.jalgebra.2009.07.016