## Abstract

The notion of determinant groupoid is a natural outgrowth of the theory of the Sato Grassmannian and thus well-known in mathematical physics. We briefly sketch here a version of the theory of determinant groupoids over an artinian local ring, taking pains to put the theory in a simple concrete form suited to number-theoretical applications. We then use the theory to give a simple proof of a reciprocity law for the Contou-Carrère symbol. Finally, we explain how from the latter to recover various classical explicit reciprocity laws on nonsingular complete curves over an algebraically closed field, namely sum-of-residues-equals-zero, Weil reciprocity, and an explicit reciprocity law due to Witt. Needless to say, we have been much influenced by the work of Tate on sum-of-residues-equals-zero and the work of Arbarello-De Concini-Kac on Weil reciprocity. We also build in an essential way on a previous work of the second-named author.

Original language | English (US) |
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Pages (from-to) | 79-102 |

Number of pages | 24 |

Journal | Communications in Algebra |

Volume | 32 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2004 |

### Bibliographical note

Funding Information:The second-named author was partially supported by DGESYC research contract BFM2000-1327 and Castilla y León regional government contract SA064/01.

## Keywords

- Contou-Carrère
- Determinant groupoids
- Explicit reciprocity laws