## Algebraic GeometryThis is a graduate-level text on algebraic geometry that provides a quick and fully self-contained development of the fundamentals, including all commutative algebra which is used. A taste of the deeper theory is given: some topics, such as local algebra and ramification theory, are treated in depth. The book culminates with a selection of topics from the theory of algebraic curves, including the Riemann-Roch theorem, elliptic curves, the zeta function of a curve over a finite field, and the Riemann hypothesis for elliptic curves. |

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### Contents

Affine Algebraic Sets and Varieties | 1 |

The Extension Theorem | 7 |

Maps of Affine Varieties | 13 |

Dimension and Products | 23 |

Local Algebra | 39 |

Properties of Affine Varieties | 61 |

Varieties | 79 |

Complete Nonsingular Curves | 97 |

Completions | 125 |

Differentials and Residues | 137 |

The RiemannRoch Theorem | 159 |

Elliptic Curves and Abelian Varieties | 175 |

The Zeta Function of a Curve | 195 |

211 | |

215 | |

Ramification | 103 |

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A-module affine algebra affine curve affine open set affine variety algebraic set algebraically closed field assume bijection Cauchy sequence characteristic closed subset closed subvariety codimension coefficients complete nonsingular curve containing coordinate ring corresponding Dedekind domain defined denote dim(X dim(y dimension discrete valuation ring elliptic curve equal equation equivalent example Exercise exists fc-algebra homomorphism fiber field F field of fractions finite separable extension flp/k follows from Proposition fractional ideals function field Galois genus Gm(A Hence implies induced map injective integral closure integral domain irreducible component isomorphic k[xi kernel Lemma Let E/F Let F linear linearly maximal ideal minimal module multiplication Noetherian local ring open set Oy,x polynomial ring prevariety prime ideal Proof prove purely inseparable ramification index rational map regular local ring ring of F S~lA separable extension subring subspace Suppose surjective topology Tx(X unique vanish vector space zero