## Algebraic Geometry: Part I: Schemes. With Examples and ExercisesAlgebraic geometry has its origin in the study of systems of polynomial equations f (x ,. . . ,x )=0, 1 1 n . . . f (x ,. . . ,x )=0. r 1 n Here the f ? k[X ,. . . ,X ] are polynomials in n variables with coe?cients in a ?eld k. i 1 n n ThesetofsolutionsisasubsetV(f ,. . . ,f)ofk . Polynomialequationsareomnipresent 1 r inandoutsidemathematics,andhavebeenstudiedsinceantiquity. Thefocusofalgebraic geometry is studying the geometric structure of their solution sets. n If the polynomials f are linear, then V(f ,. . . ,f ) is a subvector space of k. Its i 1 r “size” is measured by its dimension and it can be described as the kernel of the linear n r map k ? k , x=(x ,. . . ,x ) ? (f (x),. . . ,f (x)). 1 n 1 r For arbitrary polynomials, V(f ,. . . ,f ) is in general not a subvector space. To study 1 r it, one uses the close connection of geometry and algebra which is a key property of algebraic geometry, and whose ?rst manifestation is the following: If g = g f +. . . g f 1 1 r r is a linear combination of the f (with coe?cients g ? k[T ,. . . ,T ]), then we have i i 1 n V(f ,. . . ,f)= V(g,f ,. . . ,f ). Thus the set of solutions depends only on the ideal 1 r 1 r a? k[T ,. . . ,T ] generated by the f . |

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### Contents

1 | |

7 | |

40 | |

3 Schemes | 66 |

4 Fiber products | 93 |

5 Schemes over fields | 118 |

6 Local Properties of Schemes | 145 |

7 Quasicoherent modules | 169 |

14 Flat morphisms and dimension | 423 |

15 Onedimensional schemes | 485 |

16 Examples | 503 |

A The language of categories | 541 |

B Commutative Algebra | 547 |

C Permanence for properties of morphisms of schemes | 573 |

D Relations between properties of morphisms of schemes | 576 |

E Constructible and open properties | 578 |

8 Representable Functors | 205 |

9 Separated morphisms | 226 |

10 Finiteness Conditions | 241 |

11 Vector bundles | 286 |

12 Affine and proper morphisms | 320 |

13 Projective morphisms | 366 |

583 | |

Detailed List of Contents | 588 |

598 | |

602 | |

### Common terms and phrases

A-module abelian affine algebraic sets affine scheme ample assume base change bijective called closed immersion closed point closed subscheme closed subset constructible Corollary corresponding define Definition denote dimension dimX divisor elements equivalent Exercise exists an open faithfully flat field extension finite locally free finite presentation finite type flat morphism functor geometrically hence homogeneous induces inductive limit injective integral invertible irreducible components isomorphism k-algebra Lemma Let f line bundle locally noetherian scheme locally of finite locally ringed space maximal ideal modules morphism f morphism of schemes noetherian ring noetherian scheme normal obtain open affine open covering open immersion open neighborhood open subscheme open subset OX,x polynomial presheaf prevarieties prime ideal ProjA Proof Proposition qcqs quasi-coherent quasi-coherent OX-module quasi-compact Remark resp S-scheme scheme and let sheaf Show smooth Spec SpecA subspace surjective Theorem topological space valuation ring