Download PDF by Qing Liu: Algebraic geometry and arithmetic curves

By Qing Liu

ISBN-10: 0191547808

ISBN-13: 9780191547805

Advent; 1. a few themes in commutative algebra; 2. normal houses of schemes; three. Morphisms and base switch; four. a few neighborhood homes; five. Coherent sheaves and Cech cohmology; 6. Sheaves of differentials; 7. Divisors and purposes to curves; eight. Birational geometry of surfaces; nine. ordinary surfaces; 10. relief of algebraic curves; Bibilography; Index

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Proof We can identify the closed points of Spec A with the maximal ideals of k[T1 , . . , Tn ] containing I. Let m = (T1 − α1 , . . , Tn − αn ) be a maximal ideal of k[T1 , . . , Tn ]. Let P (T ) ∈ k[T1 , . . , Tn ]. Using the Taylor expansion of P (T ) at α := (α1 , . . , αn ), we see that P (T ) ∈ m if and only if P (α) = 0. It follows that I ⊆ m if and only if P (α) = 0 for every P (T ) ∈ I. 16. The object of algebraic geometry is the study of solutions of systems of polynomial equations over a field k.

For example, if X is the disjoint union of two non-empty open sets, then condition (5) for sheaves is not verified. 1. 5. If U is an open subset of X, every presheaf F on X induces, in an obvious way, a presheaf F|U on U by setting F|U (V ) = F(V ) for every open subset V of U . This is the restriction of F to U . If F is a sheaf, then so is F|U . 6. Let B be a base of open subsets on X (we mean that B is a set of open subsets of X, that any open subset of X is a union of open subsets in B, and that B is stable by finite intersection).

Let ρ : B → B⊗A Frac(A) be the canonical map. Let m be a maximal ideal of B ⊗A Frac(A). Then q := ρ−1 (m) is a prime ideal of B. Since ρ◦f : A → B⊗A Frac(A) factorizes into A → Frac(A) → B⊗A Frac(A), and the inverse image of m in Frac(A) is zero, we have f −1 (q) = (ρ ◦ f )−1 (m) = 0. 1. Let M be an A-module. We call the ideal {a ∈ A | aM = 0} of A the annihilator of M , and we denote it by Ann(M ). Let I ⊆ Ann(M ) be an ideal. (a) Show that M is endowed, in a natural way, with the structure of an A/I-module, and that M M ⊗A A/I.

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Algebraic geometry and arithmetic curves by Qing Liu

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