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Algebraic Geometry

By Qing Liu

ISBN-10: 0198502842

ISBN-13: 9780198502845

This e-book is a common creation to the speculation of schemes, via purposes to mathematics surfaces and to the speculation of relief of algebraic curves. the 1st half introduces easy items resembling schemes, morphisms, base swap, neighborhood houses (normality, regularity, Zariski's major Theorem). this is often by way of the extra international point: coherent sheaves and a finiteness theorem for his or her cohomology teams. Then follows a bankruptcy on sheaves of differentials, dualizing sheaves, and grothendieck's duality thought. the 1st half ends with the theory of Riemann-Roch and its program to the learn of gentle projective curves over a box. Singular curves are taken care of via an in depth examine of the Picard staff. the second one half starts off with blowing-ups and desingularization (embedded or no longer) of fibered surfaces over a Dedekind ring that leads directly to intersection idea on mathematics surfaces. Castelnuovo's criterion is proved and in addition the lifestyles of the minimum usual version. This results in the research of aid of algebraic curves. The case of elliptic curves is studied intimately. The e-book concludes with the basic theorem of sturdy relief of Deligne-Mumford. The booklet is basically self-contained, together with the required fabric on commutative algebra. the must haves are accordingly few, and the ebook may still swimsuit a graduate pupil. It comprises many examples and approximately six hundred routines

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Example text

Let X be a topological space. For any open subset U of X, let C(U ) = C0 (U, R) be the set of continuous functions from U to R. The restrictions ρU V are the usual restrictions of functions. Then C is a sheaf on X. If we let F(U ) = RU be the set of functions on U with values in R, this defines a sheaf F of which C is a subsheaf. 4. Let A be a non-trivial Abelian group. Let X be a topological space. Let AX (U ) = A and ρU V = IdA if U and V are non-empty. This defines a presheaf on X. In general, AX is not a sheaf.

1 Zariski topology Let A be a (commutative) ring (with unit). We let Spec A denote the set of prime ideals of A. We call it the spectrum of A. By convention, the unit ideal is not a prime ideal. Thus Spec{0} = ∅. We will now endow Spec A with a topological structure. For any ideal I of A, let V (I) := {p ∈ Spec A | I ⊆ p}. If f ∈ A, let D(f ) := Spec A \ V (f A). 1. Let A be a ring. We have the following properties: (a) For any pair of ideals I, J of A, we have V (I) ∪ V (J) = V (I ∩ J). (b) Let (Iλ )λ be a family of ideals of A.

System and that the canonical map ←− lim An → ←− n n (b) Let us suppose that (An , πn )n satisfies the Mittag–Leffler condition and that An = ∅ for all n. Show that An = ∅, An+1 → An is surjective, and that ←− lim An = ∅. Deduce from this that ←− lim An = ∅. 3. Formal completion 25 ρ (c) Let 0 → (An )n → (Bn )n − → (Cn )n → 0 be an exact sequence of inverse systems of Abelian groups such that (An )n satisfies the lim Cn and Xn = ρ−1 Mittag–Leffler condition. Let (cn )n ∈ ←− n (cn ), n where ρn : Bn → Cn is the nth component of ρ.

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