Algebraic geometry and arithmetic curves - download pdf or read online
By Qing Liu
This ebook is a normal creation to the idea of schemes, by means of functions to mathematics surfaces and to the idea of aid of algebraic curves. the 1st half introduces simple gadgets resembling schemes, morphisms, base swap, neighborhood homes (normality, regularity, Zariski's major Theorem). this is often via 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 examine of tender projective curves over a box. Singular curves are handled via a close learn of the Picard crew. the second one half begins with blowing-ups and desingularization (embedded or no longer) of fibered surfaces over a Dedekind ring that leads directly to intersection concept on mathematics surfaces. Castelnuovo's criterion is proved and in addition the lifestyles of the minimum typical version. This ends up in the learn of relief of algebraic curves. The case of elliptic curves is studied intimately. The booklet concludes with the basic theorem of strong aid of Deligne-Mumford. The ebook is largely self-contained, together with the mandatory fabric on commutative algebra. the necessities are consequently few, and the ebook may still go well with a graduate scholar. It includes many examples and approximately six hundred workouts
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Additional info for Algebraic geometry and arithmetic curves
Tn ] is such that F (α) = 0 for every √ α ∈ Z(I), then F ∈ I. 32 2. General properties of schemes Proof Let A = k[T1 , . . , Tn ]/I and let f denote the image of F in A. We must show that f is nilpotent. 15). Hence F (α) = 0 and f ∈ m. 18, f is indeed nilpotent. 20. This proposition says that we can recover the ideal I, up to its radical, from its set of zeros Z(I). 1. Let A = k[[T ]] be the ring of formal power series with coeﬃcients in a ﬁeld k. Determine Spec A. 2. Let ϕ : A → B be a homomorphism of ﬁnitely generated algebras over a ﬁeld.
For every x ∈ X, there exists an open neighborhood Ux of x such that s|Ux = 0, because sx = 0. As the Ux cover X when x varies, we indeed have s = 0. 10. Let F, G be two presheaves on X. A morphism of presheaves α : F → G consists of, for every open subset U of X, a group homomorphism α(U ) : F(U ) → G(U ) which is compatible with the restrictions ρU V . A morphism of presheaves α is called injective if for every open subset U of X, α(U ) is injective (take care: a surjective morphism is not deﬁned in the same way, see further on).
Let m be a maximal ideal of A. Then A/m is a ﬁnite algebraic extension of k. Proof As A/m is a ﬁnitely generated k-algebra, there exists a ﬁnite injective homomorphism A0 → A/m, where A0 = k[T1 , . . , Td ] is a polynomial ring over k. Let us suppose that d ≥ 1. We have 1/T1 ∈ A/m since A/m is a ﬁeld. Hence 1/T1 is integral over A0 . By considering an integral equation for 1/T1 over A0 , we see that T1 is invertible in A0 , which is impossible. Hence A0 = k and A/m is ﬁnite over k. This corollary makes it possible to describe the maximal ideals of a ﬁnitely generated algebra over k concretely.