By Qing Liu

This ebook is a normal creation to the idea of schemes, via functions to mathematics surfaces and to the speculation of relief of algebraic curves. the 1st half introduces uncomplicated gadgets similar to schemes, morphisms, base switch, neighborhood houses (normality, regularity, Zariski's major Theorem). this can be by way of the extra worldwide 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 conception. the 1st half ends with the concept of Riemann-Roch and its software to the examine of tender projective curves over a box. Singular curves are taken care of via an in depth learn of the Picard team. 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 concept on mathematics surfaces. Castelnuovo's criterion is proved and likewise the life of the minimum ordinary version. This ends up in the learn of aid of algebraic curves. The case of elliptic curves is studied intimately. The publication concludes with the elemental theorem of solid aid of Deligne-Mumford. The booklet is largely self-contained, together with the mandatory fabric on commutative algebra. the necessities are accordingly few, and the publication may still swimsuit a graduate pupil. It comprises many examples and approximately six hundred routines

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Tr ]] = B above. Let M , N be two I-adic A-modules. It is clear that the product topology on ˆ ⊕N ˆ. M ⊕ N = M × N is also the I-adic topology. Consequently, (M ⊕ N )∧ = M r In particular, every isomorphism of A-modules A → L canonically induces an ˆ ˆ Let M be an I-adic A-module. There exists isomorphism of A-modules Aˆr → L. ˆ . It is in general neither injective nor a canonical homomorphism M ⊗A Aˆ → M surjective. 9. Let M be a ﬁnitely generated A-module. Then M ⊗A Aˆ → M surjective. 20 1.

It follows that sx = 0 for every x ∈ U . 9) and α is injective. Next, let t ∈ G(U ). Then we can ﬁnd a covering of U by open sets Ui and sections si ∈ F(Ui ) such that α(Ui )(si ) = t|Ui . As we have just seen that α is injective, si and sj coincide on Ui ∩ Uj . , s|Ui = si ). By construction, α(U )(s) and t coincide on every Ui , and are therefore equal. This proves that α(U ) is surjective. A similar proof shows that F → G is injective if and only if Fx → Gx is injective for every x ∈ X. 13. Let α : F → G be a morphism of sheaves.

Fm ) and Hn ∈ I ∩ T n+1 A[[T ]]. It is clear that the series Gq + Gq+1 + . . tends to an element of (F1 , . . , Fm ) and that Hn tends to 0. Hence F ∈ (F1 , . . , Fm ). 8. Let A be a Noetherian ring, and let I be an ideal of A. Then the formal completion of A for the I-adic topology is a Noetherian ring. Proof Let t1 , . . , tr be a system of generators of I. Let us consider the surjective homomorphism of A-algebras φ : B = A[T1 , . . , Tr ] → A deﬁned by φ(Ti ) = ti , and endow B with the m-adic topology, where m is the ideal generated by the Ti .