By Qing Liu

Advent; 1. a few subject matters in commutative algebra; 2. common homes of schemes; three. Morphisms and base swap; 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. standard surfaces; 10. aid of algebraic curves; Bibilography; Index

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**Example text**

Let πn : Bn+1 → Bn denote the transition homomorphism. Let (cn )n ∈ ←− lim Cn . Let bn ∈ Bn be an arbitrary preimage of cn . Identifying An with a submodule of Bn , we have an := πn (bn+1 )−bn ∈ An . Hence there exists an an+1 ∈ An+1 such that πn (an+1 ) = an . It follows that πn (bn+1 − an+1 ) = bn . Thus we can modify the bn one after the other to obtain lim Cn πn (bn+1 ) = bn for all n. Consequently, (bn )n ∈ ←− lim Bn , and its image in ←− is (cn )n . Let G be a topological group deﬁned by a ﬁltration (Gn )n .

N ), 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. 15 explains why such a study corresponds to that of the spectra of ﬁnitely generated algebras over k. 17. Let k be an algebraically closed ﬁeld of characteristic diﬀerent from 2. The set of closed points of Spec k[X, Y ]/(Y 2 − X 2 (X + 1)) corresponds to the algebraic set {(x, y) ∈ k 2 | y 2 − x2 (x + 1) = 0} (see Figure 2).

For any open subset U , we let OX (U ) be the set h of holomorphic functions on U . Then (X, OX ) is a complex analytic variety. It is a ringed topological space. The property that we need to verify is that the stalks h are indeed local rings. of OX h Let z ∈ Cn . Then OX,z can be identiﬁed with the holomorphic functions deﬁned on a neighborhood of z. Let mz be the set of those which vanish in z. h h because OX,z /mz C. If a holomorphic function This is a maximal ideal of OX,z f does not vanish in z, then 1/f is still holomorphic in z.