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By Qing Liu

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

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Let 0 → F → F → F be an exact sequence of sheaves on X. Show that the sequence of Abelian groups 0 → F (X) → F(X) → F (X) is exact. 40 2. 5. (Supports of sheaves) Let F be a sheaf on X. Let Supp F = {x ∈ X | Fx = 0}. We want to show that in general, Supp F is not a closed subset of X.

2. Let F be a sheaf on X. Let s, t ∈ F(X). Show that the set of x ∈ X such that sx = tx is open in X. 3. (Sheaf associated to a presheaf ) Let us fix a topological space X. Let F be a presheaf on X. 7. (a) Let U = {Ui }i be an open covering of X. Let FU (X) = Ker d1 . For any open subset W of X, we define in the same way a group FU (W ) relative to the covering {W ∩ Ui }i of W . Show that FU is a presheaf on X and that we have a morphism of presheaves F → FU . (b) Let V = {Vk }k be another open covering of X.

A−1 (1 + δ + δ 2 + . . ) ˆ Hence this is a local ring. We already know that it is is the inverse of α in A. 20. (c) Let n ≥ 1. We have nn = mn B. Since the composition A/mn → B/mn B → ˆ n Aˆ is surjective. It remains to show ˆ A/mn Aˆ is an isomorphism, B/mn B → A/m n ˆ that it is injective; that is, that m A ∩ B = mn B. We have B = A + mB = A + m2 B = · · · = A + mn B, so every element b ∈ B can be written b = a + ε ˆ If, moreover, b ∈ mn A, ˆ then a ∈ mn Aˆ ∩ A = mn , with a ∈ A, ε ∈ mn B ⊆ mn A.

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