By Qing Liu

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

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**Example text**

Let B be a ﬂat A-algebra. (a) Show that for any ﬁnite family {Iλ }λ∈Λ of ideals of A, we have ∩λ∈Λ (Iλ B) = (∩λ∈Λ Iλ )B. (b) Let us suppose that B is faithfully ﬂat. Show that IB ∩ A = I for any ideal I of A. (c) Let k be a ﬁeld, A = k[t, s], and C = A[z]/(tz − s). By considering the ideals tA and sA, show that C is not ﬂat over A. 7. Give an example of a ﬁnitely generated ﬂat module that is not free (over a suitable ring A). 8. Let A be a Noetherian ring, M a ﬁnitely generated A-module, and N an A-module.

The rest follows immediately. In the rest of this book, given a sheaf of ideals J , we will always endow V (J ) with the structure described in this lemma. 24. Let f : Y → X be a closed immersion of ringed topological spaces. Let Z be the ringed topological space V (J ) where J = Ker f # ⊆ OX . Then f factors into an isomorphism Y Z followed by the canonical closed immersion Z → X. 2. Ringed topological spaces 39 Proof As f (Y ) is closed in X, we have 0 OY,y (f∗ OY )x = if x ∈ / f (Y ) if x = f (y).

2. Let ϕ : A → B be a homomorphism of ﬁnitely generated algebras over a ﬁeld. Show that the image of a closed point under Spec ϕ is a closed point. 3. Let k = R be the ﬁeld of real numbers. Let A = k[X, Y ]/(X 2 + Y 2 + 1). We wish to describe Spec A. Let x, y be the respective images of X, Y in A. (a) Let m be a maximal ideal of A. Show that there exist a, b, c, d ∈ k such that x2 + ax + b, y 2 + cy + d ∈ m. Using the relation x2 + y 2 + 1 = 0, show that m contains an element f = αx + βy + γ with (α, β) = (0, 0).