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Algebraic Geometry 3-Homework 11 1. a. Let O be a noetherian local unique factorization domain (UFD). Show that O is a normal ring. Show that O is a discrete valuation ring (DVR) if O has Krull dimension one. b. Let X be an integral k-scheme. X is called locally factorial if each local ring OX,x is a UFD (for example, a smooth k-scheme is locally factorial). Show that for a locally factorial k-scheme X of dimension n, the divisor map [−] : Div(X) → Zn−1 (X) is an isomorphism, inducing an isomorphism Pic(X) → CHn−1 (X). 2. a. Use the localization sequence to show that for 0 ≤ m ≤ n, CHm (Pn ) is generated by the class of any linear subspace Pm ⊂ Pn . b. Show that CHn−1 (Pn ) ∼ = Z and CH0 (Pn ) ∼ = Z, the second isomorphism n arising from the degree map degk : CH0 (P ) → CH0 (Spec k) = Z. For the first isomorphism, show that OPn (m) ∼ = OPn if and only if m = 0. 3. Let X be a k-scheme. a. Let α ∈ Zn (X) be a cycle, D, D0 ∈ Div(X) linear equivalent Cartier divisors. Show that D · α = D0 · α in CHn−1 (|α|). b. Suppose X is a proper k-scheme. Recall that for a proper k-scheme p : Y → Spec k, a 0-cycle z ∈ CH0 (Y ) has degree d over k (degk z = d) if p∗ (z) = d · [Spec k] in CH0 (Spec k). Let α ∈ Z1 (X) be a cycle, D, D0 ∈ Div(X) linear equivalent Cartier divisors. Show that degk D·α = degk D0 ·α. 4. a. Let α ∈ Z1 (P2 ) be a cycle. Define deg(α) as degk (` · α), where ` is a line in P2 considered as a Cartier divisor. Show that deg(α) is welldefined (i.e., independent of the choice of `). b. Since P2 is smooth over k, we have Div(P2 ) = Z1 (P2 ), so for each pair of 1-cycles, α, β ∈ Z1 (P2 ), the intersection produce α · β ∈ CH0 (P2 ) is defined by considering α as in Div(P2 ) and β as in Z1 (P2 ). Prove Bezout’s theorem: degk (α · β) = deg(α) · deg(β). Conclude that α · β = β · α in CH0 (P2 ). 5. Let L be a line bundle on a smooth k-scheme X of pure dimension n. We haveus the pseudo-divisor (L, X, ∅) and the fundamental class [X] ∈ Zn (X). The first Chern class of L, c1 (L), is defined as c1 (L) := (L, X, ∅) · [X] ∈ CHn−1 (X). a. Let f : Y → X be a flat morphism of smooth k-schemes, L a line bundle on X. Show that f ∗ (c1 (L)) = c1 (f ∗ L). b. Show that sending L to c1 (L) induces an isomorphism Pic(X) → CHn−1 (X), and that this is the same isomorphism as the one in 1(b). 1