Download Algebraic Geometry 3-Homework 11 1. a. Let O be a noetherian

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