2019 ALGEBRAIC GEOMETRY I FINAL EXAM
A COURSE BY CHIN-LUNG WANG AT NTU
1. [II.6 Divisors] Let X be a noetherian integral separated scheme which is regular in codimension one. Show that (a) X × A
1has the same property and Cl X × A
1∼ = Cl X, (b) X × P
1has the same property and Cl X × P
1∼ = Cl X × Z.
2. [II.7 Projective morphisms] Let X be a noetherian scheme and Y, Z be closed sub- scheme, neither one containing the other. Let ˜ X be the blowing up of X along Y ∩ Z.
Show that the strict transform ˜ Y and ˜ Z in ˜ X do not meet.
3. [II.8 Differentials] Show that (a) 0 → Ω
Pnk
→ O (− 1 )
n+1→ O → 0 is exact, (b) ω
Y∼ = O
Y( d − ( n + 1 )) for Y ⊂ P
nkbeing a non-singular hypersurface of degree d. (c) Generalized (b) to the case of for Y being a complete intersection of r equations.
4. [III.2-3 H
iof affine schemes] Let X be a noetherian scheme. Show that the following are equivalent. (i) X is affine. (ii) H
i( X, F ) = 0 for all quasi-coherent F and all i ≥ 1.
(iii) H
1( X, I ) = 0 for all coherent sheaf of ideals I .
5. [III.4-5 H
iof projective schemes] Let A be a noetherian ring and X = P
rA. Determine the structure of H
i( X, O ( n )) for all i ≥ 0 and n ∈ Z, including the perfect pairing H
0( X, O ( n )) × H
r( X, O (− n − ( r + 1 ))) → A.
6. [III.6-7 Ext and duality] (a) Let X be projective over a noetherian ring A.Given coher- ent sheaves F , G , show that ∃ n
0such that Ext
i( F , G ( n )) ∼ = Γ ( X, E xt
i( F , G ( n ))) for all n ≥ n
0. (b) Show that H
q( P
nk, Ω
p) = 0 for p 6= q and ∼ = k if 0 ≤ p = q ≤ n.
7. [III.8-9 R
if
∗and flatness] Let f : X → Y be a morphism of ringed spaces, F ∈ Mod
X, E ∈ Mod
Y. (a) For all i ≥ 0, show that R
if
∗( F ⊗ f
∗E ) ∼ = ( R
if
∗F ) ⊗ E for E being locally free of finite rank. (b) Give a counterexample for coherent E .
(*) You are allowed to replace ONE problem by an essential topic/theorem/exercise in the same labelled subsection(s).
8. (Bonus) Present an essential topic/theorem/exercise within II.6 to III.9 but not listed above nor in your replacement problem.
Each problem is of 15 points (total 120 pts). Be sure to show your answers/computations/proofs in details.
Time: pm 6:00 – 9:50, January 9, 2020 at AMB 102. Also you are allowed to check the textbook (but not your notes, overleaf, or any website) during pm 7:30-7:40 without copying anything on the exam sheets.
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