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1. [II.6 Divisors] Let X be a noetherian integral separated scheme which is regular in codimension one. Show that (a) X × A

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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

1

has the same property and Cl X × A

1

∼ = Cl X, (b) X × P

1

has 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 →

Pn

k

O (− 1 )

n+1

O → 0 is exact, (b) ω

Y

∼ = O

Y

( d − ( n + 1 )) for Y ⊂ P

nk

being 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

i

of 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

i

of projective schemes] Let A be a noetherian ring and X = P

rA

. Determine the structure of H

i

( X, O ( n )) for all i0 and nZ, 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

0

such 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

,

p

) = 0 for p 6= q and ∼ = k if 0 ≤ p = q ≤ n.

7. [III.8-9 R

i

f

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

i

f

( F ⊗ f

E ) ∼ = ( R

i

f

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.

1

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