Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 113 / 4 / c

Solution

Write

A = k [x, y]

X = Spec A

, and

U = X ∖ Z

. Since

A

is an integral domain, no nonzero global section is supported only at the origin, so

H_{Z}^{0} (X, O_{X}) = 0.

The punctured affine plane has the affine cover

U = D (x) \cup D (y)

. Its Čech cochain complex for the structure sheaf is

0 ⟶ A_{x} \oplus A_{y} (a, b) \mapsto a - b A_{x y} ⟶ 0.

Consequently

H^{0} (U, O_{U}) = A, H^{1} (U, O_{U}) = \frac{A _{x y}}{A _{x} + A _{y}}, H^{i} (U, O_{U}) = 0 (i \geq 2) .

Because

X

is affine, the higher sheaf cohomology of

O_{X}

H_{Z}^{1} (X, O_{X}) = 0, H_{Z}^{2} (X, O_{X}) ≅ \frac{k [ x ^{\pm 1} , y ^{\pm 1} ]}{k [ x ^{\pm 1} , y ] + k [ x , y ^{\pm 1} ]},

and

H_{Z}^{i} (X, O_{X}) = 0

for

i \geq 3

. The nonzero group has the

k

{x^{- a} y^{- b} : a, b \geq 1} .

There are no discussions about this article yet.

There are currently no matching articles.