Affine three-space ( $A_{k}^{3}$ )

Affine three-space is

Spec k [x_{1}, x_{2}, x_{3}]

.

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- Punctured affine three-space Affine three-space

Punctured affine three-space ( $A_{k}^{3} ∖ {0}$ )

For

U = A_{k}^{3} ∖ {0}

,

H^{0} (U, O_{U}) = k [x_{1}, x_{2}, x_{3}], H^{1} (U, O_{U}) = 0,

and

H^{2} (U, O_{U}) ≅ \frac{k [ x _{1}^{\pm 1} , x _{2}^{\pm 1} , x _{3}^{\pm 1} ]}{k [ x _{1}^{\pm 1} , x _{2}^{\pm 1} , x _{3} ] + k [ x _{1}^{\pm 1} , x _{2} , x _{3}^{\pm 1} ] + k [ x _{1} , x _{2}^{\pm 1} , x _{3}^{\pm 1} ]} .

The last group has basis represented by

x_{1}^{- a} x_{2}^{- b} x_{3}^{- c}

for

a, b, c \geq 1

, and all groups in degrees at least three vanish.

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