`A(X) = { φ∈ k(X) : φ
is regular at all points x ∈
X}.`

` I = (I(X), g _{x} : x∈ X)⊂
k[x_{1},…,x_{n}].`

` Z(I) = {x∈ X : φ is not
regular at x}, `

` 1 = ∑ u _{i} g_{xi},`

` φ = ∑ u _{i} f_{xi} ∈ A(X).`

` A(S) = [
{φ∈ k(A ^{n}):φ is regular at
all s∈ S}
/
{φ∈ k(A^{n}):φ(s)=0 at all
s∈ S}].`

- Consider the quasi-affine variety
`U=A`. Since^{1}\{0}`U`is open in the irreducible variety`A`, it is irreducible and dense. Therefore,^{1}`A(U)`is a subring of`k(A`. In order to determine which rational functions^{1})= k(x)`φ=f(x)/g(x)`are regular on`U`, we only need to determine which polynomials`g(x)`are zero only at the point`0`. Since these polynomials consist only of the powers of`x`, we see that`A(U)= k[x, x`. We recognize this ring of functions as the coordinate ring of the affine plane curve^{-1}]`V`defined by`xy=1`. Moreover, projection on the`x`-coordinate is a topological homeomorphism`V→ U`that is an isomorphism on the ring of regular functions. In this way, we can identify the quasi-affine variety`U`with the affine variety`V`. - Let's try to repeat the previous example in a larger
dimension. This time, take
`U=A`. The same analysis shows that^{2}\ {(0,0)}`A(U)`is a subring of`k(X)`. Writing a rational function`φ=f(x,y)/g(x,y)`as a quotient of relatively prime polynomials, we see that`φ`is not regular on the zero set`Z(g)`. If`g`is a nonconstant polynomial, then we have seen that this zero set is infinite. It follows that`A(U)≈ A(A`, and that this isomorphism is induced by the inclusion map (which is^{1}) = k[x,y]**not**a homeomorphism). Since coordinate rings determine their varieties up to isomorphism, it follows that the quasi-affine variety`U`is not isomorphic to any affine variety.

` A(U) = k[x _{1}, …, x_{n}, 1/f] =
A(A^{n})_{f}.`

U = | A^{n}\ Z(f) | ⊂ |
A^{n} |

↓φ ↑ψ | ↑pr | ||

V | ⊂ | A^{n+1} |

`φ(a _{1}, …, a_{n}) = (a_{1}, …, a_{n},
1/f(a_{1}, …, a_{n})),`

`ψ(a _{1},…,a_{n+1}) =
(a_{1},…,a_{n}).`

`A(U) = A(X) _{f} = A(X)[1/f].`

` P∈ X\ Z(f) ⊂ U`

`ρ(X)⊂ ρ(U) ^{-}
⊂Δ
`

`k(X) =
[{(U,f) : U is a nonempty open
subset of X and f:U→A^{1}}
/
(U,f)∼(V,g) iff f|_{U∩ V} =
g|_{U∩ V}]
`

Comments on this web site should be addressed to the author:

Kevin R. CoombesDepartment of Biomedical Informatics

The Ohio State University

Columbus, Ohio 43210