Definition 3.3 An affine variety is the locus where a collection of polynomial equations is satisfied, i.e., given F = { f j }j∈J ⊂ k[x1, . . . , xn] we define V(F) = {a ∈ An(k) : f j (a) = 0 for each j ∈ J} ⊂ An(k). These polynomials are said to define the variety.
Proposition 3.12 For any ideals I1, I2 ⊂ k[x1, . . . , xn], we have
V(I1 ∩ I2) = V(I1 I2) = V(I1) ∪ V(I2)
Proposition 3.12 For any ideals I1, I2 ⊂ k[x1, . . . , xn], we have
V(I1 ∩ I2) = V(I1 I2) = V(I1) ∪ V(I2)
Proposition 3.13 An arbitrary intersection of varieties ∩β∈BVβ is a variety. A finite union of varieties ∪N i=1Vi is a variety.
Proposition 3.14 Every variety can be defined as the locus where a finite number of polynomials vanish
(Proof : Hilbert Basis Theorem)
Proposition 3.16 Consider ideals I1 ⊂ k[x1, . . . , xn] and I2 ⊂ k[y1, . . . , ym] and
the corresponding varieties V(I1) ⊂ An(k) and V(I2) ⊂ Am(k). Let
J = I1k[x1, . . . , xn, y1, . . . , ym] + I2k[x1, . . . , xn, y1, . . . , ym],
i.e., the ideal in k[x1, . . . , xn, y1, . . . , ym] generated by I1 and I2. Then V(I1) ×
V(I2) = V(J ).
proof : We can express the product as an intersection
V(I1) × V(I2) = −1
1 (V(I1)) ∩ −1
2 (V(I2)).
Example 3.17 Fermat’s Last Theorem, as proven by Andrew Wiles and Richard
Taylor, asserts that for any integers x, y, z with
x N + yN = zN , N ≥ 3,
at least one of the three integers is zero. We may as well assume x, y, z ∈ Q; multiplying
through by the least common multiple of the denominators would yield an
integral solution. In our notation, Fermat’s Last Theorem takes the following form:
If N ≥ 3 and V = V(x N + yN − zN ) ⊂ A3(Q) then xyz ∈ I (V)
Proposition 3.14 Every variety can be defined as the locus where a finite number of polynomials vanish
(Proof : Hilbert Basis Theorem)
Proposition 3.16 Consider ideals I1 ⊂ k[x1, . . . , xn] and I2 ⊂ k[y1, . . . , ym] and
the corresponding varieties V(I1) ⊂ An(k) and V(I2) ⊂ Am(k). Let
J = I1k[x1, . . . , xn, y1, . . . , ym] + I2k[x1, . . . , xn, y1, . . . , ym],
i.e., the ideal in k[x1, . . . , xn, y1, . . . , ym] generated by I1 and I2. Then V(I1) ×
V(I2) = V(J ).
proof : We can express the product as an intersection
V(I1) × V(I2) = −1
1 (V(I1)) ∩ −1
2 (V(I2)).
Example 3.17 Fermat’s Last Theorem, as proven by Andrew Wiles and Richard
Taylor, asserts that for any integers x, y, z with
x N + yN = zN , N ≥ 3,
at least one of the three integers is zero. We may as well assume x, y, z ∈ Q; multiplying
through by the least common multiple of the denominators would yield an
integral solution. In our notation, Fermat’s Last Theorem takes the following form:
If N ≥ 3 and V = V(x N + yN − zN ) ⊂ A3(Q) then xyz ∈ I (V)
