Math 214A Notes

1.1 Closed sets Let k be an algebraically closed field. Let A = Ak = {a = (a1, . . . , an) : ai ∈ k} = k be affine space. So A is a point, and A = k. Let An = k[T1, . . . , Tn] be the k-algebra of polynomials. If f ∈ An, a ∈ A, then f(a) ∈ k. Suppose S ⊆ An be a subset. Then Z(S) = {a ∈ A : f(a) = 0,∀f ∈ S} ⊆ A. A subset Z ⊆ A is called closed if Z = Z(S) for some S ⊆ An. Example 1.1. If n = 1, Z ⊂ A is closed iff Z is finite or Z = A.