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1 Affine V ar ie ti es We will begin follo wing Kempf’s Algeb raic Var ietie s, and eventually will do things more like in Hartshorne. We will also use various sources for commutativ e algebra. What is algebrai c geometr y? Classicall y , it is the study of the zero sets of polynomials. We will now fix some notation. k will be some fixed algebraically closed field, any ring is commutative with identity, ring homomorphisms preserve identity, and a k -algebra is a ring R which contains k (i.e., we have a ring homomorphism ι : k → R). P ⊆ R an ideal is prime iff R/P is an integral domain. Algebraic Sets We define affine n-space, A n = k n = { (a 1 ,...,a n ) : a i ∈ k }. Any f = f (x 1 ,...,x n ) ∈ k[x 1 ,...,x n ] defines a function f : A n → k : (a 1 ,...,a n ) → f (a 1 ,...,a n ). Exerc ise If f, g ∈ k[x 1 ,...,x n ] define the same function then f = g as polynomials. Definition 1.1 (Algebraic Sets). Let S ⊆ k[x 1 ,...,x n ] b e any subset. The n V (S ) = { a ∈ A n : f (a) = 0 for all f ∈ S }. A subset of A n is called algebraic if it is of this form. e.g., a point {(a 1 ,...,a n )} = V (x 1 − a 1 ,...,x n − a n ). Exercises 1. I = ( S ) is the ideal generated by S . Then V (S ) = V (I ). 2. I ⊆ J ⇒ V (J ) ⊆ V (I ). 3. V (∪ α I α ) = V ( I α ) = ∩ V (I α ). 4. V (I ∩ J ) = V (I · J ) = V (I ) ∪ V (J ). Definition 1.2 (Zariski Topology). We can define a topology on A n by defining the closed subsets to be the algebraic subsets. U ⊆ A n is open iff A n \ U = V (S ) for some S ⊆ k [x 1 ,...,x n ]. Exercises 3 and 4 imply that this is a topology. The closed subsets of A 1 are the finite subsets and A 1 itself. Definition 1.3 (Ideal of a Subset). If W ⊂ A n is any subset, then I (W ) = {f ∈ k [x 1 ,...,x n ] : f (a) = 0 for all a ∈ W } Facts/Exercises 1. V ⊆ W ⇒ I (W ) ⊆ I (V ) 2. I (∅) = (1) = k [x 1 ,...,x n ] 3. I (A n ) = (0). 1

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