Chapter8 - Chapter 8 Introducing Algebraic Geometry...

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Chapter 8 Introducing Algebraic Geometry (Commutative Algebra 2) We will develop enough geometry to allow an appreciation of the Hilbert Nullstellensatz, and look at some techniques of commutative algebra that have geometric signiFcance. As in Chapter 7, unless otherwise speciFed, all rings will be assumed commutative. 8.1 Varieties 8.1.1 Defnitions and Comments We will be working in k [ X 1 ,...,X n ], the ring of polynomials in n variables over the Feld k . (Any application of the Nullstellensatz requires that k be algebraically closed, but we will not make this assumption until it becomes necessary.) The set A n = A n ( k ) of all n -tuples with components in k is called affine n -space .I f S is a set of polynomials in k [ X 1 n ], then the zero-set of S , that is, the set V = V ( S )o fa l l x A n such that f ( x ) = 0 for every f S , is called a variety . (The term “affine variety” is more precise, but we will use the short form because we will not be discussing projective varieties.) Thus a variety is the solution set of simultaneous polynomial equations. If I is the ideal generated by S , then I consists of all Fnite linear combinations g i f i with g i k [ X 1 n ] and f i S . It follows that V ( S )= V ( I ), so every variety is the variety of some ideal. We now prove that we can make A n into a topological space by taking varieties as the closed sets. 8.1.2 Proposition (1) If V α = V ( I α ) for all α T , then T V α = V ( S I α ). Thus an arbitrary intersection of varieties is a variety. 1
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2 CHAPTER 8. INTRODUCING ALGEBRAIC GEOMETRY (2) If V j = V ( I j ), j =1 ,...,r , then S r j =1 V j = V ( { f 1 ··· f r : f j I j , 1 j r } ). Thus a Fnite union of varieties is a variety. (3) A n = V (0) and = V (1), so the entire space and the empty set are varieties. Consequently, there is a topology on A n , called the Zariski topology , such that the closed sets and the varieties coincide. Proof. (1) If x A n , then x T V α i± every polynomial in every I α vanishes at x x V ( S I α ). (2) x S r j =1 V j i± for some j , every f j I j vanishes at x x V ( { f 1 f r : f j I j for all j } ). (3) The zero polynomial vanishes everywhere and a nonzero constant polynomial van- ishes nowhere. Note that condition (2) can also be expressed as r j =1 V j = V r Y j =1 I j = V ( r j =1 I j ) . [See (7.6.1) for the deFnition of a product of ideals.] We have seen that every subset of k [ X 1 ,...,X n ], in particular every ideal, determines a variety. We can reverse this process as follows. 8.1.3 Defnitions and Comments If X is an arbitrary subset of A n , we deFne the ideal of X as I ( X )= { f k [ X 1 n ]: f vanishes on X } . By deFnition we have: (4) If X Y then I ( X ) I ( Y ); if S T then V ( S ) V ( T ). Now if S is any set of polynomials, deFne IV ( S )as I ( V ( S )), the ideal of the zero-set of S ; we are simply omitting parentheses for convenience. Similarly, if X is any subset of A n , we can deFne VI ( X ), IVI ( X ), VIV ( S ), and so on. ²rom the deFnitions we have: (5) ( S ) S ; ( X ) X .
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Chapter8 - Chapter 8 Introducing Algebraic Geometry...

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