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Unformatted text preview: Appendix A Algebraic Geometry We develop the aspects of algebraic geometry needed for the study of algebraic groups over C in this book. Although we give selfcontained proofs of almost all of the results stated, we do not attempt to give an introduction to the field of alge braic geometry or to give motivating examples. We refer the interested reader to Cox, Little, and O’Shea [1992], Harris [1992], Shafarevich [1994], and Zariski and Samuel [1958] for more details. A.1 Affine Algebraic Sets A.1.1 Basic Properties Let V be a finitedimensional vector space over C . A complexvalued function f on V is a polynomial of degree ≤ k if for some basis { e 1 , . . ., e n } of V one has f (∑ n i =1 x i e i ) = ∑  I ≤ k a I x I Here for a multiindex I = ( i 1 , . . ., i n ) ∈ N n we write x I = x i 1 1 ··· x i n n . This definition is obviously independent of the choice of basis for V . If there exists a multiindex I with  I  = k and a I negationslash = 0 , then we say that f has degree k . Let P ( V ) be the set of all polynomials on V , P k ( V ) the polynomials of degree ≤ k, and P k ( V ) the polynomials of degree k . Then P ( V ) is a commutative algebra, relative to pointwise multiplication of functions. It is freely generated as an algebra by the linear coordinate functions x 1 , . . ., x n . A choice of a basis for V thus gives rise to an algebra isomorphism P ( V ) ∼ = C [ x 1 , . . ., x n ] , the polynomial ring in n variables. Definition A.1.1. A subset X ⊂ V is an affine algebraic set if there exist functions f j ∈ P ( V ) such that X = { v ∈ V : f j ( v ) = 0 for j = 1 , . . ., m } . When X is an affine algebraic set, we define the affine ring O [ X ] of X to be the functions on X that are restrictions of polynomials on V : O [ X ] = { f  X : f ∈ P ( V ) } . 605 606 APPENDIX A. ALGEBRAIC GEOMETRY We call these functions the regular functions on X . Define I X = { f ∈ P ( V ) : f  X = 0 } . Then I X is an ideal in P ( V ) , and O [ X ] ∼ = P ( V ) / I X as a commutative algebra. Theorem A.1.2 (Hilbert basis theorem) . Let I ⊂ P ( V ) be an ideal. Then there is a finite set of polynomials { f 1 , . . ., f d } ⊂ I so that every g ∈ I can be written as g = g 1 f 1 + ··· + g d f d with g i ∈ P ( V ) . Proof. Let dim V = n . Then P ( V ) ∼ = C [ x 1 , . . ., x n ] . Since a polynomial in x 1 , . . ., x n with coefficients in C can be written uniquely as a polynomial in x n with coefficients that are polynomials in x 1 , . . ., x n 1 , there is a ring isomorphism P ( V ) ∼ = R [ x n ] , with R = C [ x 1 , . . ., x n 1 ] . (A.1) We call an arbitrary commutative ring R Noetherian if every ideal in R is finitely generated. For example, the field C is Noetherian since its only ideals are { } and C . To prove the theorem, we see from (A.1) that it suffices to prove the following: (N) If a ring R is Noetherian, then the polynomial ring R [ x ] is Noetherian We first show that the Noetherian property for a ring...
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This note was uploaded on 02/28/2012 for the course MATH 207C taught by Professor N.r.wallach during the Spring '10 term at Colorado.
 Spring '10
 N.R.Wallach
 Algebra, Geometry

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