appendixA - Appendix A Algebraic Geometry We develop the...

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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 self-contained 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 finite-dimensional vector space over C . A complex-valued 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 multi-index 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 multi-index 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
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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 { 0 } 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 R is equivalent to the as- cending chain condition for ideals in R : (ACC) If I 1 I 2 ⊂ · · · ⊂ R is an ascending chain of ideals in R , then there exists an index p so that I j = I p for all j p .
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