Algebraic Spaces
This chapter gives the basic results concerning solutions of polynomial equations in several variables over a field k. First it will be proved that if such equations have a common zero in some field, then they have a common zero in the algebraic closure o
- PDF / 3,366,036 Bytes
- 36 Pages / 439 x 666 pts Page_size
- 47 Downloads / 242 Views
IX
Algebraic Spaces
This chapter gives the bas ic results concerning solutions of polynomial equations in several variables over a field k. First it will be proved that if such equations have a common zero in some field, then they have a common zero in the algebraic closure of k, and such a zero can be obtained by the process known as specialization. However, it is useful to deal with transcendental extensions of k as welL Indeed, if p is a prime ideal in k[X] == k[XJ, ... , X n ], then k[X]/p is a finitely generated ring over k, and the images Xi of Xi in this ring may be transcendental over k, so we are led to consider such rings. Even if we want to deal only with polynomial equations over a field, we are led in a natural way to deal with equations over the integers Z. Indeed, if the equations are homogeneous in the variables, then we shall prove in §3 and §4 that there are universal polynomials in their coefficients which determine whether these equations have a common zero or not. "Universal" means that the coefficients are integers, and any given special case comes from specializing these universal polynomials to the special case. Being led to consider polynomial equations over Z, we then consider ideals a in Z[X] . The zeros of such an ideal form what is called an algebraic space . If p is a prime ideal , the zeros of p form what is called an arithmetic variety. We shall meet the first example in the discussion of elimination theory, for which I follow van der Waerden 's treatment in the first two editions of his Moderne Algebra , Chapter XI. However, when taking the polynomial ring Z[X]/a for some ideal a, it usually happens that such a factor ring has divisors of zero, or even nilpotent elements . Thus it is also natural to consider arbitrary commutative rings, and to lay the foundations of algebraic geometry over arbitrary commutative rings as did Grothendieck . We give some basic definitions for this purpose in §5 . Whereas the present chapter gives the flavor of algebraic geometry dealing with specific polynomial ideals , the next chapter gives the flavor of geometry developing from commutative algebra, and its systematic application to the more general cases just mentioned .
377 S. Lang, Algebra © Springer Science+Business Media LLC 2002
378
IX, §1
ALGEBRAIC SPACES
The present chapter and the next will also serve the purp ose of giving the reader an introduction to books on algebraic geometry, notably Hartshorne' s systematic basic account. For instan ce , I have included tho se result s which are needed for Hartshorne' s Chapter I and II.
§1 .
HILBERT'S NULLSTELLENSATZ
The Nullstellensatz has to do with a special cas e of the extensio n theorem for homomorphisms, applied to finitely generated ring s over fields.
Theorem 1.1. Let k be a field, and let k[x] = k[x l , . . . , x n] be a finitely generated ring over k. Let tp : k --+ L be an embedding of k into an algebraically closed field L. Then there exists an extension of cp to a homomorphism of k[x] into L. Proof. Let 9Jl be a max imal ide
Data Loading...
