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Unformatted text preview: THE CATEGORY OF AFFINE k-VARIETIES PETE L. CLARK 1. The category of affine varieties over a field k Let k be an arbitrary field. We wish to define a category of affine varieties over k . Recall the most classical definition: if k = k , then an affine subvariety of k n is given by the zero locus of a set (which, by the Hilbert basis theorem, can be taken to be finite) of polynomials P 1 ( x ) , . . . , P r ( x ), where x = ( x 1 , . . . , x n ). Even in the classical case, this is a somewhat creaky definition since it is so ex- trinsic: it is like defining a complex manifold as a certain type of subset of C n . But in the case of a general field, it is manifestly inadequate: we would like to regard the equation x 2 + y 2 =- 1 as defining an affine curve in R 2 , but this curve has no (real) points. There are several rather elaborate remedies for this: we could switch to the theory of schemes, or we could introduce the theory of Galois descent. But let us try something more modest: we switch our arrows around. Namely, let’s try to define a category Aff op k , the opposite of the category of affine varieties. In this category, the objects will be affine k-algebras , i.e., finitely generated k- algebras A . In other words, A is a ring, equipped with a morphism of rings k → A , which is isomorphic, as a k-algebra to k [ x 1 , . . . , x n ] /I for some n ∈ Z + . Of course there will be many such isomorphisms, and we do not privilege any one of them as part of the definition. A morphism of affine algebras is just what is sounds like: a morphism of rings which respects the k-module structure. Then an affine variety V k will be identified with a corresponding affine alge- bra A V . However, a morphism V → W of affine varieties is precisely a morphism W k → V k , i.e., a morphism of the corresponding affine algebras, but in the other direction. Exercise: Show that the category of affine k-algebras is closed under quotients, finite Cartesian products, and finite tensor products (over k ), but is not closed under passage to arbitrary subalgebras. Before we go any farther, let us introduce the following construction: let k , → k be any field extension. Then we have a functor from the category of affine k-algebras to the category of affine k-algebras: we map A to A k := A ⊗ k k . The only thing we need to check is that if A is finitely generated over k , then A k is finitely generated over k , but this is immediate: we can write A = k [ x ] /I and then A k ∼ = k [ x ] /I . 1 2 PETE L. CLARK We refer to this process as base extension ....
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This note was uploaded on 10/26/2011 for the course MATH 8320 taught by Professor Clark during the Fall '10 term at UGA.
- Fall '10