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8320notes2 - THE CATEGORY OF AFFINE k-VARIETIES PETE L...

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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
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2 PETE L. CLARK We refer to this process as base extension . Now we can use commutative algebra to define “geometric properties” of affine varieties. For instance, recall that a commutative ring A is called reduced if it has no nonzero nilpotent elements: equivalently, its nilradical rad( A ) – the intersec- tion of all prime ideals – is equal to 0. If we have an affine algebra which is not
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