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Maths > Algebraic geometry > Functor of points

Functors, affine varieties and Yoneda

Posted by Martin Orr on Wednesday, 02 September 2009 at 22:51

In this article, I will examine in more detail the functor of points of an affine variety, which I defined in the last article. I shall show that this functor is the same as a Hom-functor on the category of k-algebras, and that morphisms of varieties correspond to natural transformations of functors.

Points of an affine variety and Hom-functors

First we give a categorical description of the functor of points of an affine variety. Recall that the category of affine k-varieties is dual to the category of finitely-generated integral-domain k-algebras. In particular, the coordinate ring B of an affine k-variety X is a k-algebra, and the variety is determined by its coordinate ring.

Hence there is one obvious choice of (covariant) functor k\textbf{-Alg} \to \textbf{Set} associated with X, namely \mathop{\mathrm{Hom}}(B, -). It is necessary to prove that this is the same as the functor of points of X, but it is not hard.

Lemma 1. For every k-algebra A, there is a bijection between A-points of X and elements of \mathop{\mathrm{Hom}}(B, A) (where the Hom is in the category of k-algebras).

Proof. This is just a matter of unpacking definitions.

First we unpack "the A-points of X".

Suppose that X is embedded in affine n-space over k.

Then X is defined by an ideal I in k[x_1, \ldots, x_n], and the A-points of X are those points in A^n at which all polynomials in I vanish.

Furthermore, B = k[x_1, \ldots, x_n] / I. So we have to prove:

Claim. There is a bijection between points of A^n at which all polynomials in I vanish, and elements of \mathop{\mathrm{Hom}}(k[x_1, \ldots, x_n] / I, A).

Now a k-algebra homomorphism f : k[x_1, \ldots, x_n] \to A is determined by giving n elements of A, namely f(x_1, \ldots, x_n). This gives a bijection between A^n and \mathop{\mathrm{Hom}}(k[x_1, \ldots, x_n], A) (the case I = 0 of the claim).

And all the polynomials in I vanish at a given point in A^n iff I is contained in the kernel of the corresponding homomorphism k[x_1, \ldots, x_n] \to A, establishing the claim.

End of proof.

This shows that the functor of points of X and the functor \mathop{\mathrm{Hom}}(B, -) agree on objects. Verifying that they agree on morphisms is not hard.

Morphisms and natural transformations

Now we discuss morphisms. The obvious definition of a morphism between two functors is a natural transformation. We shall prove that in fact, morphisms between affine varieties are the same as natural transformations between their functors of points. The tool we use to do this is the Yoneda lemma (this proof illustrates a general method of applying the Yoneda lemma).

Let X, Y be affine k-varieties with coordinate rings A, B.

The Yoneda lemma tells us that for any functor F : k\textbf{-Alg} \to \textbf{Set}, there is a natural bijection between \mathop{\mathrm{Nat}}(\mathop{\mathrm{Hom}}(A, -), F) and F(A).

In particular, taking F = Hom(B, -), and using the duality between the categories of affine k-varieties and finitely-generated integral-domain k-algebras, we get that

\mathop{\mathrm{Nat}}(\mathop{\mathrm{Hom}}(A, -), \mathop{\mathrm{Hom}}(B, -)) = \mathop{\mathrm{Hom}}(B, A) = \mathop{\mathrm{Mor}}(X, Y).

Since \mathop{\mathrm{Hom}}(A, -) and \mathop{\mathrm{Hom}}(B, -) are just the functors of points of X and Y respectively, this says that natural transformations between the functors of points biject with morphisms between affine k-varieties.

Note that this fact as an important corollary: If two affine k-varieties have the same functor of points, then they are isomorphic as varieties. This follows by applying the above bijection to the isomorphism between the functors, getting an isomorphism between the varieties.

Thus we have seen that the functor of points of an affine variety has a simple categorical description, and that we can easily describe morphisms between affine varieties via the functors. This suggests that an approach to algebraic geometry based on functors of points is not just going to make things hopelessly complicated.

Tags alg-geom, maths, points-func, yoneda

Trackbacks

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