Points and Morphisms, from the Classical and Scheme-theoretic Point of View
نویسنده
چکیده
Naively, given a field k and some n > 0, an affine algebraic set in kn is the zero set of some finite collection of polynomial equations with n variables and coefficients in k. A morphism between two affine algebraic sets is a map defined by a tuple of polynomials. Armed with Hilbert’s Nullstellensatz, it isn’t too hard to make this precise in the case that k is algebraically closed, so we begin with this. First, an affine algebraic set V ⊆ kn is a subset obtained as the common zero set of some f1, . . . , fs ∈ k[x1, . . . , xn]. Given a second affine algebraic set W ⊆ km, defined as the zero set of g1, . . . , gt, a morphism V → W is defined as follows. A m-tuple (h1, . . . , hm) of polynomials in k[x1, . . . , xn] defines a map ~h : k → k by sending ~c := (c1, . . . , cn) to (h1(~c), . . . , hm(~c)). A morphism is then a function V →W obtained as the restriction of some ~h as above. Now, the above definitions are very naive and set-theoretic, but in the the algebraically closed case, they turn out to behave well. Starting with the definition of algebraic set, one might wonder if we are losing information by only looking at the zero set, and not keeping track of the original defining polynomials. The answer is that we are losing very little information. Namely, it is easy to see that (irrespective of the algebraic closure hypothesis) V only depends on the ideal generated by the fi, and in fact only on the radical of this ideal. So the best we could hope to do is to recover the radical of the ideal generated by the fi from V , and Hilbert’s Nullstellensatz says that in the algebraically closed case, we can do precisely that: if we simply take all polynomials which vanish on V , we will get precisely the radical of the ideal generated by the fi. Thus, in working with algebraic sets over algebraic closed fields, it is irrelevant whether we think of them in terms of the sets themselves, or in terms of the radical ideals of polynomials used to define them.
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