The purpose of this post is to prove Chevalley’s theorem: If is a finite surjective morphism of noetherian separated schemes, with affine, then is affine.
We will follow the outline in Hartshorne (III.3 Problems 1 & 2 and III.4 Problems 1 & 2).
Theorem 1 Let be an affine morphism of noetherian schemes. Then for any coherent sheaf on , there are natural isomorphisms for all ,
Proof: According to (II, Ex. 5.17), when is affine, the direct image functor induces an equivalence from the category of coherent -modules to the category of coherent -modules. Moreover, an equivalence of abelian categories (i.e. an additive functor which is also an equivalence) is exact. Therefore, if is a left additive functor, by the uniqueness of the -functor extending a given left additive functor, it follows that there exists a natural isomorphism for each .
Theorem 2 Let be a noetherian scheme. Then is affine if and only if is.
Proof: Clearly is affine if is affine.
Conversely, suppose is affine. We prove that has cohomological dimension , hence it is affine by Serre’s theorem (III.3.7). Let be a quasi-coherent sheaf on . As indicated in the hint, we let denote the sheaf of nilpotents of and we consider the filtration
of . Since is noetherian, there exists an such that , so the filtration is finite.
We prove by descending induction on that is acyclic. For , it is trivial. Now consider the exact sequence of quasi-coherent sheaves on ,
The quasi-coherent sheaf is naturally a quasi-coherent -module, and its cohomology can be calculated either as an -module or as an module by Theorem 1 (using the fact that the reduction morphism is affine). Therefore, it is acyclic, since is affine by assumption. The sheaf is acyclic by the inductive hypothesis. By the long exact sequence of cohomology, we see that is also acyclic.
Theorem 3 Let be a reduced scheme. Then is affine if and only if each irreducible component of is affine.
Proof: The irreducible components of are closed subschemes of , hence they are affine if is affine. Conversely, suppose that every irreducible component of is affine. We prove that has cohomological dimension .
We proceed by induction on the number of irreducible components of . If is irreducible, then the statement is vacuously true. Now suppose it holds for noetherian schemes with irreducible components. Suppose that has irreducible components, and write it as where is irreducible. Let be a quasi-coherent sheaf on . Denote the inclusion and the inclusion , where each closed subscheme is given the canonical reduced closed subscheme structure. Since is Noetherian, is also a quasi-coherent sheaf on , supported on . There is a canonical morphism , and . (Each of these two morphisms is a unit of the “inverse image – direct image” adjunction). Let
be their sum. It is easy to see that this morphism is surjective, and an isomorphism away from the intersection. Let . Then is quasi-coherent and supported in . Therefore we have an exact sequence
Since is affine by the induction hypothesis, is affine, being a closed subscheme of an affine scheme. Now, since , the cohomology of can be calculated either as an -module or as an -module, and therefore it vanishes. Similarily the sheaf is acyclic because and are affine. Therefore, by the long exact sequence of cohomology, is also acyclic.
Lemma 4 Let be a finite surjective morphism of integral noetherian schemes. Then there is a coherent sheaf on , and a morphism of sheaves for some , such that is an isomorphism at the generic point of .
Proof: Let be the function field of and be the function field of . Then the morphism gives rise to a field homomorphism . Since is finite and surjective, is finite over , say of degree . Let be a basis for over . Each can be represented as a section of over an open set . Let be the inclusion. Let be the sheaf on . Obviously is coherent (in fact free of rank ). Let . Then is quasi-coherent on since is noetherian; since is finite, is in fact coherent. Let . Define the morphism by the global sections of (using the fact that represents the global sections functor ). Then, by construction, is an isomorphism of -vector spaces at the generic point of .
Lemma 5 Let be a finite surjective morphism of integral noetherian schemes. Then for any coherent sheaf on , there exists a coherent sheaf on , and a a morphism which is an isomorphism at the generic point of .
Proof: We take , where is the sheaf and is the morphism of Lemma 4:
Remark that . Moreover, the sheaf naturally has a structure of -module. By (II, Ex. 5.17), when is an affine morphism, induces an equivalence between the category of coherent -modules and the category of coherent -modules. Therefore is isomorphic to an -module of the form , where is a coherent -module. Thus has the form .
Moreover, it follows from the fact that a coherent sheaf on a noetherian scheme is finitely presented that on such a scheme, taking sheaf commutes with taking stalks of morphisms; therefore is also an isomorphism at the generic point of .
Now we are ready to prove Chevalley’s theorem.
Theorem 6 (Chevalley’s theorem). Let be a finite surjective morphism of noetherian separated schemes, where is affine. Then is affine.
Proof: By Theorems 2 and 3, we may suppose that and are reduced and irreducible. We prove by contradiction that is affine. Let be the collection of closed subschemes of which are not affine. Suppose it not empty; then it contains a minimal element , which we may view as having the reduced induced subscheme structure. Since finite morphisms are stable under base change, we may in fact suppose that (what this means is that we are replacing by its restriction to if necessary). Therefore, we suppose that every proper closed subscheme of is affine.
Let be a coherent sheaf on . By Lemma , there exists a coherent sheaf on and a morphism which is generically an isomorphism (and which is therefore surjective, since is irreducible). Thus, if , we have an exact sequence of sheaves on
Now, as in the proof of Theorem 3, we view as a quasi-coherent sheaf on the proper closed subscheme . By the minimality of , is affine and therefore is acyclic. Moreover, since a finite morphism is affine, we can apply Theorem 1 to see that is also acyclic. Therefore, by the long exact sequence of cohomology, is acyclic, so is acyclic.
Thefore, has cohomological dimension , which contradicts the assumption that it is not affine.