# Chevalley’s theorem

The purpose of this post is to prove Chevalley’s theorem: If ${f: X \rightarrow Y}$ is a finite surjective morphism of noetherian separated schemes, with ${X}$ affine, then ${Y}$ is affine.

We will follow the outline in Hartshorne (III.3 Problems 1 & 2 and III.4 Problems 1 & 2).

Theorem 1 Let ${f: X \rightarrow Y}$ be an affine morphism of noetherian schemes. Then for any coherent sheaf ${\mathcal F}$ on ${X}$, there are natural isomorphisms for all ${i \geq 0}$,

$\displaystyle H^i(X, \mathcal F) \simeq H^i(Y, f_* \mathcal F).$

Proof: According to (II, Ex. 5.17), when ${f}$ is affine, the direct image functor ${f_*}$ induces an equivalence from the category of coherent ${\mathcal O_X}$-modules to the category of coherent ${f_*\mathcal O_X}$-modules. Moreover, an equivalence ${\tau : A \rightarrow B}$ of abelian categories (i.e. an additive functor which is also an equivalence) is exact. Therefore, if ${F: B \rightarrow \text{Ab}}$ is a left additive functor, by the uniqueness of the ${\delta}$-functor extending a given left additive functor, it follows that there exists a natural isomorphism ${R^i(F \circ \tau) \simeq R^i F \circ \tau}$ for each ${i}$. $\Box$

Theorem 2 Let ${X}$ be a noetherian scheme. Then ${X}$ is affine if and only if ${X_{\text{red}}}$ is.

Proof: Clearly ${X_\text{red}}$ is affine if ${X}$ is affine.

Conversely, suppose ${X_{\text{red}}}$ is affine. We prove that ${X}$ has cohomological dimension ${0}$, hence it is affine by Serre’s theorem (III.3.7). Let ${\mathcal F}$ be a quasi-coherent sheaf on ${X}$. As indicated in the hint, we let ${\mathcal N}$ denote the sheaf of nilpotents of ${X}$ and we consider the filtration

$\displaystyle \mathcal F \supseteq \mathcal N \cdot \mathcal F \supseteq \mathcal N^2 \cdot \mathcal F \supseteq \dots$

of ${\mathcal F}$. Since ${X}$ is noetherian, there exists an ${n>0}$ such that ${\mathcal N^n = 0}$, so the filtration is finite.

We prove by descending induction on ${j}$ that ${ \mathcal N^j \cdot \mathcal F}$ is acyclic. For ${j=n}$, it is trivial. Now consider the exact sequence of quasi-coherent sheaves on ${X}$,

$\displaystyle 0 \rightarrow \mathcal N^j \cdot \mathcal F\rightarrow \mathcal N^{j-1} \cdot \mathcal F \rightarrow (\mathcal N^{j-1} \cdot \mathcal F) / (\mathcal N^j \cdot \mathcal F) \rightarrow 0.$

The quasi-coherent sheaf ${(\mathcal N^{j-1} \cdot \mathcal F) / (\mathcal N^j \cdot \mathcal F)}$ is naturally a quasi-coherent ${\mathcal O_X / \mathcal N \simeq \mathcal O_{X_{\text{red}}}}$-module, and its cohomology can be calculated either as an ${\mathcal O_X}$-module or as an ${\mathcal O_{X_{\text{red}}}}$ module by Theorem 1 (using the fact that the reduction morphism ${X_{\text{red}} \to X}$ is affine). Therefore, it is acyclic, since ${X_{\text{red}}}$ is affine by assumption. The sheaf ${\mathcal N^j \cdot \mathcal F}$ is acyclic by the inductive hypothesis. By the long exact sequence of cohomology, we see that ${\mathcal N^{j-1} \cdot \mathcal F}$ is also acyclic. $\Box$

Theorem 3 Let ${X}$ be a reduced scheme. Then ${X}$ is affine if and only if each irreducible component of ${X}$ is affine.

Proof: The irreducible components of ${X}$ are closed subschemes of ${X}$, hence they are affine if ${X}$ is affine. Conversely, suppose that every irreducible component of ${X}$ is affine. We prove that ${X}$ has cohomological dimension ${0}$.

We proceed by induction on the number of irreducible components of ${X}$. If ${X}$ is irreducible, then the statement is vacuously true. Now suppose it holds for noetherian schemes with ${n-1}$ irreducible components. Suppose that ${X}$ has ${n}$ irreducible components, and write it as ${X=Y \cup X'}$ where ${Y}$ is irreducible. Let ${\mathcal F}$ be a quasi-coherent sheaf on ${X}$. Denote ${\tau}$ the inclusion ${Y \hookrightarrow X}$ and ${\iota}$ the inclusion ${X' \hookrightarrow X}$, where each closed subscheme is given the canonical reduced closed subscheme structure. Since ${Y}$ is Noetherian, ${ \tau_* \tau^* \mathcal F}$ is also a quasi-coherent sheaf on ${X}$, supported on ${Y}$. There is a canonical morphism ${\mathcal F \rightarrow \tau_* \tau^* \mathcal F}$, and ${\mathcal F \rightarrow \iota_* \iota^* \mathcal F }$. (Each of these two morphisms is a unit of the “inverse image – direct image” adjunction). Let

$\displaystyle g : \mathcal F \rightarrow \tau_* \tau^* \mathcal F \oplus \iota_* \iota^* \mathcal F$

be their sum. It is easy to see that this morphism is surjective, and an isomorphism away from the intersection. Let ${\mathcal G= \ker g}$. Then ${\mathcal G}$ is quasi-coherent and supported in ${Y \cap X'}$. Therefore we have an exact sequence

$\displaystyle 0 \rightarrow \mathcal G \rightarrow \mathcal F \rightarrow \tau_* \tau^* \mathcal F \oplus \iota_* \iota^* \mathcal F \rightarrow 0$

Since ${X'}$ is affine by the induction hypothesis, ${Y \cap X'}$ is affine, being a closed subscheme of an affine scheme. Now, since ${\text{Supp }\mathcal G \subseteq Y \cap X'}$, the cohomology of ${\mathcal G}$ can be calculated either as an ${\mathcal O_{(Y \cap X')}}$-module or as an ${\mathcal O_X}$-module, and therefore it vanishes. Similarily the sheaf ${\tau_* \tau^* \mathcal F \oplus \iota_* \iota^* \mathcal F}$ is acyclic because ${Y}$ and ${X'}$ are affine. Therefore, by the long exact sequence of cohomology, ${\mathcal F}$ is also acyclic. $\Box$

Lemma 4 Let ${f: X \rightarrow Y}$ be a finite surjective morphism of integral noetherian schemes. Then there is a coherent sheaf ${\mathcal M}$ on ${X}$, and a morphism of sheaves ${\alpha : \mathcal O_Y^r \rightarrow f_* \mathcal M}$ for some ${r>0}$, such that ${\alpha}$ is an isomorphism at the generic point of ${Y}$.

Proof: Let ${L}$ be the function field of ${X}$ and ${K}$ be the function field of ${Y}$. Then the morphism ${f}$ gives rise to a field homomorphism ${K \hookrightarrow L}$. Since ${f}$ is finite and surjective, ${L}$ is finite over ${K}$, say of degree ${r}$. Let ${\{x_1, \dots, x_r\}}$ be a basis for ${L}$ over ${K}$. Each ${x_j}$ can be represented as a section ${s_j}$ of ${\mathcal O_X}$ over an open set ${U_j}$. Let ${\tau_j : U_j \hookrightarrow X}$ be the inclusion. Let ${\mathcal E_j}$ be the sheaf ${\mathcal E_j = s_j \cdot \mathcal O_{U_j}}$ on ${U_j}$. Obviously ${\mathcal E_j}$ is coherent (in fact free of rank ${1}$). Let ${\mathcal F_j = (\tau_j)_*(\mathcal E_j)}$. Then ${\mathcal F_j}$ is quasi-coherent on ${X}$ since ${U_j}$ is noetherian; since ${f}$ is finite, ${\mathcal F_j}$ is in fact coherent. Let ${\mathcal M = \bigoplus_j \mathcal F_j}$. Define the morphism ${\alpha : \mathcal O^r_Y \rightarrow f_*\mathcal M}$ by the global sections ${x_j}$ of ${f_*\mathcal M}$ (using the fact that ${\mathcal O_Y}$ represents the global sections functor ${\Gamma(Y, -)}$). Then, by construction, ${\alpha}$ is an isomorphism of ${K}$-vector spaces ${K^r \cong L}$ at the generic point of ${Y}$. $\Box$

Lemma 5 Let ${f: X \rightarrow Y}$ be a finite surjective morphism of integral noetherian schemes. Then for any coherent sheaf ${\mathcal F}$ on ${Y}$, there exists a coherent sheaf ${\mathcal G}$ on ${X}$, and a a morphism ${\beta : f_* \mathcal G \rightarrow \mathcal F^r}$ which is an isomorphism at the generic point of ${Y}$.

Proof: We take ${\beta = \mathcal{H}\text{om}(\alpha, \mathcal F)}$, where ${\mathcal{H}\text{om}}$ is the sheaf ${\mathcal{H}\text{om}}$ and ${\alpha}$ is the morphism of Lemma 4:

$\displaystyle \beta: \mathcal{H}\text{om}(f_*\mathcal M, \mathcal F) \rightarrow \mathcal{H}\text{om}(\mathcal O_Y^r, \mathcal F).$

Remark that ${\mathcal{H}\text{om}(\mathcal O_Y^r, \mathcal F) \simeq \mathcal F^r}$. Moreover, the sheaf ${\mathcal{H}\text{om}(f_*\mathcal M, \mathcal F)}$ naturally has a structure of ${f_*\mathcal O_X}$-module. By (II, Ex. 5.17), when ${f}$ is an affine morphism, ${f_*}$ induces an equivalence between the category of coherent ${\mathcal O_Y}$-modules and the category of coherent ${f_*\mathcal O_X}$-modules. Therefore ${\mathcal{H}\text{om}(f_*\mathcal M, \mathcal F)}$ is isomorphic to an ${\mathcal O_Y}$-module of the form ${f_*\mathcal G}$, where ${\mathcal G}$ is a coherent ${\mathcal O_X}$-module. Thus ${\beta}$ has the form ${f_* \mathcal G \rightarrow \mathcal F^r}$.

Moreover, it follows from the fact that a coherent sheaf on a noetherian scheme is finitely presented that on such a scheme, taking sheaf ${\mathcal{H}\text{om}}$ commutes with taking stalks of morphisms; therefore ${\beta}$ is also an isomorphism at the generic point of ${Y}$. $\Box$

Now we are ready to prove Chevalley’s theorem.

Theorem 6 (Chevalley’s theorem). Let ${f: X \rightarrow Y}$ be a finite surjective morphism of noetherian separated schemes, where ${X}$ is affine. Then ${Y}$ is affine.

Proof: By Theorems 2 and 3, we may suppose that ${X}$ and ${Y}$ are reduced and irreducible. We prove by contradiction that ${Y}$ is affine. Let ${\Sigma}$ be the collection of closed subschemes of ${Y}$ which are not affine. Suppose it not empty; then it contains a minimal element ${Z \hookrightarrow X}$, 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 ${Z=Y}$ (what this means is that we are replacing ${f}$ by its restriction to ${f^{-1}(Z)}$ if necessary). Therefore, we suppose that every proper closed subscheme of ${Y}$ is affine.

Let ${\mathcal F}$ be a coherent sheaf on ${X}$. By Lemma ${5}$, there exists a coherent sheaf ${\mathcal G}$ on ${X}$ and a morphism ${\beta: f_* \mathcal G \rightarrow \mathcal F^r}$ which is generically an isomorphism (and which is therefore surjective, since ${Y}$ is irreducible). Thus, if ${\mathcal D = \ker \beta}$, we have an exact sequence of sheaves on ${Y}$

$\displaystyle 0 \rightarrow \mathcal D \rightarrow f_* \mathcal G \rightarrow \mathcal F^r \rightarrow 0.$

Now, as in the proof of Theorem 3, we view ${\mathcal D}$ as a quasi-coherent sheaf on the proper closed subscheme ${\text{Supp }\mathcal D}$. By the minimality of ${Y}$, ${\text{Supp }\mathcal D}$ is affine and therefore ${\mathcal D}$ is acyclic. Moreover, since a finite morphism is affine, we can apply Theorem 1 to see that ${f_* \mathcal G}$ is also acyclic. Therefore, by the long exact sequence of cohomology, ${\mathcal F^r}$ is acyclic, so ${\mathcal F}$ is acyclic. $\Box$

Thefore, ${Y}$ has cohomological dimension ${0}$, which contradicts the assumption that it is not affine.

# The Mayer-Vietoris sequence in sheaf cohomology

In this post, I will prove the Mayer-Vietoris Sequence for sheaf cohomology.

In what follows, ${X}$ is a topological space and ${\mathcal F, \mathcal G, \mathcal H}$ are sheaves of abelian groups on ${X}$. Let ${Z}$ be a closed subset of ${X}$. We let ${\Gamma_Z(X,\mathcal F)}$ denote the global sections of ${\mathcal F}$ with support in ${Z}$. The functor ${\Gamma_Z(X, -)}$ is a left-exact additive functor from sheaves on ${X}$ to abelian groups, and its right derived functors, denoted ${H^i_Z(X, -)}$, is the ${i}$-th cohomology of ${X}$ with support in ${Z}$. If ${\mathcal F}$ is a sheaf, the presheaf ${U \mapsto \Gamma_{Y \cap U}(U, \mathcal F)}$ is also a sheaf on ${X}$, denoted ${\mathcal H^0_Y(\mathcal F)}$ and called the “subsheaf of ${\mathcal F}$ with support in ${Y}$“.

The Mayer-Vietoris sequence, for a sheaf ${\mathcal F}$ and for a pair of closed subsets ${Y,Z \subseteq X}$, is the long exact sequence of cohomology with supports

$\displaystyle \dots \rightarrow H^i_{Y \cap Z}(X, \mathcal F) \rightarrow H^i_Y(X, \mathcal F) \oplus H^i_Z(X, \mathcal F) \rightarrow H^i_{Y \cup Z}(X, \mathcal F) \rightarrow H^{i+1}_{Y \cap Z}(X, \mathcal F) \rightarrow \dots$

We will prove the existence of this sequence in several steps.

Lemma 1 Let ${\mathcal E}$ be a flasque sheaf, ${Y}$ a closed subset of ${X}$, and ${U=X-Y}$. Then the sequence

$\displaystyle 0 \rightarrow \Gamma_Y(X, \mathcal E) \rightarrow \Gamma(X, \mathcal E) \rightarrow \Gamma(U, \mathcal E) \rightarrow 0$

is exact.

Proof: Trivial. $\Box$

Lemma 2 Let ${\mathcal E}$ be a flasque sheaf, and let ${Y, Z}$ be closed subsets of ${X}$. Then the sequence

$\displaystyle 0 \rightarrow \Gamma_{Y \cap Z}(X, \mathcal E) \rightarrow \Gamma_Y(X, \mathcal E) \oplus \Gamma_Z(X, \mathcal E) \rightarrow \Gamma_{Y \cup Z}(X, \mathcal E) \rightarrow 0$

is exact (where the first map is the diagonal embedding, and the second map is ${(s, t) \mapsto s-t}$).

Proof: Exactness is clear except possibly on the right. Let $U=X-Y, V=X-Z$, and Let ${D, E}$ be the short exact sequences

$\displaystyle 0 \rightarrow \Gamma(X, \mathcal E) \rightarrow \Gamma(X, \mathcal E) \oplus \Gamma(X, \mathcal E) \rightarrow \Gamma(X, \mathcal E) \rightarrow 0$

and

$\displaystyle 0 \rightarrow \Gamma(U \cup V, \mathcal E) \rightarrow \Gamma(U, \mathcal E) \oplus \Gamma(V, \mathcal E) \rightarrow \Gamma(U \cap V, \mathcal E) \rightarrow 0$

where the maps are defined similarily as in the statement of the Lemma. There is an obvious morphism of short exact sequences ${D \rightarrow E}$. Since ${\mathcal E}$ is flasque, this morphism is surjective onto each term of ${E}$. By the snake lemma, and using Lemma 1, we get the desired short exact sequence. $\Box$

Now we are ready to prove the existence of the Mayer-Vietoris sequence for ${\mathcal F}$. Let

$\displaystyle 0 \rightarrow \mathcal F \rightarrow \mathcal E^0 \rightarrow \mathcal E^1 \rightarrow \dots$

be a flasque resolution of ${\mathcal F}$. By the lemma, we have a short exact sequence of complexes

$\displaystyle 0 \rightarrow \Gamma_{Y \cap Z}(X, \mathcal E^\bullet) \rightarrow \Gamma_Y(X, \mathcal E^\bullet) \oplus \Gamma_Z(X, \mathcal E^\bullet) \rightarrow \Gamma_{Y \cup Z}(X, \mathcal E^\bullet) \rightarrow 0.$

The long exact sequence of cohomology associated to this short exact sequences of complexes is precisely the Mayer-Vietoris sequence.