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.