Prove Chevalley’s theorem: Let \(f: X \to Y\) be a finite surjective morphism of noetherian separated schemes, with \(X\) affine. Then \(Y\) is affine.
Let \(f: X \to Y\) be a finite surjective morphism of integral noetherian schemes. Show that there is a coherent sheaf \(\mcm\) on \(X\), and a morphism of sheaves \(\alpha: \mco_Y^r \to f_* \mcm\) for some \(r>0\), such that \(\alpha\) is an isomorphism at the generic point of \(Y\).
With \(f\) as in (a), for any coherent sheaf \(\mcf\) on \(Y\), show that there is a coherent sheaf \(\mcg\) on \(X\), and a morphism \(\beta: f_* \mcg \to \mcf^r\) which is an isomorphism at the generic point of \(Y\).
Now prove Chevalley’s theorem. First use (Ex. 3.1) and (Ex. 3.2) to reduce to the case \(X\) and \(Y\) integral. Then use (3.7), (Ex. 4.1), consider \(\operatorname{ker}\beta\) and \(\coker\beta\), and use noetherian induction on \(Y\).
Hint
For (b), apply \(\sheafhom(\wait,\mcf)\) to \(\alpha\) and use (II, Ex. 5.17e).
Solution
In (a) and (b), put \(K=K(Y)\) and \(L=K(X)\), and let \(\eta\) be the generic point of \(Y\). The finite surjective morphism gives an inclusion \(K\hookrightarrow L\) and a finite field extension of degree \(r=[L:K]>0\). All occurrences of a power of a sheaf in the maps \(\alpha\) and \(\beta\) mean finite direct sums.
1. A \(K\)-basis \(b_1,\ldots,b_r\) of \(L\) gives the coherent sheaf and morphism required in (a).
Let \(\mathcal K_X\) be the sheaf of rational functions on the integral scheme \(X\). It is quasi-coherent: on \(\Spec B\subseteq X\) it is the associated sheaf of the \(B\)-module \(\operatorname{Frac}B=L\), whose localizations at nonzero elements are again \(L\). Each \(b_i\) is a global section of this sheaf. Define \(\mcm\) to be the \(\OO_X\)-submodule generated by these finitely many sections. It is a quasi-coherent image of \(\OO_X^{\oplus r}\) and is of finite type, hence is coherent on the noetherian scheme.
The sections \(b_i\in\Gamma(X,\mcm)=\Gamma(Y,f_*\mcm)\) define \[ \alpha:\OO_Y^{\oplus r}\longrightarrow f_*\mcm, \qquad(a_1,\ldots,a_r)\longmapsto\sum_i a_i b_i. \] At \(\eta\), the inverse image is \(\Spec L\). Indeed, over an affine open \(\Spec A\subseteq Y\), the finite inverse image has coordinate ring a domain \(B\) finite over \(A\); its localization \(B\otimes_A K\) is a finite-dimensional \(K\)-domain, hence a field, and is \(L\). The sheaf \(\mcm\) on this fibre is \(L\), since its nonzero generators span it as an \(L\)-module. Consequently \(\alpha_\eta\) is the basis isomorphism \(K^{\oplus r}\to L\). This proves (a), without requiring \(L/K\) to be separable.
2. The coherent sheaf\[ \mathcal H=\sheafhom_{\OO_Y}(f_*\mcm,\mcf) \]is the direct image of a coherent sheaf \(\mcg\) on \(X\).
Finite pushforward preserves coherence on noetherian schemes by P-AGH255PUSHCOH, part (c). Thus \(f_*\mcm\) and \(\mcf\) are coherent. Their sheaf Hom is coherent: locally a finite presentation of the first module identifies Hom with the kernel of a map between two finite direct sums of the second module. This description also proves compatibility with localization.
Put \(\mathcal B=f_*\OO_X\). The sheaf \(f_*\mcm\) is a \(\mathcal B\)-module, so give \(\mathcal H\) the action \[ (b\cdot h)(m)=h(bm). \] Commutativity of \(\mathcal B\) makes this a left \(\mathcal B\)-module structure compatible with its \(\OO_Y\)-module structure. The affine-morphism equivalence of P-AGH2517AFFMOR, part (e), therefore gives a quasi-coherent \(\mcg\) on \(X\) with \(f_*\mcg\cong\mathcal H\).
To see coherence, work over \(\Spec A\subseteq Y\), with finite inverse image \(\Spec B\). The module corresponding to \(\mathcal H\) is finite over \(A\) by its coherence. Its finite \(A\)-generating set also generates it as a \(B\)-module, since the \(B\)-action extends the \(A\)-action. The ring \(B\) is noetherian, so the corresponding sheaf on \(\Spec B\) is coherent. These opens cover \(X\), proving coherence of \(\mcg\).
3. Precomposition with \(\alpha\) gives the morphism in (b), which is an isomorphism at \(\eta\).
1.It is therefore an isomorphism there for every finite-dimensional \(K\)-vector space \(\mcf_\eta\), including zero.This proves (b).
4. In the setting of the theorem, suppose in addition that \(X,Y\) are integral and every proper reduced closed subscheme of \(Y\) is affine.Then \(Y\) is affine.
For any coherent \(\mcf\) on \(Y\), take \(\beta\) from step 3 and set \[ \mathcal K=\ker\beta,\qquad\mathcal C=\coker\beta, \qquad\mathcal Q=\im\beta. \] These are coherent. The first two have proper closed support, since their generic stalks are zero. Each is a coherent sheaf on the closed subscheme defined by its annihilator. That subscheme has affine reduction by the hypothesis on proper reduced closed subschemes, hence is itself affine by P-AGH331REDAFFINE. The case of a zero sheaf uses the empty affine subscheme. Closed pushforward preserves cohomology [@Har10a, Lemma III.2.10], so affine vanishing gives \[ H^i(Y,\mathcal K)=H^i(Y,\mathcal C)=0\qquad(i>0). \]
Since \(f\) is affine and \(X,Y\) are noetherian and separated, P-AGH341AFFINEMORPH gives \[ H^i(Y,f_*\mcg)\cong H^i(X,\mcg)=0\qquad(i>0). \] The exact sequence \(0\to\mathcal K\to f_*\mcg\to\mathcal Q\to0\) now gives \(H^1(Y,\mathcal Q)=0\), using \(H^1(Y,f_*\mcg)=0\) and \(H^2(Y,\mathcal K)=0\). The second exact sequence \[ 0\longrightarrow\mathcal Q\longrightarrow\mcf^{\oplus r} \longrightarrow\mathcal C\longrightarrow0 \] then gives \(H^1(Y,\mcf^{\oplus r})=0\). Cohomology commutes with finite direct sums, and \(r>0\), so \(H^1(Y,\mcf)=0\). This holds for every coherent sheaf, in particular every coherent ideal sheaf. Serre’s criterion T-5IOUR proves that \(Y\) is affine.
5. Noetherian induction proves (c) for the original finite surjective morphism.
Fix the morphism in the theorem, and induct on closed subsets \(T\subseteq Y\) to prove that their reduced induced subschemes \(T_{\mathrm{red}}\) are affine. The empty subset is affine. Suppose the assertion holds for every proper closed subset of a nonempty \(T\). The base change \[ X_T=X\times_Y T_{\mathrm{red}}\longrightarrow T_{\mathrm{red}} \] is finite and surjective, and \(X_T\) is affine because it is a closed subscheme of the affine scheme \(X\). Its reduction is still affine and finite surjective over \(T_{\mathrm{red}}\).
If \(T\) is reducible, each of its finitely many irreducible components is a proper closed subset, and its reduced subscheme is affine by induction. Then P-AGH332COMPONENTAFFINE gives affineness of \(T_{\mathrm{red}}\). If \(T\) is irreducible, \(T_{\mathrm{red}}\) is integral. The finitely many irreducible components of \((X_T)_{\mathrm{red}}\) have closed images under the finite morphism, and those images cover \(T\). Irreducibility forces one image to be all of \(T\). Choose that component with its reduced structure; it is integral and affine, being closed in \(X_T\), and maps finite surjectively to \(T_{\mathrm{red}}\). Every proper reduced closed subscheme of \(T_{\mathrm{red}}\) is affine by induction. Step 4 applies to this component and proves affineness of \(T_{\mathrm{red}}\).
Thus the induction closes on the noetherian space \(Y\). Taking \(T=Y\) proves that \(Y_{\mathrm{red}}\) is affine, and P-AGH331REDAFFINE finally gives affineness of \(Y\). This retains any nonreduced structure in the original schemes and in the supports used in step 4.
6. Q.E.D.
Step 1 proves (a), steps <1>2–<1>3 prove (b), and steps <1>4–<1>5 prove (c) and the theorem.