A closed immersion is a morphism of finite type.
A quasi-compact open immersion is of finite type.
A composition of two morphisms of finite type is of finite type.
Morphisms of finite type are stable under base extension.
If \(X\) and \(Y\) are schemes of finite type over \(S\), then \(\fiberprod{X}{S}{Y}\) is of finite type over \(S\).
If \(X \mapsvia{f} Y \mapsvia{g} Z\) are two morphisms, and if \(f\) is quasi-compact and \(g \circ f\) is of finite type, then \(f\) is of finite type.
If \(f: X \to Y\) is a morphism of finite type and \(Y\) is noetherian, then \(X\) is noetherian.
Solution
1. A closed immersion is of finite type.
Let \[ i:Z\hookrightarrow X \] be a closed immersion and let \[ U=\Spec A\subseteq X \] be affine. By Hartshorne II.3.11(b), \[ i^{-1}(U)=\Spec(A/I) \] for some ideal \(I\subseteq A\).
The quotient \(A/I\) is generated as an \(A\)-algebra by the image of \(1\), indeed by no extra algebra generators at all. Thus it is a finitely generated \(A\)-algebra. Hartshorne II.3.3(b) shows that \(i\) is of finite type.
2. An open immersion is locally of finite type.
Let \[ j:U\hookrightarrow X \] be an open immersion and let \(u\in U\). Choose an affine neighborhood \[ V=\Spec A\subseteq X \] of \(u\). The intersection \(U\cap V\) is open in \(V\), so choose a distinguished neighborhood \[ u\in D(f)\subseteq U\cap V. \] Then \[ D(f)\cong\Spec A_f, \] and \(A_f=A[f^{-1}]\) is generated as an \(A\)-algebra by the single element \(f^{-1}\). Hence \(j\) is locally of finite type.
3. A quasi-compact open immersion is of finite type.
By 2 it is locally of finite type. It is quasi-compact by hypothesis. Hartshorne II.3.3(a) therefore implies that it is of finite type.
4. The composition of two locally finite type morphisms is locally of finite type.
Let \[ X\xrightarrow{f}Y\xrightarrow{g}Z \] be locally of finite type. Take an affine open \[ W=\Spec C\subseteq Z. \] By Hartshorne II.3.1, \(g^{-1}(W)\) has an affine cover \[ V_j=\Spec B_j \] with every \(B_j\) finitely generated over \(C\).
Again by II.3.1, each \(f^{-1}(V_j)\) has an affine cover \[ U_{jk}=\Spec A_{jk} \] with \(A_{jk}\) finitely generated over \(B_j\).
If \[ B_j=C[b_1,\ldots,b_m] \] and \[ A_{jk}=B_j[a_1,\ldots,a_n], \] then \[ A_{jk}=C[b_1,\ldots,b_m,a_1,\ldots,a_n]. \] Thus \(A_{jk}\) is finitely generated over \(C\). These affines cover \((g\circ f)^{-1}(W)\), proving local finite type.
5. The composition of two quasi-compact morphisms is quasi-compact.
Let \(W\subseteq Z\) be affine. Since \(g\) is quasi-compact, \[ g^{-1}(W) \] is quasi-compact. Cover it by finitely many affine opens \[ V_1,\ldots,V_r. \] Since \(f\) is quasi-compact, every \(f^{-1}(V_i)\) is quasi-compact. Their finite union is \[ (g\circ f)^{-1}(W), \] so this inverse image is quasi-compact. Hartshorne II.3.2 gives the assertion.
6. The composition of two finite type morphisms is of finite type.
By Hartshorne II.3.3(a), finite type means locally finite type plus quasi-compact. Apply 4 to the local finite type parts and 5 to the quasi-compact parts.
7. A base change of a locally finite type morphism is locally of finite type.
It is enough to work affinely. Suppose \[ \Spec A\longrightarrow\Spec B \] comes from a finitely generated \(B\)-algebra \(A\), and let \[ \Spec B'\longrightarrow\Spec B \] be any affine base change. Then the pullback is \[ \Spec(A\otimes_BB')\longrightarrow\Spec B'. \] If \[ A=B[a_1,\ldots,a_n], \] then \[ A\otimes_BB'=B'[a_1\otimes1,\ldots,a_n\otimes1]. \] Thus the base-changed algebra is finitely generated. Covering source and target by such affine charts gives the general statement.
8. A base change of a quasi-compact morphism is quasi-compact.
Let \(f:X\to Y\) be quasi-compact and \(Y'\to Y\) any morphism. It is enough to test the base change \[ f':X\times_YY'\longrightarrow Y' \] over an affine open \(V'\subseteq Y'\). Cover the image of \(V'\) in \(Y\) by affine opens \(V_i\) and shrink to a finite cover of \(V'\) by distinguished affine opens mapping into individual \(V_i\).
Over such an affine \(W'\to V_i\), the preimage is the base change of the quasi-compact scheme \(f^{-1}(V_i)\) along \(W'\to V_i\). Choose a finite affine cover of \(f^{-1}(V_i)\); its base changes are affine and finitely many, so the preimage over \(W'\) is quasi-compact. A finite union gives quasi-compactness over \(V'\).
9. A base change of a finite type morphism is of finite type.
Finite type is locally finite type plus quasi-compact by II.3.3(a). Apply 7 and 8.
10. If \(X\) and \(Y\) are of finite type over \(S\), then\[ X\times_SY\longrightarrow S \]is of finite type.
9. The morphism \(Y\to S\) is finite type by hypothesis. Their composition is the structure morphism\[ X\times_SY\to S, \]which is finite type by
6.
11. Suppose\[ X\xrightarrow{f}Y\xrightarrow{g}Z \]and \(g\circ f\) is locally of finite type. Then \(f\) is locally of finite type.
Fix \(x\in X\). Choose an affine open \[ W=\Spec C\subseteq Z \] containing \(g(f(x))\). Since \(g\circ f\) is locally of finite type, choose an affine neighborhood \[ U=\Spec A\subseteq(g\circ f)^{-1}(W) \] of \(x\) such that \(A\) is finitely generated over \(C\).
Choose an affine neighborhood \[ V=\Spec B\subseteq g^{-1}(W) \] of \(f(x)\). Shrink \(U\) to a distinguished affine neighborhood \[ U'=\Spec A_a\subseteq U\cap f^{-1}(V) \] of \(x\).
The ring maps factor as \[ C\longrightarrow B\longrightarrow A_a. \] Since \(A_a\) is finitely generated over \(C\), say \[ A_a=C[\alpha_1,\ldots,\alpha_n], \] the same elements also generate it over \(B\): \[ B[\alpha_1,\ldots,\alpha_n] \supseteq C[\alpha_1,\ldots,\alpha_n] =A_a, \] while the left side is a subring of \(A_a\), so equality holds. Thus \(f\) is locally of finite type at \(x\). Since \(x\) was arbitrary, \(f\) is locally of finite type.
12. If \(f\) is quasi-compact and \(g\circ f\) is of finite type, then \(f\) is of finite type.
The composite \(g\circ f\) is locally of finite type, so 11 shows that \(f\) is locally of finite type. Together with the assumed quasi-compactness of \(f\), II.3.3(a) gives that \(f\) is of finite type.
13. Let \(f:X\to Y\) be finite type and let \(Y\) be noetherian. Then \(X\) has a finite affine cover\[ X=\bigcup_{i,j}\Spec A_{ij} \]in which every \(A_{ij}\) is noetherian.
Because \(Y\) is noetherian, it is quasi-compact and has a finite affine cover \[ Y=\bigcup_{i=1}^mV_i, \qquad V_i=\Spec B_i, \] with every \(B_i\) noetherian.
By II.3.3(b), each \(f^{-1}(V_i)\) has a finite affine cover \[ \Spec A_{ij} \] with each \(A_{ij}\) finitely generated over \(B_i\). Hilbert’s basis theorem therefore implies that each \(A_{ij}\) is noetherian. Altogether these form a finite affine cover of \(X\).
14. The scheme \(X\) in 13 is noetherian.
Each affine chart \(\Spec A_{ij}\) is a noetherian topological space because \(A_{ij}\) is a noetherian ring. A finite union of noetherian open subspaces is noetherian. Thus the underlying topological space of \(X\) is noetherian, and its affine coordinate rings are noetherian. Hence \(X\) is a noetherian scheme.
15. Q.E.D.
Steps 1, 3, 6, 9, 10, 12, and 14 prove parts (a)–(g), respectively.