Proper + Affine = Finite

Another argument I really like. We first present two auxiliary lemmata.

Lemma 1. Let $f \colon A \ra B$ be an injective ring homomorphism such that the associated map $\phi \colon \Spec B \ra \Spec A$ is closed. Then $f^{-1}(B\x) = A\x$ .

Proof. Since $f$ is injective, the map $\phi$ is dominant. Hence, it is a closed surjective map. For such a map, a closed set $Z \sbq \Spec A$ is empty if and only if $\phi^{-1}(Z)$ is empty. Applying this to $Z = V(a)$ for $a \in A$ , we get $V(a) = \varnothing$ if and only if $V(f(a)) = \varnothing$ , i.e. $a \in A\x$ if and only if $f(a) \in B\x$ . $\qedsymbol$

Lemma 2. Let $f \colon A \ra B$ be an injective ring homomorphism such that the associated map $\phi \colon \A^1_B \ra \A^1_A$ (induced by $A[x] \ra B[x]$ ) is closed. Then $f$ is integral.

Proof. Let $b \in B$ . Consider the ideal $I = (xb - 1) \sbq B[x]$ , and let $J = I \cap A[x]$ . Note that $B[x]/I = B[\tfrac{1}{b}]$ , and $A[x]/J$ is the image $C$ of the composite map $A[x] \ra B[x] \ra B[\tfrac{1}{b}]$ (first isomorphism theorem). Write $g$ for the inclusion $C \ra B[\tfrac{1}{b}]$ . Note that the map

$\psi \colon \Spec B\left[\tfrac{1}{b}\right] \ra \Spec C$

induced by $g$ is just the restriction of $\phi$ to the closed subsets $V(I) \ra V(J)$ , hence it is a closed map. Since $g$ is injective, Lemma 1 asserts that

$g^{-1}\left(B\left[\tfrac{1}{b}\right]\x\right) = C\x.$

But $\tfrac{1}{b}$ is invertible in $B[\tfrac{1}{b}]$ (its inverse is $b$ ), and it lies in $C$ since it is the image of $x$ under $A[x] \ra B[\tfrac{1}{b}]$ . Hence, it is invertible in $C$ , i.e. $b \in C$ . That is, we can write

$b = a_0 + a_1 \left(\frac{1}{b}\right) + \ldots + a_n \left(\frac{1}{b}\right)^n$

for certain $a_0, \ldots, a_n \in A$ . Hence, $b^{n+1} - a_0 b^n - \ldots - a_n = 0$ in $B\left[\tfrac{1}{b}\right]$ . Hence, some multiple $b^m \left(b^{n+1} - a_0 b^n - \ldots - a_n\right)$ is $0$ in $B$ , proving that $b$ is integral over $A$ . $\qedsymbol$

Corollary. Let $f \colon A \ra B$ be a ring homomorphism such that the associated map $\Spec B \ra \Spec A$ is proper. Then $f$ is finite.

Proof. Let $I$ be the kernel of $f$ ; then $\Spec B \ra \Spec A$ lands in $\Spec A/I$ . Then $\Spec B \ra \Spec A/I$ is proper (Hartshorne, Corollary II.4.8(e)). Moreover, the map

$\bar{f} \colon A/I \ra B$

is injective, hence integral by Lemma 2. Since $\Spec B \ra \Spec A/I$ is proper, it is of finite type. Hence, $\bar{f}$ is integral and of finite type, hence finite. Hence so is $f$ . $\qedsymbol$

A more geometric version is the following:

Theorem. Let $\phi \colon X \ra Y$ be a morphism of schemes that is both affine and proper. Then $\phi$ is finite.

Proof. Let $U = \Spec A$ be an affine open in $Y$ . Then $V = \phi^{-1}(U)$ is affine (since $\phi$ is an affine morphism); say $V = \Spec B$ . Then the restriction $\phi|_V \colon V \ra U$ is proper (properness is local on the target), hence $B$ is finite over $A$ by the theorem above. This proves that $\phi$ is finite. $\qedsymbol$

Remark. The converse is also true, and in fact much easier: a finite morphism is affine by definition, and proper since in the affine case it is in fact projective. The question whether finite morphisms are always projective turns out to be somewhat subtle: it is true in the EGA sense of projective (it is given by the relative Proj of some sheaf of graded algebras, i.e. it embeds into some projective bundle), but not in the sense of Hartshorne (it embeds into $\P^n_Y$ for some $n$ , i.e. it embeds into a trivial projective bundle).

I believe that the following is an example of the last statement (i.e. a finite morphism which is not H-projective). Let $K$ be a field of infinite degree of imperfection, i.e. $\char K = p$ and $K$ is an infinite extension of $K^p$ . Then Theorem 2 in [B-McL] says that for each $n \in \N$ there exists a finite field extension $L_n$ of $K$ which cannot be generated by fewer than $n$ elements¹. Correspondingly, $\Spec L_n \ra \Spec K$ cannot be embedded into $\P^m_K$ for $m < n$ . Then consider $Y$ an infinite disjoint union of $\Spec K$ (labelled by $\N$ ), and $X$ the disjoint union of $\Spec L_n$ for all $n$ . Then $Y$ is certainly projective in EGA’s sense (although any projective bundle it embeds into cannot have constant dimension). Yet it cannot be embedded into any $\P^n_X$ .

References.
[B-McL] M.F. Becker, S. MacLane, The minimum number of generators for inseparable algebraic extensions. Bull. Amer. Math. Soc. Volume 46-2 (1940), p. 182-186.

¹It is probably not very hard to actually come up with an example of such a field $K$ with extensions $L_n$ . I think that $K = \F_p(x_1, x_2, \ldots)$ and $L_n = K(\sqrt[\uproot{3}p]{x_1}, \ldots, \sqrt[\uproot{3}p]{x_n})$ should do the trick.

I think the following is a counterexample. Let $Y$ be an infinite chain of $\mathbf P^1$ s over a finite field $k$ , with irreducible components $Y_0, Y_1, \ldots$ . Say that the point $\infty$ in $Y_i$ is attached to the point $0$ in $Y_{i+1}$ . For notational reasons, it is easier to remove $0$ from $Y_0$ , so let’s do so. I will define $X$ as another tree of $\mathbf P^1$ s based on some local model.

For a subset $J \subseteq \mathbf N$ , write $Y_J$ for the open subset $Y \setminus \bigcup_{i \not\in J} Y_i$ . Note that $Y_{\{i\}}$ is not the irreducible component $Y_i$ , but rather the open subset $\mathbf P^1 \setminus \{0,\infty\}$ of $Y_i = \mathbf P^1$ . Moreover, $Y$ is the union of the opens $Y_{i,i+1} := Y_{\{i,i+1\}}$ .

For the local model, note that $Y_{i,i+1}$ is isomorphic to $S = \operatorname{Spec} k[x,y]/(xy)$ , where the $x$ -axis $V(y)$ corresponds to $Y_i$ and the $y$ -axis $V(x)$ to $Y_{i+1}$ . Write $T = \operatorname{Spec} k[x,y,z]/(xy,xz,yz)$ , the union of the three axes in $\mathbf A^3$ . The map $\mathbf A^3 \to \mathbf A^2$ given by $(x,y,z) \mapsto (x,y+z)$ induces a finite map $T \to S$ . Note that over $Y_{\{i\}}$ , this is an isomorphism, whereas over $Y_{\{i+1\}}$ it is two copies of $Y_{\{i+1\}}$ .

Now to construct $X$ , let $X_{i,i+1}$ be the disjoint union of $2^i$ copies of $T$ , with its natural map to $S = Y_{i,i+1}$ . Above $Y_{\{i\}}$ , this is $2^i$ copies of $Y_{\{i\}}$ , and above $Y_{\{i+1\}}$ it is $2^{i+1}$ copies of $Y_{\{i+1\}}$ . This gives the glueing maps between $X_{i,i+1}$ and $X_{i+1,i+2}$ , and we let $X$ be the union.

Then $X \to Y$ is finite since this is a local condition. But it cannot be embedded into $\mathbf P^n_Y$ for any $n$ , since $\mathbf P^n(k)$ is finite and the fibres of $X_{\{i\}} \to Y_{\{i\}}$ get arbitrarily many $k$ -points. $\qedsymbol$

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3 thoughts on “Proper + Affine = Finite”

Anonymous on 17 May 2024 at 02:25 said:

But is there a connected counterexample of the last statement?

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- Remy on 17 May 2024 at 09:06 said:
  
  I think the following is a counterexample. Let $Y$ be an infinite chain of $\mathbf P^1$ s over a finite field $k$ , with irreducible components $Y_0, Y_1, \ldots$ . Say that the point $\infty$ in $Y_i$ is attached to the point $0$ in $Y_{i+1}$ . For notational reasons, it is easier to remove $0$ from $Y_0$ , so let’s do so. I will define $X$ as another tree of $\mathbf P^1$ s based on some local model.
  
  For a subset $J \subseteq \mathbf N$ , write $Y_J$ for the open subset $Y \setminus \bigcup_{i \not\in J} Y_i$ . Note that $Y_{\{i\}}$ is not the irreducible component $Y_i$ , but rather the open subset $\mathbf P^1 \setminus \{0,\infty\}$ of $Y_i = \mathbf P^1$ . Moreover, $Y$ is the union of the opens $Y_{i,i+1} := Y_{\{i,i+1\}}$ .
  
  For the local model, note that $Y_{i,i+1}$ is isomorphic to $S = \operatorname{Spec} k[x,y]/(xy)$ , where the $x$ -axis $V(y)$ corresponds to $Y_i$ and the $y$ -axis $V(x)$ to $Y_{i+1}$ . Write $T = \operatorname{Spec} k[x,y,z]/(xy,xz,yz)$ , the union of the three axes in $\mathbf A^3$ . The map $\mathbf A^3 \to \mathbf A^2$ given by $(x,y,z) \mapsto (x,y+z)$ induces a finite map $T \to S$ . Note that over $Y_{\{i\}}$ , this is an isomorphism, whereas over $Y_{\{i+1\}}$ it is two copies of $Y_{\{i+1\}}$ .
  
  Now to construct $X$ , let $X_{i,i+1}$ be the disjoint union of $2^i$ copies of $T$ , with its natural map to $S = Y_{i,i+1}$ . Above $Y_{\{i\}}$ , this is $2^i$ copies of $Y_{\{i\}}$ , and above $Y_{\{i+1\}}$ it is $2^{i+1}$ copies of $Y_{\{i+1\}}$ . This gives the glueing maps between $X_{i,i+1}$ and $X_{i+1,i+2}$ , and we let $X$ be the union.
  
  Then $X \to Y$ is finite since this is a local condition. But it cannot be embedded into $\mathbf P^n_Y$ for any $n$ , since $\mathbf P^n(k)$ is finite and the fibres of $X_{\{i\}} \to Y_{\{i\}}$ get arbitrarily many $k$ -points. $\qedsymbol$
  
  Reply ↓
- Remy on 17 May 2024 at 09:19 said:
  
  That said, in cases of interest this is of course true. For instance, if $Y$ is quasi-projective over an affine scheme, then the two notions of projectivity agree; see [Tag 087S].
  
  If $Y$ has the resolution property [Tag 0F85], then every quasi-coherent sheaf of finite type is a quotient of a locally free sheaf, so then at least you can always embed $X$ into $\mathbf P(\mathscr E)$ for some locally free sheaf $\mathscr E$ . If $Y$ is connected, then $\mathscr E$ has constant rank.
  
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Lovely little lemmas

Or maybe ‘amusing observations’

3 thoughts on “Proper + Affine = Finite”

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