The Stacks project

Lemma 3.9.8. Let $f : X \to Y$ be a monomorphism of schemes. If at least one of the following properties holds, then $\text{size}(X) \leq \text{size}(Y)$:

  1. $f$ is quasi-compact,

  2. $f$ is locally of finite presentation,

  3. add more here as needed.

But the bound does not hold for monomorphisms which are locally of finite type.

Proof. Let $Y = \bigcup _{j \in J} V_ j$ be an affine open covering of $Y$ with $|J| \leq \text{size}(Y)$. By Lemma 3.9.5 it suffices to bound the size of the inverse image of $V_ j$ in $X$. Hence we reduce to the case that $Y$ is affine, say $Y = \mathop{\mathrm{Spec}}(B)$. For any affine open $\mathop{\mathrm{Spec}}(A) \subset X$ we have $|A| \leq \max (|B|, \aleph _0) = \text{size}(Y)$, see remark above and Lemma 3.9.4. Thus it suffices to show that $X$ has at most $\text{size}(Y)$ affine opens. This is clear if $X$ is quasi-compact, whence case (1) holds. In case (2) the number of isomorphism classes of $B$-algebras $A$ that can occur is bounded by $\text{size}(B)$, because each $A$ is of finite type over $B$, hence isomorphic to an algebra $B[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$ for some $n, m$, and $f_ j \in B[x_1, \ldots , x_ n]$. However, as $X \to Y$ is a monomorphism, there is a unique morphism $\mathop{\mathrm{Spec}}(A) \to X$ over $Y = \mathop{\mathrm{Spec}}(B)$ if there is one, hence the number of affine opens of $X$ is bounded by the number of these isomorphism classes.

To prove the final statement of the lemma consider the ring $B = \prod _{n \in \mathbf{N}} \mathbf{F}_2$ and set $Y = \mathop{\mathrm{Spec}}(B)$. For every ultrafilter $\mathcal{U}$ on $\mathbf{N}$ we obtain a maximal ideal $\mathfrak m_\mathcal {U}$ with residue field $\mathbf{F}_2$; the map $B \to \mathbf{F}_2$ sends the element $(x_ n)$ to $\mathop{\mathrm{lim}}\nolimits _\mathcal {U} x_ n$. Details omitted. The morphism of schemes $X = \coprod _\mathcal {U} \mathop{\mathrm{Spec}}(\mathbf{F}_2) \to Y$ is a monomorphism as all the points are distinct. However the cardinality of the set of affine open subschemes of $X$ is equal to the cardinality of the set of ultrafilters on $\mathbf{N}$ which is $2^{2^{\aleph _0}}$. We conclude as $|B| = 2^{\aleph _0} < 2^{2^{\aleph _0}}$. $\square$

Comments (2)

Comment #610 by Wei Xu on

I am sorry. I donot understand, isn't the example for the proof of the final statement a counterexample of (2)?

From the proof of (2), (2) should be locally of finite presentation.

Comment #615 by on

Thank you, thank you, thank you! Hopefully I got it right this time. You can read the change to the tex files here.

There are also:

  • 3 comment(s) on Section 3.9: Constructing categories of schemes

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