The Stacks project

35.12 Local properties of schemes

It often happens one can prove the members of a covering of a scheme have a certain property. In many cases this implies the scheme has the property too. For example, if $S$ is a scheme, and $f : S' \to S$ is a surjective flat morphism such that $S'$ is a reduced scheme, then $S$ is reduced. You can prove this by looking at local rings and using Algebra, Lemma 10.164.2. We say that the property of being reduced descends through flat surjective morphisms. Some results of this type are collected in Algebra, Section 10.164 and for schemes in Section 35.16. Some analogous results on descending properties of morphisms are in Section 35.11.

On the other hand, there are examples of surjective flat morphisms $f : S' \to S$ with $S$ reduced and $S'$ not, for example the morphism $\mathop{\mathrm{Spec}}(k[x]/(x^2)) \to \mathop{\mathrm{Spec}}(k)$. Hence the property of being reduced does not ascend along flat morphisms. Having infinite residue fields is a property which does ascend along flat morphisms (but does not descend along surjective flat morphisms of course). Some results of this type are collected in Algebra, Section 10.163.

Finally, we say that a property is local for the flat topology if it ascends along flat morphisms and descends along flat surjective morphisms. A somewhat silly example is the property of having residue fields of a given characteristic. To be more precise, and to tie this in with the various topologies on schemes, we make the following formal definition.

Definition 35.12.1. Let $\mathcal{P}$ be a property of schemes. Let $\tau \in \{ fpqc, \linebreak[0] fppf, \linebreak[0] syntomic, \linebreak[0] smooth, \linebreak[0] {\acute{e}tale}, \linebreak[0] Zariski\} $. We say $\mathcal{P}$ is local in the $\tau $-topology if for any $\tau $-covering $\{ S_ i \to S\} _{i \in I}$ (see Topologies, Section 34.2) we have

\[ S \text{ has }\mathcal{P} \Leftrightarrow \text{each }S_ i \text{ has }\mathcal{P}. \]

To be sure, since isomorphisms are always coverings we see (or require) that property $\mathcal{P}$ holds for $S$ if and only if it holds for any scheme $S'$ isomorphic to $S$. In fact, if $\tau = fpqc, \linebreak[0] fppf, \linebreak[0] syntomic, \linebreak[0] smooth, \linebreak[0] {\acute{e}tale}$, or $Zariski$, then if $S$ has $\mathcal{P}$ and $S' \to S$ is flat, flat and locally of finite presentation, syntomic, smooth, étale, or an open immersion, then $S'$ has $\mathcal{P}$. This is true because we can always extend $\{ S' \to S\} $ to a $\tau $-covering.

We have the following implications: $\mathcal{P}$ is local in the fpqc topology $\Rightarrow $ $\mathcal{P}$ is local in the fppf topology $\Rightarrow $ $\mathcal{P}$ is local in the syntomic topology $\Rightarrow $ $\mathcal{P}$ is local in the smooth topology $\Rightarrow $ $\mathcal{P}$ is local in the étale topology $\Rightarrow $ $\mathcal{P}$ is local in the Zariski topology. This follows from Topologies, Lemmas 34.4.2, 34.5.2, 34.6.2, 34.7.2, and 34.9.6.

Lemma 35.12.2. Let $\mathcal{P}$ be a property of schemes. Let $\tau \in \{ fpqc, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $. Assume that

  1. the property is local in the Zariski topology,

  2. for any morphism of affine schemes $S' \to S$ which is flat, flat of finite presentation, étale, smooth or syntomic depending on whether $\tau $ is fpqc, fppf, étale, smooth, or syntomic, property $\mathcal{P}$ holds for $S'$ if property $\mathcal{P}$ holds for $S$, and

  3. for any surjective morphism of affine schemes $S' \to S$ which is flat, flat of finite presentation, étale, smooth or syntomic depending on whether $\tau $ is fpqc, fppf, étale, smooth, or syntomic, property $\mathcal{P}$ holds for $S$ if property $\mathcal{P}$ holds for $S'$.

Then $\mathcal{P}$ is $\tau $ local on the base.

Remark 35.12.3. In Lemma 35.12.2 above if $\tau = smooth$ then in condition (3) we may assume that the morphism is a (surjective) standard smooth morphism. Similarly, when $\tau = syntomic$ or $\tau = {\acute{e}tale}$.


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