# etale localization reference request

I'm looking for a reference for the following statement:

Let $P$ be a property of morphisms of schemes local on the target in the etale topology. Let $f : X\rightarrow Y$ be a morphism of schemes which is locally of finite presentation, and such that for all points $y\in Y$, the restriction of $f$ to the strict henselian local ring $\mathcal{O}_{Y,y}^{sh}$ has property $P$. Then $f$ has property $P$.

(Is the LoFP condition necessary?)

This seems to follow from various results in the stacks project, but I can't find it explicitly stated anywhere except for the specific case where P = "flat" (tag 05VL).

I need this result for an appendix in my thesis. Currently I've stitched a proof together using various results of the stacks project, but it seems more elegant to cite a reference that proves this explicitly if possible.

• You could contribute it to the stacks project. Mar 22, 2016 at 6:32
• There exist people who do not like the stacks project as a reference because it has not been refereed. I am not suggesting that I am one of those people, but refereeing is part of the academic process -- although probably far more so in other sciences. The argument is that "I am written by a smart guy" should not be enough -- look at the ABC fiasco for example.
– znt
Mar 22, 2016 at 7:53

## 1 Answer

That is not true without further hypotheses, but it is true with one additional hypothesis. First, here is a counterexample. Let $P$ be the property that the morphism $f$ is quasi-compact. This property is local for the étale topology, and even for the fpqc topology, cf. http://stacks.math.columbia.edu/tag/02KQ for instance.

Let $Y$ be $\text{Spec}\ \mathbb{Z}$, and let $X$ be the disjoint union over all primes $p$ of $\text{Spec}\ \mathbb{Z}/p\mathbb{Z}$. There is a unique morphism $f:X\to Y$. This morphism is not quasi-compact since $X$ is not quasi-compact. The morphism $f$ is locally of finite presentation since every point of $X$ has an open neighborhood that is isomorphic to $\text{Spec}\ \mathbb{Z}/p\mathbb{Z}$ for some prime $p$.

On the other hand, for every closed point $y=\langle p\rangle$ of $Y$ the base change of $f$ over the local ring $\mathcal{O}_{Y,y} = \mathbb{Z}_{\langle p \rangle}$ is the quasi-compact, even affine, morphism corresponding to the ring homomorphism $\mathbb{Z}_{\langle p \rangle} \to \mathbb{Z}_{\langle p \rangle}/p\mathbb{Z}_{\langle p \rangle}$. Thus the base change of $f$ to every strict Henselization is also quasi-compact.

The simplest additional hypothesis guaranteeing your result is "compatible with filtered limits of schemes / filtered colimits of rings": for every affine scheme $Y=\text{Spec}\ A$ that is a filtered colimit $A = \varinjlim A_\lambda$ of rings $(A_\lambda)_{\lambda\in I}$, for every compatible family of morphisms $$f_\lambda:X_\lambda \to \text{Spec}\ A_\lambda, \ \ \phi_{\mu,\lambda}:X_\mu \xrightarrow{\cong} X_{\lambda}\otimes_{A_\lambda} A_\mu,$$ with filtered limit $f:X\to \text{Spec}\ A$, then $f$ has property $P$ if and only if there exists some $\lambda$ in $I$ such that for all $\mu>\lambda$, $f_\mu$ has property $P$. There are many examples of such properties in EGA $\textrm{IV}_3$, Section 8.10, pp. 36-41. (Please note, in EGA, there is a standing hypothesis that every $f_\lambda$ is finitely presented, i.e., locally finitely presented and quasi-compact. Also I have stated "compatible with limits" slightly incorrectly because I am only considering the case that $Y$ is affine. The correct formulation is in EGA. If your property is étale local on the base, then you can always reduce to the affine case.)

Since the strict henselization $A$ of a Noetherian ring $B$ at a prime $\mathfrak{p}$ is a filtered colimit of finitely presented, étale $B$-algebras, if your property is compatible with limits and is also étale local, then it holds on $\text{Spec} B$ if and only if it holds after base change to each strict Henselization $\text{Spec}\ B_{\mathfrak{p}}^{sh}$.

• The definition of "finitely presented" requires "quasi-separated" too (as J.S. knows). Also, it is worth noting that EGA has a many more instances of such properties P later on, such as flatness (11.2.6), smooth, etale, unramified (IV$_4$, 17.7.8), and differential smoothness (17.12.6); overall EGA remains the best source on such limit results. But for the original question with P = "flat" none of this discussion (nor LFP) is needed, since the OP's hypotheses (with all $y\in Y$) makes it an elementary exercise with flatness because each $O_y \rightarrow O_y^{\rm{sh}}$ is faithfully flat. Mar 22, 2016 at 14:57
• The stacks project so far has P = "affine", "finite", "unramified", "closed immersion", "separated", "flat", "finite locally free", "smooth", "etale", "isomorphism", "monomorphism", "surjective", "syntomic", "proper", "quasi-finite", and "at-worst-nodal of relative dimension 1". Most can be found in stacks.math.columbia.edu/tag/081C and it is usually straighforward to add new ones. Mar 22, 2016 at 16:00
• @CountDracula: Is the proof for flatness in the Stacks Project simpler than (or much different from) the one in EGA? (There are very many back-references to chase down in the SP version, so I can't tell at a glance how it compares with Raynaud's proof in EGA.) Mar 22, 2016 at 16:08
• @nfdc23 I think it is about the same complexity and essentially the same. I think in both the key ingredient is openness of flatness for fp ring map, but I read the proof in EGA a really long time ago so I am not completely sure. Still the graph stacks.math.columbia.edu/tag/02JO/graph/force of logical implication does not look too big! Mar 22, 2016 at 16:54
• @wyc: No, you definitely need a finite presentation assumption to spread-out from local rings to ambient affine base schemes, for example; it is not just a device to bootstrap beyond the affine case. Jason Starr gives you an example of failure of spreading-out from stalks on $Y$ with $Y$ affine. Mar 22, 2016 at 23:46