## 59.41 Push and pull

Let $f : X \to Y$ be a morphism of schemes. Here is a list of conditions we will consider in the following:

1. For every étale morphism $U \to X$ and $u \in U$ there exist an étale morphism $V \to Y$ and a disjoint union decomposition $X \times _ Y V = W \amalg W'$ and a morphism $h : W \to U$ over $X$ with $u$ in the image of $h$.

2. For every $V \to Y$ étale, and every étale covering $\{ U_ i \to X \times _ Y V\}$ there exists an étale covering $\{ V_ j \to V\}$ such that for each $j$ we have $X \times _ Y V_ j = \coprod W_{ij}$ where $W_{ij} \to X \times _ Y V$ factors through $U_ i \to X \times _ Y V$ for some $i$.

3. For every $U \to X$ étale, there exists a $V \to Y$ étale and a surjective morphism $X \times _ Y V \to U$ over $X$.

It turns out that each of these properties has meaning in terms of the behaviour of the functor $f_{small, *}$. We will work this out in the next few sections.

Comment #238 by Keenan Kidwell on

It seems based on the next few tags that the three conditions should be labeled A, B, and C instead of 1, 2, and 3.

Comment #239 by on

This is actually a known issue, and is related only to the translation from TeX to HTML. The pdf version renders this in the correct way, so the Stacks project itself is not to blame, it's the website. Given that someone noticed I should try to fix this :).

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