The Stacks project

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.