## 59.72 Auxiliary lemmas on morphisms

Some lemmas that are useful for proving functoriality properties of constructible sheaves.

Lemma 59.72.1. Let $U \to X$ be an étale morphism of quasi-compact and quasi-separated schemes (for example an étale morphism of Noetherian schemes). Then there exists a partition $X = \coprod _ i X_ i$ by constructible locally closed subschemes such that $X_ i \times _ X U \to X_ i$ is finite étale for all $i$.

Proof. If $U \to X$ is separated, then this is More on Morphisms, Lemma 37.45.4. In general, we may assume $X$ is affine. Choose a finite affine open covering $U = \bigcup U_ j$. Apply the previous case to all the morphisms $U_ j \to X$ and $U_ j \cap U_{j'} \to X$ and choose a common refinement $X = \coprod X_ i$ of the resulting partitions. After refining the partition further we may assume $X_ i$ affine as well. Fix $i$ and set $V = U \times _ X X_ i$. The morphisms $V_ j = U_ j \times _ X X_ i \to X_ i$ and $V_{jj'} = (U_ j \cap U_{j'}) \times _ X X_ i \to X_ i$ are finite étale. Hence $V_ j$ and $V_{jj'}$ are affine schemes and $V_{jj'} \subset V_ j$ is closed as well as open (since $V_{jj'} \to X_ i$ is proper, so Morphisms, Lemma 29.41.7 applies). Then $V = \bigcup V_ j$ is separated because $\mathcal{O}(V_ j) \to \mathcal{O}(V_{jj'})$ is surjective, see Schemes, Lemma 26.21.7. Thus the previous case applies to $V \to X_ i$ and we can further refine the partition if needed (it actually isn't but we don't need this). $\square$

In the Noetherian case one can prove the preceding lemma by Noetherian induction and the following amusing lemma.

Lemma 59.72.2. Let $f: X \to Y$ be a morphism of schemes which is quasi-compact, quasi-separated, and locally of finite type. If $\eta$ is a generic point of an irreducible component of $Y$ such that $f^{-1}(\eta )$ is finite, then there exists an open $V \subset Y$ containing $\eta$ such that $f^{-1}(V) \to V$ is finite.

Proof. This is Morphisms, Lemma 29.51.1. $\square$

The statement of the following lemma can be strengthened a bit.

Lemma 59.72.3. Let $f : Y \to X$ be a quasi-finite and finitely presented morphism of affine schemes.

1. There exists a surjective morphism of affine schemes $X' \to X$ and a closed subscheme $Z' \subset Y' = X' \times _ X Y$ such that

1. $Z' \subset Y'$ is a thickening, and

2. $Z' \to X'$ is a finite étale morphism.

2. There exists a finite partition $X = \coprod X_ i$ by locally closed, constructible, affine strata, and surjective finite locally free morphisms $X'_ i \to X_ i$ such that the reduction of $Y'_ i = X'_ i \times _ X Y \to X'_ i$ is isomorphic to $\coprod _{j = 1}^{n_ i} (X'_ i)_{red} \to (X'_ i)_{red}$ for some $n_ i$.

Proof. Setting $X' = \coprod X'_ i$ we see that (2) implies (1). Write $X = \mathop{\mathrm{Spec}}(A)$ and $Y = \mathop{\mathrm{Spec}}(B)$. Write $A$ as a filtered colimit of finite type $\mathbf{Z}$-algebras $A_ i$. Since $B$ is an $A$-algebra of finite presentation, we see that there exists $0 \in I$ and a finite type ring map $A_0 \to B_0$ such that $B = \mathop{\mathrm{colim}}\nolimits B_ i$ with $B_ i = A_ i \otimes _{A_0} B_0$, see Algebra, Lemma 10.127.8. For $i$ sufficiently large we see that $A_ i \to B_ i$ is quasi-finite, see Limits, Lemma 32.18.2. Thus we reduce to the case of finite type algebras over $\mathbf{Z}$, in particular we reduce to the Noetherian case. (Details omitted.)

Assume $X$ and $Y$ Noetherian. In this case any locally closed subset of $X$ is constructible. By Lemma 59.72.2 and Noetherian induction we see that there is a finite partition $X = \coprod X_ i$ of $X$ by locally closed strata such that $Y \times _ X X_ i \to X_ i$ is finite. We can refine this partition to get affine strata. Thus after replacing $X$ by $X' = \coprod X_ i$ we may assume $Y \to X$ is finite.

Assume $X$ and $Y$ Noetherian and $Y \to X$ finite. Suppose that we can prove (2) after base change by a surjective, flat, quasi-finite morphism $U \to X$. Thus we have a partition $U = \coprod U_ i$ and finite locally free morphisms $U'_ i \to U_ i$ such that $U'_ i \times _ X Y \to U'_ i$ is isomorphic to $\coprod _{j = 1}^{n_ i} (U'_ i)_{red} \to (U'_ i)_{red}$ for some $n_ i$. Then, by the argument in the previous paragraph, we can find a partition $X = \coprod X_ j$ with locally closed affine strata such that $X_ j \times _ X U_ i \to X_ j$ is finite for all $i, j$. By Morphisms, Lemma 29.48.2 each $X_ j \times _ X U_ i \to X_ j$ is finite locally free. Hence $X_ j \times _ X U'_ i \to X_ j$ is finite locally free (Morphisms, Lemma 29.48.3). It follows that $X = \coprod X_ j$ and $X_ j' = \coprod _ i X_ j \times _ X U'_ i$ is a solution for $Y \to X$. Thus it suffices to prove the result (in the Noetherian case) after a surjective flat quasi-finite base change.

Applying Morphisms, Lemma 29.48.6 we see we may assume that $Y$ is a closed subscheme of an affine scheme $Z$ which is (set theoretically) a finite union $Z = \bigcup _{i \in I} Z_ i$ of closed subschemes mapping isomorphically to $X$. In this case we will find a finite partition of $X = \coprod X_ j$ with affine locally closed strata that works (in other words $X'_ j = X_ j$). Set $T_ i = Y \cap Z_ i$. This is a closed subscheme of $X$. As $X$ is Noetherian we can find a finite partition of $X = \coprod X_ j$ by affine locally closed subschemes, such that each $X_ j \times _ X T_ i$ is (set theoretically) a union of strata $X_ j \times _ X Z_ i$. Replacing $X$ by $X_ j$ we see that we may assume $I = I_1 \amalg I_2$ with $Z_ i \subset Y$ for $i \in I_1$ and $Z_ i \cap Y = \emptyset$ for $i \in I_2$. Replacing $Z$ by $\bigcup _{i \in I_1} Z_ i$ we see that we may assume $Y = Z$. Finally, we can replace $X$ again by the members of a partition as above such that for every $i, i' \subset I$ the intersection $Z_ i \cap Z_{i'}$ is either empty or (set theoretically) equal to $Z_ i$ and $Z_{i'}$. This clearly means that $Y$ is (set theoretically) equal to a disjoint union of the $Z_ i$ which is what we wanted to show. $\square$

Comment #5474 by Andrés on

In Lemma 03S1, “generic point of on irreducible component” should read “generic point of an irreducible component”.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).