Lemma 37.70.1. Let $f : X \to Y$ be a locally quasi-finite morphism. Let $w : X \to \mathbf{Z}$ be a weighting of $f$. Then the level sets of the function $w$ are locally constructible in $X$.

## 37.70 More on weightings

We prove a few more basic properties of weightings. Allthough at first it appears that weightings can be very wild, it actually turns out the condition imposed in Definition 37.69.2 is rather strong.

**Proof.**
The question is Zariski local on $X$ and $Y$, hence we may assume $X$ and $Y$ are affine. (Observe that the restriction of a weighting to an open is a weighting of the restriction of $f$ to that open.) By Lemma 37.69.3 after base change we still have a weighthing. If we can find a surjective morphism $Y' \to Y$ of finite presentation such that the level sets of $w$ pull back to locally constructible subsets of $X' = Y' \times _ Y X$, then we conclude by Morphisms, Theorem 29.22.3. We may choose an immersion $X \to T$ where $T \to Y$ is finite, see Lemma 37.39.3. By Morphisms, Lemma 29.48.6 after replacing $Y$ by $Y'$ surjective finite locally free over $Y$, replacing $X$ by $Y' \times _ Y X$ and $T$ by a scheme finite locally free over $Y'$ containing $Y' \times _ Y T$ as a closed subscheme, we may assume $T$ is finite locally free over $Y$, contains closed subschemes $T_ i$ mapping isomorphically to $Y$ such that $T = \bigcup _{i = 1, \ldots , n} T_ i$ (set theoretically). Observe that for $I \subset \{ 1, \ldots , n\} $ the intersection $\bigcap _{i \in I} T_ i$ is a constructible closed subset of $T$ (and hence maps to a constructible closed subset of $Y$).

For a disjoint union decomposition $\{ 1, \ldots , n\} = I_1 \amalg \ldots \amalg I_ r$ with nonempty parts consider the subset $Y_{I_1, \ldots , I_ r} \subset Y$ consisting of points $y \in Y$ such that $T_ y = \{ x_1, \ldots , x_ r\} $ consists of exactly $r$ points with $x_ j \in T_ i \Leftrightarrow i \in I_ j$. This is a constructible partition of $Y$, hence we may assume that $Y = Y_{I_1, \ldots , I_ r}$ for some disjoint union decomposition $\{ 1, \ldots , n\} = I_1 \amalg \ldots \amalg I_ r$. (You can base change by an affine scheme $Y'$ of finite presentation over $Y$ such that the image of $Y' \to Y$ is exactly $Y_{I_1, \ldots , I_ r}$, see Algebra, Lemma 10.29.4.) Thus $T = T(1) \amalg \ldots \amalg T(r)$ with $T(j) = \bigcap _{i \in I_ j} T_ i$ is a decomposition of $T$ into disjoint closed (and hence open) subsets. Intersecting with the locally closed subscheme $X$ we obtain an analogous decomposition $X = X(1) \amalg \ldots \amalg X(r)$ into open and closed parts. The morphism $X(j) \to Y$ an immersion. Since $w$ is a weigthing, it follows that $w|_{X(j)}$ is locally constant^{1} and we conclude.
$\square$

Lemma 37.70.2. Let $f : X \to Y$ be a locally quasi-finite morphism of finite presentation. Let $w : X \to \mathbf{Z}$ be a weighting of $f$. Then the level sets of the function $\int _ f w$ are locally constructible in $Y$.

**Proof.**
By Lemma 37.69.1 formation of the function $\int _ f w$ commutes with arbitrary base change and by Lemma 37.69.3 after base change we still have a weighthing. This means that if we can find $Y' \to Y$ surjective and of finite presentation, then it suffices to prove the result after base change to $Y'$, see Morphisms, Theorem 29.22.3.

The question is local on $Y$ hence we may assume $Y$ is affine. Then $X$ is quasi-compact and quasi-separated (as $f$ is of finite presentation). Suppose that $X = U \cup V$ are quasi-compact open. Then we have

Thus if we know the result for $w|_ U$, $w|_ V$, $w|_{U \cap V}$ then we know the result for $w$. By the induction principle (Cohomology of Schemes, Lemma 30.4.1) it suffices to prove the lemma when $X$ is affine.

Assume $X$ and $Y$ are affine. We may choose an open immersion $X \to T$ where $T \to Y$ is finite, see Lemma 37.39.3. Because we may still base change with a suitable $Y' \to Y$ we can use Morphisms, Lemma 29.48.6 to reduce to the case where all residue field extensions induced by the morphism $T \to Y$ (and a foriori induced by $X \to Y$) are trivial. In this situation $\int _ f w$ is just taking the sums of the values of $w$ in fibres. The level sets of $w$ are locally constructible in $X$ (Lemma 37.70.1). The function $w$ only takes a finite number of values by Properties, Lemma 28.2.7. Hence we conclude by Morphisms, Theorem 29.22.3 and some elementary arguments on sums of integers. $\square$

Lemma 37.70.3. Let $f : X \to Y$ be a locally quasi-finite morphism. Let $w : X \to \mathbf{Z}_{> 0}$ be a positive weighting of $f$. Then $w$ is upper semi-continuous.

**Proof.**
Let $x \in X$ with image $y \in Y$. Choose an étale neighbourhood $(V, v) \to (Y, y)$ and an open $U \subset X_ V$ such that $\pi : U \to V$ is finite and there is a unique point $u \in U$ mapping to $v$ with $\kappa (u)/\kappa (v)$ purely inseparable. See Lemma 37.37.3. Then $(\int _\pi w|_ U)(v) = w(u)$. It follows from Definition 37.69.2 that after replacing $V$ by a neighbourhood of $v$ we we have $w|_ U(u') \leq w|_ U(u) = w(x)$ for all $u' \in U$. Namely, $w|_ U(u')$ occurs as a summand in the expression for $(\int _\pi w|_ U)(\pi (u'))$. This proves the lemma because the étale morphism $U \to X$ is open.
$\square$

Lemma 37.70.4. Let $f : X \to Y$ be a separated, locally quasi-finite morphism with finite fibres. Let $w : X \to \mathbf{Z}_{> 0}$ be a positive weighting of $f$. Then $\int _ f w$ is lower semi-continuous.

**Proof.**
Let $y \in Y$. Let $x_1, \ldots , x_ r \in X$ be the points lying over $y$. Apply Lemma 37.37.5 to get an étale neighbourhood $(U, u) \to (Y, y)$ and a decomposition

as in locus citatus. Observe that $(\int _ f w)(y) = \sum w(v_{i, j})$ where $w(v_{i, j}) = w(x_ i)$. Since $\int _{V_{i, j} \to U} w|_{V_{i, j}}$ is locally constant by definition, we may after shrinking $U$ assume these functions are constant with value $w(v_{i, j})$. We conclude that

This is $\geq (\int _ f w)(y)$ and we conclude because $U \to Y$ is open and formation of the integral commutes with base change (Lemma 37.69.1). $\square$

Lemma 37.70.5. Let $f : X \to Y$ be a locally quasi-finite morphism with $X$ quasi-compact. Let $w : X \to \mathbf{Z}$ be a weighting of $f$. Then $\int _ f w$ attains its maximum.

**Proof.**
It follows from Lemma 37.70.1 and Properties, Lemma 28.2.7 that $w$ only takes a finite number of values on $X$. It follows from Morphisms, Lemma 29.56.9 that $X \to Y$ has bounded geometric fibres. This shows that $\int _ f w$ is bounded.
$\square$

Lemma 37.70.6. Let $f : X \to Y$ be a separated, locally quasi-finite morphism. Let $w : X \to \mathbf{Z}_{> 0}$ be a positive weighting of $f$. Assume $\int _ w f$ attains its maximum $d$ and let $Y_ d \subset Y$ be the open set of points $y$ with $(\int _ f w)(y) = d$. Then the morphism $f^{-1}(Y_ d) \to Y_ d$ is finite.

**Proof.**
Observe that $Y_ d$ is open by Lemma 37.70.4. Let $y \in Y_ d$. Say $x_1, \ldots , x_ n$ are the points of $X$ lying over $y$. Apply Lemma 37.37.5 to get an étale neighbourhood $(U, u) \to (Y, y)$ and a decomposition

as in locus citatus. Observe that $d = \sum w(v_{i, j})$ where $w(v_{i, j}) = w(x_ i)$. Since $\int _{V_{i, j} \to U} w|_{V_{i, j}}$ is locally constant by definition, we may after shrinking $U$ assume these functions are constant with value $w(v_{i, j})$. We conclude that

This is $\geq (\int _ f w)(y) = d$ and we conclude that $W$ must be the emptyset. Thus $U \times _ Y X \to U$ is finite. By Descent, Lemma 35.20.23 this implies that $X \to Y$ is finite over the image of the open morphism $U \to Y$. In other words, we see that $f$ is finite over an open neighbourhood of $y$ as desired. $\square$

Lemma 37.70.7. Let $A \to B$ be a ring map which is finite and of finite presentation. There exists a finitely presented ring map $A \to A_{univ}$ and an idempotent $e_{univ} \in B \otimes _ A A_{univ}$ such that for any ring map $A \to A'$ and idempotent $e \in B \otimes _ A A'$ there is a ring map $A_{univ} \to A'$ mapping $e_{univ}$ to $e$.

**Proof.**
Choose $b_1, \ldots , b_ n \in B$ generating $B$ as an $A$-module. For each $i$ choose a monic $P_ i \in A[x]$ such that $P_ i(b_ i) = 0$ in $B$, see Algebra, Lemma 10.36.3. Thus $B$ is a quotient of the finite free $A$-algebra $B' = A[x_1, \ldots , x_ n]/(P_1(x_1), \ldots , P_ n(x_ n))$. Let $J \subset B'$ be the kernel of the surjection $B' \to B$. Then $J =(f_1, \ldots , f_ m)$ is finitely generated as $B$ is a finitely generated $A$-algebra, see Algebra, Lemma 10.6.2. Choose an $A$-basis $b'_1, \ldots , b'_ N$ of $B'$. Consider the algebra

where $I$ is the ideal generated by the coefficients in $A[z_1, \ldots , z_ n, y_1, \ldots , y_ m]$ of the basis elements $b'_1, \ldots , b'_ N$ of the expresssion

in $B'[z_1, \ldots , z_ N, y_1, \ldots , y_ m]$. By construction the element $\sum z_ j b'_ j$ maps to an idempotent $e_{univ}$ in the algebra $B \otimes _ A A_{univ}$. Moreover, if $e \in B \otimes _ A A'$ is an idempotent, then we can lift $e$ to an element of the form $\sum b'_ j \otimes a'_ j$ in $B' \otimes _ A A'$ and we can find $a''_ k \in A'$ such that

is zero in $B' \otimes _ A A'$. Hence we get an $A$-algebra map $A_{univ} \to A$ sending $z_ j$ to $a'_ j$ and $y_ k$ to $a''_ k$ mapping $e_{univ}$ to $e$. This finishes the proof. $\square$

Lemma 37.70.8. Let $X \to Y$ be a morphism of affine schemes which is quasi-finite and of finite presentation. There exists a morphism $Y_{univ} \to Y$ of finite presentation and an open subscheme $U_{univ} \subset Y_{univ} \times _ Y X$ such that $U_{univ} \to Y_{univ}$ is finite with the following property: given any morphism $Y' \to Y$ of affine schemes and an open subscheme $U' \subset Y' \times _ Y X$ such that $U' \to Y'$ is finite, there exists a morphism $Y' \to Y_{univ}$ such that the inverse image of $U_{univ}$ is $U'$.

**Proof.**
Recall that a finite type morphism is quasi-finite if and only if it has relative dimension $0$, see Morphisms, Lemma 29.29.5. By Lemma 37.31.9 applied with $d = 0$ we reduce to the case where $X$ and $Y$ are Noetherian. We may choose an open immersion $X \to X'$ such that $X' \to Y$ is finite, see Algebra, Lemma 10.123.14. Note that if we have $Y' \to Y$ and $U'$ as in (2), then

is open immersion between schemes finite over $Y'$ and hence is closed as well. We conclude that $U'$ corresponds to an idempotent in

whose corresponding open and closed subset is contained in the open $Y' \times _ Y X$. Let $Y'_{univ} \to Y$ and idempotent

be the pair constructed in Lemma 37.70.7 for the ring map $\Gamma (Y, \mathcal{O}_ Y) \to \Gamma (X', \mathcal{O}_{X'})$ (here we use that $Y$ is Noetherian to see that $X'$ is of finite presentation over $Y$). Let $U'_{univ} \subset Y'_{univ} \times _ Y X'$ be the corresponding open and closed subscheme. Then we see that

is a closed subset of $U'_{univ}$ and hence has closed image $T \subset Y'_{univ}$. If we set $Y_{univ} = Y'_{univ} \setminus T$ and $U_{univ}$ the restriction of $U'_{univ}$ to $Y_{univ} \times _ Y X$, then we see that the lemma is true. $\square$

Lemma 37.70.9. Let $Y = \mathop{\mathrm{lim}}\nolimits Y_ i$ be a directed limit of affine schemes. Let $0 \in I$ and let $f_0 : X_0 \to Y_0$ be a morphism of affine schemes which is quasi-finite and of finite presentation. Let $f : X \to Y$ and $f_ i : X_ i \to Y_ i$ for $i \geq 0$ be the base changes of $f_0$. If $w : X \to \mathbf{Z}$ is a weighting of $f$, then for sufficiently large $i$ there exists a weighting $w_ i : X_ i \to \mathbf{Z}$ of $f_ i$ whose pullback to $X$ is $w$.

**Proof.**
By Lemma 37.70.1 the level sets of $w$ are constructible subsets $E_ k$ of $X$. This implies the function $w$ only takes a finite number of values by Properties, Lemma 28.2.7. Thus there exists an $i$ such that $E_ k$ descends to a construcible subset $E_{i, k}$ in $X_ i$ for all $k$; moreover, we may assume $X_ i = \coprod E_{i, k}$. This follows as the topological space of $X$ is the limit in the category of topological spaces of the spectral spaces $X_ i$ along a directed system with spectral transition maps. See Limits, Section 32.4 and Topology, Section 5.24. We define $w_ i : X_ i \to \mathbf{Z}$ such that its level sets are the constructible sets $E_{i, k}$.

Choose $Y_{i, univ} \to Y_ i$ and $U_{i, univ} \subset Y_{i, univ} \times _{Y_ i} X_ i$ as in Lemma 37.70.8. By the universal property of the construction, in order to show that $w_ i$ is a weighting, it would suffice to show that

is locally constant on $Y_{i, univ}$. By Lemma 37.70.2 this function has constructible level sets but it may not (yet) be locally constant. Set $Y_{univ} = Y_{i, univ} \times _{Y_ i} Y$ and let $U_{univ} \subset Y_{univ} \times _ Y X$ be the inverse image of $U_{i, univ}$. Then, since the pullback of $w$ to $Y_{univ} \times _ Y X$ is a weighting for $Y_{univ} \times _ Y X \to Y_{univ}$ (Lemma 37.69.3) we do have that

is locally constant on $Y_{univ}$. Thus the level sets of $\tau $ are open and closed. Finally, we have $Y_{univ} = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} Y_{i', univ}$ and the level sets of $\tau $ are the inverse limits of the level sets of $\tau _{i'}$ (similarly defined). Hence the references above imply that for sufficiently large $i'$ the level sets of $\tau _{i'}$ are open as well. For such an index $i'$ we conclude that $w_{i'}$ is a weighting of $f_{i'}$ as desired. $\square$

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