The Stacks project

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.

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 $X$.

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 constant1 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

\[ \textstyle {\int }_ f w = \textstyle {\int }_{f|_ U} w|_ U + \textstyle {\int }_{f|_ V} w|_ V - \textstyle {\int }_{f|_{U \cap V}} w|_{U \cap V} \]

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

\[ U \times _ Y X = W \amalg \ \coprod \nolimits _{i = 1, \ldots , n} \ \coprod \nolimits _{j = 1, \ldots , m_ i} V_{i, j} \]

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

\[ \textstyle {\int }_{U \times _ Y X \to U} w|_{U \times _ Y X} = \textstyle {\int }_{W \to U} w|_ W + \sum \textstyle {\int }_{V_{i, j} \to U} w|_{V_{i, j}} = \textstyle {\int }_{W \to U} w|_ W + (\int _ f w)(y) \]

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

\[ U \times _ Y X = W \amalg \ \coprod \nolimits _{i = 1, \ldots , n} \ \coprod \nolimits _{j = 1, \ldots , m_ i} V_{i, j} \]

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

\[ \textstyle {\int }_{U \times _ Y X \to U} w|_{U \times _ Y X} = \textstyle {\int }_{W \to U} w|_ W + \sum \textstyle {\int }_{V_{i, j} \to U} w|_{V_{i, j}} = \textstyle {\int }_{W \to U} w|_ W + (\int _ f w)(y) \]

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

\[ A_{univ} = A[z_1, \ldots , z_ N, y_1, \ldots , y_ m]/I \]

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

\[ (\sum z_ j b'_ j)^2 - \sum z_ j b'_ j + \sum y_ k f_ k \]

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

\[ (\sum b'_ j \otimes a'_ j)^2 - \sum b'_ j \otimes a'_ j + \sum f_ k \otimes a''_ k \]

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

\[ U' \to Y' \times _ Y X \to Y' \times _ Y X' \]

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

\[ \Gamma (Y', \mathcal{O}_{Y'}) \otimes _{\Gamma (Y, \mathcal{O}_ Y)} \Gamma (X', \mathcal{O}_{X'}) \]

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

\[ e'_{univ} \in \Gamma (Y_{univ}, \mathcal{O}_{Y_{univ}}) \otimes _{\Gamma (Y, \mathcal{O}_ Y)} \Gamma (X', \mathcal{O}_{X'}) \]

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

\[ U'_{univ} \setminus Y'_{univ} \times _ Y X \]

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

\[ \tau _ i = \textstyle {\int }_{U_{i, univ} \to Y_{i, univ}} w_ i|_{U_{i, univ}} \]

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

\[ \tau = \textstyle {\int }_{U_{univ} \to Y_{univ}} w_ i|_{U_{univ}} \]

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. $\square$

[1] In fact, if $f : X \to Y$ is an immersion and $w$ is a weighting of $f$, then $f$ restricts to an open map on the locus where $w$ is nonzero.

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