The Stacks project

Lemma 37.73.9. 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 37.73.1 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 37.73.8. 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 37.73.2 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 37.72.3) 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. $\square$


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