Lemma 62.7.7. Let $S$ be a locally Noetherian scheme. Let $f : X \to S$ be a locally quasi-finite morphism of schemes. Let $\alpha \in z(X/S, 0)$. The map $w_\alpha : X \to \mathbf{Z}$ constructed above is a weighting. Conversely, if $X$ is quasi-compact, then given a weighting $w : X \to \mathbf{Z}$ there exists an integer $n > 0$ such that $nw = w_\alpha $ for some $\alpha \in z(X/S, 0)$. Finally, the integer $n$ may be chosen to be a power of the prime $p$ if $S$ is a scheme over $\mathbf{F}_ p$.
Proof. First, let us show that the construction is compatible with base change: if $g : S' \to S$ is a morphism of locally Noetherian schemes, then $w_{g^*\alpha } = w_\alpha \circ g'$ where $g' : X' \to X$ is the projection $X' = S' \times _ S X \to X$. Namely, let $x' \in X'$ with images $s', s, x$ in $S', S, X$. Then the coefficient of $[x']$ in the base change of $[x]$ by $\kappa (s')/\kappa (s)$ is the length of the local ring $(\kappa (s') \otimes _{\kappa (s)} \kappa (x))_\mathfrak q$. Here $\mathfrak q$ is the prime ideal corresponding to $x'$. Thus compatibility with base change follows if
\[ [\kappa (x) : \kappa (s)]_ i = \text{length}((\kappa (s') \otimes _{\kappa (s)} \kappa (x))_\mathfrak q) [\kappa (x') : \kappa (s')]_ i \]
Let $k/\kappa (s')$ be an algebraically closure. Choose a prime $\mathfrak p \subset k \otimes _{\kappa (s)} \kappa (x)$ lying over $\mathfrak q$. Suppose we can show that
\[ [\kappa (x) : \kappa (s)]_ i = \text{length}((k \otimes _{\kappa (s)} \kappa (x))_\mathfrak p) \quad \text{and}\quad [\kappa (x') : \kappa (s')]_ i = \text{length}((k \otimes _{\kappa (s')} \kappa (x'))_\mathfrak p) \]
Then we win because
\[ \text{length}((\kappa (s') \otimes _{\kappa (s)} \kappa (x))_\mathfrak q) \text{length}((k \otimes _{\kappa (s')} \kappa (x'))_\mathfrak p) = \text{length}((k \otimes _{\kappa (s)} \kappa (x))_\mathfrak p) \]
by Algebra, Lemma 10.52.13 and flatness of $\kappa (s') \otimes _{\kappa (s)} \kappa (x) \to k \otimes _{\kappa (s)} \kappa (x)$. To show the two equalities, it suffices to prove the first. Let $\kappa (x)/\kappa /\kappa (s)$ be the subfield constructed in Fields, Lemma 9.14.6. Then we see that
\[ k \otimes _{\kappa (s)} \kappa (x) = \prod \nolimits _{\sigma : \kappa \to k} k \otimes _{\sigma , \kappa } \kappa (x) \]
and each of the factors is local of degree $[\kappa (x) : \kappa ] = [\kappa (x) : \kappa (s)]_ i$ as desired.
Let $\alpha \in z(X/S, 0)$ and choose a diagram
\[ \xymatrix{ X \ar[d]_ f & U \ar[l]^ h \ar[d]^\pi \\ Y & V \ar[l]_ g } \]
as in More on Morphisms, Definition 37.75.2. Denote $\beta \in z(U/V, 0)$ the restriction of the base change $g^*\alpha $. By the compatibility with base change above we have $w_\beta = w_\alpha \circ h$ and it suffices to show that $\int _\pi w_\beta $ is locally constant on $V$. Next, note that
\begin{align*} \left( \int _\pi w_\beta \right)(v) & = \sum \nolimits _{u \in U, \pi (u) = v} \beta (u) [\kappa (u) : \kappa (v)]_ i [\kappa (u) : \kappa (v)]_ s \\ & = \sum \nolimits _{u \in U, \pi (u) = v} \beta (u)[\kappa (u) : \kappa (v)] \end{align*}
This last expression is the coefficient of $v$ in $\pi _*\beta \in z(V/V, 0)$. By Lemma 62.6.11 this function is locally constand on $V$.
Conversely, let $w : X \to S$ be a weighting and $X$ quasi-compact. Choose a sufficiently divisible integer $n$. Let $\alpha $ be the family of $0$-cycles on fibres of $X/S$ such that for $s \in S$ we have
\[ \alpha _ s = \sum \nolimits _{f(x) = s} \frac{n w(x)}{[\kappa (x) : \kappa (s)]_ i} [x] \]
as a zero cycle on $X_ s$. This makes sense since the fibres of $f$ are universally bounded (Morphisms, Lemma 29.57.9) hence we can find $n$ such that the right hand side is an integer for all $s \in S$. The final statement of the lemma also follows, provided we show $\alpha $ is a relative $0$-cycle. To do this we have to show that $\alpha $ is compatible with specializations along discrete valuation rings. By the first paragraph of the proof our construction is compatible with base change (small detail omitted; it is the “inverse” construction we are discussing here). Also, the base change of a weighting is a weighting, see More on Morphisms, Lemma 37.75.3. Thus we reduce to the problem studied in the next paragraph.
Assume $S$ is the spectrum of a discrete valuation ring with generic point $\eta $ and closed point $0$. Let $w : X \to S$ be a weighting with $X$ quasi-finite over $S$. Let $\alpha $ be the family of $0$-cycles on fibres of $X/S$ constructed in the previous paragraph (for a suitable $n$). We have to show that $sp_{X/S}(\alpha _\eta ) = \alpha _0$. Let $\beta \in z(X/S, 0)$ be the relative $0$-cycle on $X/S$ with $\beta _\eta = \alpha _\eta $ and $\beta _0 = sp_{X/S}(\alpha _\eta )$. Then $w' = w_\beta - nw : X \to \mathbf{Z}$ is a weighting (using the result above) and zero in the points of $X$ which map to $\eta $. Now it is easy to see that a weighting which is zero on all points of $X$ mapping to $\eta $ has to be zero; details omitted. Hence $w' = 0$, i.e., $w_\beta = nw$, hence $\alpha = \beta $ as desired. $\square$
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