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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