Definition 62.7.1. Let $f : X \to S$ be a morphism of schemes. Assume $S$ is locally Noetherian and $f$ is locally of finite type. Let $r \geq 0$ be an integer. We say a relative $r$-cycle $\alpha $ on $X/S$ equidimensional if the support of $\alpha $ (Remark 62.5.6) is contained in a closed subset $W \subset X$ whose relative dimension over $S$ is $\leq r$. The group of all equidimensional relative $r$-cycles on $X/S$ is denoted $z_{equi}(X/S, r)$.
62.7 Equidimensional relative cycles
Here is the definition.
Example 62.7.2. There exist relative $r$-cycles which are not equidimensional. Namely, let $k$ be a field and let $X = \mathop{\mathrm{Spec}}(k[x, y, t])$ over $S = \mathop{\mathrm{Spec}}(k[x, y])$. Let $s$ be a point of $S$ and denote $a, b \in \kappa (s)$ the images of $x$ and $y$. Consider the family $\alpha $ of $0$-cycles on $X/S$ defined by
$\alpha _ s = 0$ if $b = 0$ and otherwise
$\alpha _ s = [p] - [q]$ where $p$, resp. $q$ is the $\kappa (s)$-rational point of $\mathop{\mathrm{Spec}}(\kappa (s)[t])$ with $t = a/b$, resp. $t = (a + b^2)/b$.
We leave it to the reader to show that this is compatible with specializations; the idea is that $a/b$ and $(a + b^2)/b = a/b + b$ limit to the same point in $\mathbf{P}^1$ over the residue field of any valuation $v$ on $\kappa (s)$ with $v(b) > 0$. On the other hand, the closure of the support of $\alpha $ contains the whole fibre over $(0, 0)$.
Lemma 62.7.3. Let $f : X \to S$ be a morphism of schemes. Assume $S$ is locally Noetherian and $f$ is locally of finite type. Let $r \geq 0$ be an integer. Let $\alpha $ be a relative $r$-cycle on $X/S$. If $\alpha $ is equidimensional, then any restriction, base change, or flat pullback of $\alpha $ is equidimensional.
Proof. Omitted. $\square$
Lemma 62.7.4. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r \geq 0$ be an integer. Let $\alpha $ be a relative $r$-cycle on $X/S$. Then to check that $\alpha $ is equidimensional we may work Zariski locally on $X$ and $S$.
Proof. Namely, the condition that $\alpha $ is equidimensional just means that the closure of the support of $\alpha $ has relative dimension $\leq r$ over $S$. Since taking closures commutes with restriction to opens, the lemma follows (small detail omitted). $\square$
Lemma 62.7.5. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r \geq 0$ be an integer. Let $\alpha $ be a relative $r$-cycle on $X/S$. Let $\{ g_ i : S_ i \to S\} $ be an fppf covering. Then $\alpha $ is equidimensional if and only if each base change $g_ i^*\alpha $ is equidimensional.
Proof. If $\alpha $ is equidimensional, then each $g_ i^*\alpha $ is too by Lemma 62.7.3. Assume each $g_ i^*\alpha $ is equidimensional. Denote $W$ the closure of $\text{Supp}(\alpha )$ in $X$. Since $g_ i : S_ i \to S$ is universally open (being flat and locally of finite presentation), so is the morphism $f_ i : X_ i = S_ i \times _ S X \to X$. Denote $\alpha _ i = g_ i^*\alpha $. We have $\text{Supp}(\alpha _ i) = f_ i^{-1}(\text{Supp}(\alpha ))$ by Lemma 62.5.7. Since $f_ i$ is open, we see that $W_ i = f_ i^{-1}(W)$ is the closure of $\text{Supp}(\alpha _ i)$. Hence by assumption the morphism $W_ i \to S_ i$ has relative dimension $\leq r$. By Morphisms, Lemma 29.28.3 (and the fact that the morphisms $S_ i \to S$ are jointly surjective) we conclude that $W \to S$ has relative dimension $\leq r$. $\square$
Lemma 62.7.6. Let $f : X \to S$ be a morphism of schemes. Assume $S$ locally Noetherian and $f$ locally of finite type. Let $r, e \geq 0$ be integers. Let $\alpha $ be a relative $r$-cycle on $X/S$. Let $\{ f_ i : X_ i \to X\} $ be a jointly surjective family of flat morphisms, locally of finite type, and of relative dimension $e$. Then $\alpha $ is equidimensional if and only if each flat pullback $f_ i^*\alpha $ is equidimensional.
Proof. Omitted. Hint: As in the proof of Lemma 62.7.5 one shows that the inverse image by $f_ i$ of the closure $W$ of the support of $\alpha $ is the closure $W_ i$ of the support of $f_ i^*\alpha $. Then $W \to S$ has relative dimension $\leq r$ holds if $W_ i \to S$ has relative dimension $\leq r + e$ for all $i$. $\square$
Let $S$ be a locally Noetherian scheme. Let $f : X \to S$ be a locally quasi-finite morphism of schemes. Then we have $z(X/S, 0) = z_{equi}(X/S, 0)$ and $z(X/S, r) = 0$ for $r > 0$. Given $\alpha \in z(X/S, 0)$ let us define a map
\[ w_\alpha : X \longrightarrow \mathbf{Z},\quad x \mapsto \alpha (x) [\kappa (x) : \kappa (s)]_ i \quad \text{where }s = f(x) \]
Here $\alpha (x)$ denotes the coefficient of $x$ in the $0$-cycle $\alpha _ s$ on the fibre $X_ s$ and $[K : k]_ i$ denotes the inseparable degree of a finite field extension. The following lemma shows that this map is a weighting of $f$ (More on Morphisms, Definition 37.75.2) and that every weighting is of this form up to taking a multiple.
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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