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 $ containes 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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)