## 42.21 Rational equivalence and the projective line

Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Given any closed subscheme $Z \subset X \times _ S \mathbf{P}^1_ S = X \times \mathbf{P}^1$ we let $Z_0$, resp. $Z_\infty$ be the scheme theoretic closed subscheme $Z_0 = \text{pr}_2^{-1}(D_0)$, resp. $Z_\infty = \text{pr}_2^{-1}(D_\infty )$. Here $D_0$, $D_\infty$ are as in (42.18.1.1).

Lemma 42.21.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $W \subset X \times _ S \mathbf{P}^1_ S$ be an integral closed subscheme of $\delta$-dimension $k + 1$. Assume $W \not= W_0$, and $W \not= W_\infty$. Then

1. $W_0$, $W_\infty$ are effective Cartier divisors of $W$,

2. $W_0$, $W_\infty$ can be viewed as closed subschemes of $X$ and

$[W_0]_ k \sim _{rat} [W_\infty ]_ k,$
3. for any locally finite family of integral closed subschemes $W_ i \subset X \times _ S \mathbf{P}^1_ S$ of $\delta$-dimension $k + 1$ with $W_ i \not= (W_ i)_0$ and $W_ i \not= (W_ i)_\infty$ we have $\sum ([(W_ i)_0]_ k - [(W_ i)_\infty ]_ k) \sim _{rat} 0$ on $X$, and

4. for any $\alpha \in Z_ k(X)$ with $\alpha \sim _{rat} 0$ there exists a locally finite family of integral closed subschemes $W_ i \subset X \times _ S \mathbf{P}^1_ S$ as above such that $\alpha = \sum ([(W_ i)_0]_ k - [(W_ i)_\infty ]_ k)$.

Proof. Part (1) follows from Divisors, Lemma 31.13.13 since the generic point of $W$ is not mapped into $D_0$ or $D_\infty$ under the projection $X \times _ S \mathbf{P}^1_ S \to \mathbf{P}^1_ S$ by assumption.

Since $X \times _ S D_0 \to X$ is a closed immersion, we see that $W_0$ is isomorphic to a closed subscheme of $X$. Similarly for $W_\infty$. The morphism $p : W \to X$ is proper as a composition of the closed immersion $W \to X \times _ S \mathbf{P}^1_ S$ and the proper morphism $X \times _ S \mathbf{P}^1_ S \to X$. By Lemma 42.18.2 we have $[W_0]_ k \sim _{rat} [W_\infty ]_ k$ as cycles on $W$. Hence part (2) follows from Lemma 42.20.3 as clearly $p_*[W_0]_ k = [W_0]_ k$ and similarly for $W_\infty$.

The only content of statement (3) is, given parts (1) and (2), that the collection $\{ (W_ i)_0, (W_ i)_\infty \}$ is a locally finite collection of closed subschemes of $X$. This is clear.

Suppose that $\alpha \sim _{rat} 0$. By definition this means there exist integral closed subschemes $V_ i \subset X$ of $\delta$-dimension $k + 1$ and rational functions $f_ i \in R(V_ i)^*$ such that the family $\{ V_ i\} _{i \in I}$ is locally finite in $X$ and such that $\alpha = \sum (V_ i \to X)_*\text{div}(f_ i)$. Let

$W_ i \subset V_ i \times _ S \mathbf{P}^1_ S \subset X \times _ S \mathbf{P}^1_ S$

be the closure of the graph of the rational map $f_ i$ as in Lemma 42.18.2. Then we have that $(V_ i \to X)_*\text{div}(f_ i)$ is equal to $[(W_ i)_0]_ k - [(W_ i)_\infty ]_ k$ by that same lemma. Hence the result is clear. $\square$

Lemma 42.21.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $Z$ be a closed subscheme of $X \times \mathbf{P}^1$. Assume

1. $\dim _\delta (Z) \leq k + 1$,

2. $\dim _\delta (Z_0) \leq k$, $\dim _\delta (Z_\infty ) \leq k$, and

3. for any embedded point $\xi$ (Divisors, Definition 31.4.1) of $Z$ either $\xi \not\in Z_0 \cup Z_\infty$ or $\delta (\xi ) < k$.

Then $[Z_0]_ k \sim _{rat} [Z_\infty ]_ k$ as $k$-cycles on $X$.

Proof. Let $\{ W_ i\} _{i \in I}$ be the collection of irreducible components of $Z$ which have $\delta$-dimension $k + 1$. Write

$[Z]_{k + 1} = \sum n_ i[W_ i]$

with $n_ i > 0$ as per definition. Note that $\{ W_ i\}$ is a locally finite collection of closed subsets of $X \times _ S \mathbf{P}^1_ S$ by Divisors, Lemma 31.26.1. We claim that

$[Z_0]_ k = \sum n_ i[(W_ i)_0]_ k$

and similarly for $[Z_\infty ]_ k$. If we prove this then the lemma follows from Lemma 42.21.1.

Let $Z' \subset X$ be an integral closed subscheme of $\delta$-dimension $k$. To prove the equality above it suffices to show that the coefficient $n$ of $[Z']$ in $[Z_0]_ k$ is the same as the coefficient $m$ of $[Z']$ in $\sum n_ i[(W_ i)_0]_ k$. Let $\xi ' \in Z'$ be the generic point. Set $\xi = (\xi ', 0) \in X \times _ S \mathbf{P}^1_ S$. Consider the local ring $A = \mathcal{O}_{X \times _ S \mathbf{P}^1_ S, \xi }$. Let $I \subset A$ be the ideal cutting out $Z$, in other words so that $A/I = \mathcal{O}_{Z, \xi }$. Let $t \in A$ be the element cutting out $X \times _ S D_0$ (i.e., the coordinate of $\mathbf{P}^1$ at zero pulled back). By our choice of $\xi ' \in Z'$ we have $\delta (\xi ) = k$ and hence $\dim (A/I) = 1$. Since $\xi$ is not an embedded point by assumption (3) we see that $A/I$ is Cohen-Macaulay. Since $\dim _\delta (Z_0) = k$ we see that $\dim (A/(t, I)) = 0$ which implies that $t$ is a nonzerodivisor on $A/I$. Finally, the irreducible closed subschemes $W_ i$ passing through $\xi$ correspond to the minimal primes $I \subset \mathfrak q_ i$ over $I$. The multiplicities $n_ i$ correspond to the lengths $\text{length}_{A_{\mathfrak q_ i}}(A/I)_{\mathfrak q_ i}$. Hence we see that

$n = \text{length}_ A(A/(t, I))$

and

$m = \sum \text{length}_ A(A/(t, \mathfrak q_ i)) \text{length}_{A_{\mathfrak q_ i}}(A/I)_{\mathfrak q_ i}$

Thus the result follows from Lemma 42.3.2. $\square$

Lemma 42.21.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $\mathcal{F}$ be a coherent sheaf on $X \times \mathbf{P}^1$. Let $i_0, i_\infty : X \to X \times \mathbf{P}^1$ be the closed immersion such that $i_ t(x) = (x, t)$. Denote $\mathcal{F}_0 = i_0^*\mathcal{F}$ and $\mathcal{F}_\infty = i_\infty ^*\mathcal{F}$. Assume

1. $\dim _\delta (\text{Supp}(\mathcal{F})) \leq k + 1$,

2. $\dim _\delta (\text{Supp}(\mathcal{F}_0)) \leq k$, $\dim _\delta (\text{Supp}(\mathcal{F}_\infty )) \leq k$, and

3. for any embedded associated point $\xi$ of $\mathcal{F}$ either $\xi \not\in (X \times \mathbf{P}^1)_0 \cup (X \times \mathbf{P}^1)_\infty$ or $\delta (\xi ) < k$.

Then $[\mathcal{F}_0]_ k \sim _{rat} [\mathcal{F}_\infty ]_ k$ as $k$-cycles on $X$.

Proof. Let $\{ W_ i\} _{i \in I}$ be the collection of irreducible components of $\text{Supp}(\mathcal{F})$ which have $\delta$-dimension $k + 1$. Write

$[\mathcal{F}]_{k + 1} = \sum n_ i[W_ i]$

with $n_ i > 0$ as per definition. Note that $\{ W_ i\}$ is a locally finite collection of closed subsets of $X \times _ S \mathbf{P}^1_ S$ by Lemma 42.10.1. We claim that

$[\mathcal{F}_0]_ k = \sum n_ i[(W_ i)_0]_ k$

and similarly for $[\mathcal{F}_\infty ]_ k$. If we prove this then the lemma follows from Lemma 42.21.1.

Let $Z' \subset X$ be an integral closed subscheme of $\delta$-dimension $k$. To prove the equality above it suffices to show that the coefficient $n$ of $[Z']$ in $[\mathcal{F}_0]_ k$ is the same as the coefficient $m$ of $[Z']$ in $\sum n_ i[(W_ i)_0]_ k$. Let $\xi ' \in Z'$ be the generic point. Set $\xi = (\xi ', 0) \in X \times _ S \mathbf{P}^1_ S$. Consider the local ring $A = \mathcal{O}_{X \times _ S \mathbf{P}^1_ S, \xi }$. Let $M = \mathcal{F}_\xi$ as an $A$-module. Let $t \in A$ be the element cutting out $X \times _ S D_0$ (i.e., the coordinate of $\mathbf{P}^1$ at zero pulled back). By our choice of $\xi ' \in Z'$ we have $\delta (\xi ) = k$ and hence $\dim (\text{Supp}(M)) = 1$. Since $\xi$ is not an associated point of $\mathcal{F}$ by assumption (3) we see that $M$ is a Cohen-Macaulay module. Since $\dim _\delta (\text{Supp}(\mathcal{F}_0)) = k$ we see that $\dim (\text{Supp}(M/tM)) = 0$ which implies that $t$ is a nonzerodivisor on $M$. Finally, the irreducible closed subschemes $W_ i$ passing through $\xi$ correspond to the minimal primes $\mathfrak q_ i$ of $\text{Ass}(M)$. The multiplicities $n_ i$ correspond to the lengths $\text{length}_{A_{\mathfrak q_ i}}M_{\mathfrak q_ i}$. Hence we see that

$n = \text{length}_ A(M/tM)$

and

$m = \sum \text{length}_ A(A/(t, \mathfrak q_ i)A) \text{length}_{A_{\mathfrak q_ i}}M_{\mathfrak q_ i}$

Thus the result follows from Lemma 42.3.2. $\square$

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