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. $\square$

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).