Lemma 42.20.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be schemes locally of finite type over $S$. Let $p : X \to Y$ be a proper morphism. Suppose $\alpha , \beta \in Z_ k(X)$ are rationally equivalent. Then $p_*\alpha$ is rationally equivalent to $p_*\beta$.

Proof. What do we have to show? Well, suppose we are given a collection

$i_ j : W_ j \longrightarrow X$

of closed immersions, with each $W_ j$ integral of $\delta$-dimension $k + 1$ and rational functions $f_ j \in R(W_ j)^*$. Moreover, assume that the collection $\{ i_ j(W_ j)\} _{j \in J}$ is locally finite on $X$. Then we have to show that

$p_*\left(\sum i_{j, *}\text{div}(f_ j)\right)$

is rationally equivalent to zero on $X$.

Note that the sum is equal to

$\sum p_*i_{j, *}\text{div}(f_ j).$

Let $W'_ j \subset Y$ be the integral closed subscheme which is the image of $p \circ i_ j$. The collection $\{ W'_ j\}$ is locally finite in $Y$ by Lemma 42.11.2. Hence it suffices to show, for a given $j$, that either $p_*i_{j, *}\text{div}(f_ j) = 0$ or that it is equal to $i'_{j, *}\text{div}(g_ j)$ for some $g_ j \in R(W'_ j)^*$.

The arguments above therefore reduce us to the case of a since integral closed subscheme $W \subset X$ of $\delta$-dimension $k + 1$. Let $f \in R(W)^*$. Let $W' = p(W)$ as above. We get a commutative diagram of morphisms

$\xymatrix{ W \ar[r]_ i \ar[d]_{p'} & X \ar[d]^ p \\ W' \ar[r]^{i'} & Y }$

Note that $p_*i_*\text{div}(f) = i'_*(p')_*\text{div}(f)$ by Lemma 42.12.2. As explained above we have to show that $(p')_*\text{div}(f)$ is the divisor of a rational function on $W'$ or zero. There are three cases to distinguish.

The case $\dim _\delta (W') < k$. In this case automatically $(p')_*\text{div}(f) = 0$ and there is nothing to prove.

The case $\dim _\delta (W') = k$. Let us show that $(p')_*\text{div}(f) = 0$ in this case. Let $\eta \in W'$ be the generic point. Note that $c : W_\eta \to \mathop{\mathrm{Spec}}(K)$ is a proper integral curve over $K = \kappa (\eta )$ whose function field $K(W_\eta )$ is identified with $R(W)$. Here is a diagram

$\xymatrix{ W_\eta \ar[r] \ar[d]_ c & W \ar[d]^{p'} \\ \mathop{\mathrm{Spec}}(K) \ar[r] & W' }$

Let us denote $f_\eta \in K(W_\eta )^*$ the rational function corresponding to $f \in R(W)^*$. Moreover, the closed points $\xi$ of $W_\eta$ correspond $1 - 1$ to the closed integral subschemes $Z = Z_\xi \subset W$ of $\delta$-dimension $k$ with $p'(Z) = W'$. Note that the multiplicity of $Z_\xi$ in $\text{div}(f)$ is equal to $\text{ord}_{\mathcal{O}_{W_\eta , \xi }}(f_\eta )$ simply because the local rings $\mathcal{O}_{W_\eta , \xi }$ and $\mathcal{O}_{W, \xi }$ are identified (as subrings of their fraction fields). Hence we see that the multiplicity of $[W']$ in $(p')_*\text{div}(f)$ is equal to the multiplicity of $[\mathop{\mathrm{Spec}}(K)]$ in $c_*\text{div}(f_\eta )$. By Lemma 42.18.3 this is zero.

The case $\dim _\delta (W') = k + 1$. In this case Lemma 42.18.1 applies, and we see that indeed $p'_*\text{div}(f) = \text{div}(g)$ for some $g \in R(W')^*$ as desired. $\square$

In the second paragraph from the end, I can see that closed points of the fiber correspond to closed integral subschemes of codimension 1 (using (EGA, IV, 5.6.5)), but are these the same as those of dimension k? When W is "biequidimensional", one has dim(Z) + codim(Z, W) + dim(W) (EGA, O_IV, 14.3.3), for example when W is irreducible and locally of finite type over a field (EGA, IV, 5.2.1). But I'm confused about what the correct argument is in general.

This issue is significant because for an integral closed subscheme Z of codimension 1, f(Z) has a nonzero coefficient in the push-forward of the cycle when dim(f(Z)) = dim(Z), and if we had the dim + codim formula then it would follow that dim(Z) = k, hence dim(f(Z)) = dim(Z) iff f(Z) = f(W). So we would only care about closed integral Z with codim(Z,W) = 1 and f(Z) = W, which we do know are in bijection with the closed points of the generic fiber. But again I don't see how to argue when the dim + codim formula doesn't hold.

Sorry, I meant p instead of f everywhere above. And I meant dim(Z) + codim(Z, W) = dim(W) of course.

Comment #171 by on

First of all, there was a typo (switching $f$ and $p'$) which I have fixed, see here.

Say $div(f) = \sum n_Z[Z]$. Note that $\dim_\delta(Z) = k$ for the $Z$ occurring in this sum, see 42.16.1.

In that paragraph we only care about $Z$ in $W$ with $p(Z) = W'$. The reason for this is that we are using $\delta$-dimension. The $\delta$-dimension is gotten by adding the transcendence degree of $\kappa(\xi)$ over $\kappa(s)$ to $\delta(s)$ where $s \in S$ is the image of the generic point of $Z$. The only way the $\delta$-dimension of $p(Z)$ can be equal to $k$ in the situation of the proof is if $p(Z) = W$. To see this you use that $\delta$ (on $W'$) decreases under specialization (see discussion in Section 42.7) and that $\dim_\delta(W) = k$.

In other words, the disaster that you mention in your remark can probably happen if you do not assume the base scheme $S$ is universally catenary and comes with a dimension function as we do! I hope you agree!

I see, thanks for the clarification. So 42.7.5 is a generalization of (EGA, IV, 5.2.1) from S = Spec(k) to locally noetherian, universally catenary, Jacobson schemes, and in particular it means that, if one takes the dimension function $\delta$ on S to be the standard combinatorial dimension, then the canonical dimension function on X also coincides with the combinatorial dimension. Is that right?

Would you mind also briefly explaining the advantages of developing this theory for arbitrary dimension functions instead of just the standard dimension?

Comment #174 by on

The advantage is that it is more general. If $S$ is not Jacobson, then it can happen that $S$ is universally catenary and has a dimension function, but that what you call the standard dimension is not. It would take some work to find an example though.

Overall, the idea behind writing this chapter was to weaken the usual hypotheses of intersection theory to an absolute minimum as a preliminary to doing things for algebraic stacks (and especially Deligne-Mumford stacks). For example, we don't assume the schemes we work on are separated or quasi-compact and all the definitions work out fine. Some pretty weird things can happen though, e.g., one of my students (Zachary Maddock) showed that you can have a connected, universally closed, smooth scheme $X$ over a field with $A_0(X) = 0$, which is a bit counter intuitive as you'd think points cannot escape'' from such an $X$.

I see, thank you.

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