The Stacks project

80.16 Rational equivalence and push and pull

This section is the analogue of Chow Homology, Section 42.20. In this section we show that flat pullback and proper pushforward commute with rational equivalence.

Lemma 80.16.1. In Situation 80.2.1 let $X, Y/B$ be good. Assume $Y$ integral with $\dim _\delta (Y) = k$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Then for $g \in R(Y)^*$ we have

\[ f^*\text{div}_ Y(g) = \sum m_{X', X} (X' \to X)_*\text{div}_{X'}(g \circ f|_{X'}) \]

as $(k + r - 1)$-cycles on $X$ where the sum is over the irreducible components $X'$ of $X$ and $m_{X', X}$ is the multiplicity of $X'$ in $X$.

Proof. Observe that any irreducible component of $X$ dominates $Y$ (Lemma 80.9.1) and hence the composition $g \circ f|_{X'}$ is defined (Morphisms of Spaces, Section 65.47). We will reduce this to the case of schemes. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Picture

\[ \xymatrix{ U \ar[r]_ a \ar[d]_ h & X \ar[d]^ f \\ V \ar[r]^ b & Y } \]

Since $a$ is surjective and étale it follows from Lemma 80.10.3 that it suffices to prove the equality of cycles after pulling back by $a$. We can use Lemma 80.13.2 to write

\[ b^*\text{div}_ Y(g) = \sum (V' \to V)_*\text{div}_{V'}(g \circ b|_{V'}) \]

where the sum is over the irreducible components $V'$ of $V$. Using Lemma 80.11.1 we find

\[ h^*b^*\text{div}_ Y(g) = \sum (V' \times _ V U \to U)_*(h')^*\text{div}_{V'}(g \circ b|_{V'}) \]

where $h' : V' \times _ V U \to V'$ is the projection. We may apply the lemma in the case of schemes (Chow Homology, Lemma 42.20.1) to the morphism $h' : V' \times _ V U \to V'$ to see that we have

\[ (h')^*\text{div}_{V'}(g \circ b|_{V'}) = \sum m_{U', V' \times _ V U} (U' \to V' \times _ V U)_*\text{div}_{U'}(g \circ b|_{V'} \circ h'|_{U'}) \]

where the sum is over the irreducible components $U'$ of $V' \times _ V U$. Each $U'$ occurring in this sum is an irreducible component of $U$ and conversely every irreducible component $U'$ of $U$ is an irreducible component of $V' \times _ V U$ for a unique irreducible component $V' \subset V$. Given an irreducible component $U' \subset U$, denote $\overline{a(U')} \subset X$ the “image” in $X$ (Lemma 80.7.1); this is an irreducible component of $X$ for example by Lemma 80.9.1. The muplticity $m_{U', V' \times _ V U}$ is equal to the multiplicity $m_{\overline{a(U')}, X}$. This follows from the equality $h^*a^*[Y] = b^*f^*[Y]$ (Lemma 80.10.4), the definitions, and Lemma 80.10.3. Combining all of what we just said we obtain

\[ a^*f^*\text{div}_ Y(g) = h^*b^*\text{div}_ Y(g) = \sum m_{\overline{a(U')}, X} (U' \to U)_*\text{div}_{U'}(g \circ (f \circ a)|_{U'}) \]

Next, we analyze what happens with the right hand side of the formula in the statement of the lemma if we pullback by $a$. First, we use Lemma 80.11.1 to get

\[ a^*\sum m_{X', X} (X' \to X)_*\text{div}_{X'}(g \circ f|_{X'}) = \sum m_{X', X} (X' \times _ X U \to U)_*(a')^*\text{div}_{X'}(g \circ f|_{X'}) \]

where $a' : X' \times _ X U \to X'$ is the projection. By Lemma 80.13.2 we get

\[ (a')^*\text{div}_{X'}(g \circ f|_{X'}) = \sum (U' \to X' \times _ X U)_*\text{div}_{U'}(g \circ (f \circ a)|_{U'}) \]

where the sum is over the irreducible components $U'$ of $X' \times _ X U$. These $U'$ are irreducible components of $U$ and in fact are exactly the irreducible components of $U$ such that $\overline{a(U')} = X'$. Comparing with what we obtained above we conclude. $\square$

Lemma 80.16.2. In Situation 80.2.1 let $X, Y/B$ be good. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\alpha \sim _{rat} \beta $ be rationally equivalent $k$-cycles on $Y$. Then $f^*\alpha \sim _{rat} f^*\beta $ as $(k + r)$-cycles on $X$.

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

\[ i_ j : W_ j \longrightarrow Y \]

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

\[ f^*(\sum i_{j, *}\text{div}(g_ j)) = \sum f^*i_{j, *}\text{div}(g_ j) \]

is rationally equivalent to zero on $X$. The sum on the right makes sense by Lemma 80.9.2.

Consider the fibre products

\[ i'_ j : W'_ j = W_ j \times _ Y X \longrightarrow X. \]

and denote $f_ j : W'_ j \to W_ j$ the first projection. By Lemma 80.11.1 we can write the sum above as

\[ \sum i'_{j, *}(f_ j^*\text{div}(g_ j)) \]

By Lemma 80.16.1 we see that each $f_ j^*\text{div}(g_ j)$ is rationally equivalent to zero on $W'_ j$. Hence each $i'_{j, *}(f_ j^*\text{div}(g_ j))$ is rationally equivalent to zero. Then the same is true for the displayed sum by the discussion in Remark 80.15.3. $\square$

Lemma 80.16.3. In Situation 80.2.1 let $X, Y/B$ be good. 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 subspace which is the image of $p \circ i_ j$, see Lemma 80.7.1. The collection $\{ W'_ j\} $ is locally finite in $Y$ by Lemma 80.7.5. 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 subspace $W \subset X$ 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 80.8.2. 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. Since $(p')_*\text{div}(f)$ is a $k$-cycle, we see that $(p')_*\text{div}(f) = n[W']$ for some $n \in \mathbf{Z}$. In order to prove that $n = 0$ we may replace $W'$ by a nonempty open subspace. In particular, we may and do assume that $W'$ is a scheme. Let $\eta \in W'$ be the generic point. Let $K = \kappa (\eta ) = R(W')$ be the function field. Consider the base change diagram

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

Observe that $c$ is proper. Also $|W_\eta |$ has dimension $1$: use Decent Spaces, Lemma 66.18.6 to identify $|W_\eta |$ as the subspace of $|W|$ of points mapping to $\eta $ and note that since $\dim _\delta (W) = k + 1$ and $\delta (\eta ) = k$ points of $W_\eta $ must have $\delta $-value either $k$ or $k + 1$. Thus the local rings have dimension $\leq 1$ by Decent Spaces, Lemma 66.20.2. By Spaces over Fields, Lemma 70.9.3 we find that $W_\eta $ is a scheme. Since $\mathop{\mathrm{Spec}}(K)$ is the limit of the nonempty affine open subschemes of $W'$ we conclude that we may assume that $W$ is a scheme by Limits of Spaces, Lemma 68.5.11. Then finally by the case of schemes (Chow Homology, Lemma 42.20.3) we find that $n = 0$.

The case $\dim _\delta (W') = k + 1$. In this case Lemma 80.14.1 applies, and we see that indeed $p'_*\text{div}(f) = \text{div}(g)$ for some $g \in R(W')^*$ as desired. $\square$


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