The Stacks project

81.16 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 81.16.1. In Situation 81.2.1 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 81.9.1) and hence the composition $g \circ f|_{X'}$ is defined (Morphisms of Spaces, Section 66.47). 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 81.10.3 that it suffices to prove the equality of cycles after pulling back by $a$. We can use Lemma 81.13.2 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 81.11.1 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 81.7.1); this is an irreducible component of $X$ for example by Lemma 81.9.1. 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 81.10.4), the definitions, and Lemma 81.10.3. 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 81.11.1 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 81.13.2 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 81.16.2. In Situation 81.2.1 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 81.9.2.

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 81.11.1 we can write the sum above as

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

By Lemma 81.16.1 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 81.15.3. $\square$

Lemma 81.16.3. In Situation 81.2.1 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 81.7.1. The collection $\{ W'_ j\} $ is locally finite in $Y$ by Lemma 81.7.5. 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 81.8.2. 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 67.18.6 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 67.20.2. By Spaces over Fields, Lemma 71.9.3 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 69.5.11. 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 81.14.1 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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