The Stacks project

42.25 Intersecting with an invertible sheaf and push and pull

In this section we prove that the operation $c_1(\mathcal{L}) \cap -$ commutes with flat pullback and proper pushforward.

Lemma 42.25.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Assume $Y$ is integral and $n = \dim _\delta (Y)$. Let $s$ be a nonzero meromorphic section of $\mathcal{L}$. Then we have

\[ f^*\text{div}_\mathcal {L}(s) = \sum n_ i\text{div}_{f^*\mathcal{L}|_{X_ i}}(s_ i) \]

in $Z_{n + r - 1}(X)$. Here the sum is over the irreducible components $X_ i \subset X$ of $\delta $-dimension $n + r$, the section $s_ i = f|_{X_ i}^*(s)$ is the pullback of $s$, and $n_ i = m_{X_ i, X}$ is the multiplicity of $X_ i$ in $X$.

Proof. To prove this equality of cycles, we may work locally on $Y$. Hence we may assume $Y$ is affine and $s = p/q$ for some nonzero sections $p \in \Gamma (Y, \mathcal{L})$ and $q \in \Gamma (Y, \mathcal{O})$. If we can show both

\[ f^*\text{div}_\mathcal {L}(p) = \sum n_ i\text{div}_{f^*\mathcal{L}|_{X_ i}}(p_ i) \quad \text{and}\quad f^*\text{div}_\mathcal {O}(q) = \sum n_ i\text{div}_{\mathcal{O}_{X_ i}}(q_ i) \]

(with obvious notations) then we win by the additivity, see Divisors, Lemma 31.27.5. Thus we may assume that $s \in \Gamma (Y, \mathcal{L})$. In this case we may apply the equality (42.24.3.1) to see that

\[ [Z(f^*(s))]_{k + r - 1} = \sum n_ i\text{div}_{f^*\mathcal{L}|_{X_ i}}(s_ i) \]

where $f^*(s) \in f^*\mathcal{L}$ denotes the pullback of $s$ to $X$. On the other hand we have

\[ f^*\text{div}_\mathcal {L}(s) = f^*[Z(s)]_{k - 1} = [f^{-1}(Z(s))]_{k + r - 1}, \]

by Lemmas 42.23.2 and 42.14.4. Since $Z(f^*(s)) = f^{-1}(Z(s))$ we win. $\square$

Lemma 42.25.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Let $\alpha $ be a $k$-cycle on $Y$. Then

\[ f^*(c_1(\mathcal{L}) \cap \alpha ) = c_1(f^*\mathcal{L}) \cap f^*\alpha \]

in $\mathop{\mathrm{CH}}\nolimits _{k + r - 1}(X)$.

Proof. Write $\alpha = \sum n_ i[W_ i]$. We will show that

\[ f^*(c_1(\mathcal{L}) \cap [W_ i]) = c_1(f^*\mathcal{L}) \cap f^*[W_ i] \]

in $\mathop{\mathrm{CH}}\nolimits _{k + r - 1}(X)$ by producing a rational equivalence on the closed subscheme $f^{-1}(W_ i)$ of $X$. By the discussion in Remark 42.19.5 this will prove the equality of the lemma is true.

Let $W \subset Y$ be an integral closed subscheme of $\delta $-dimension $k$. Consider the closed subscheme $W' = f^{-1}(W) = W \times _ Y X$ so that we have the fibre product diagram

\[ \xymatrix{ W' \ar[r] \ar[d]_ h & X \ar[d]^ f \\ W \ar[r] & Y } \]

We have to show that $f^*(c_1(\mathcal{L}) \cap [W]) = c_1(f^*\mathcal{L}) \cap f^*[W]$. Choose a nonzero meromorphic section $s$ of $\mathcal{L}|_ W$. Let $W'_ i \subset W'$ be the irreducible components of $\delta $-dimension $k + r$. Write $[W']_{k + r} = \sum n_ i[W'_ i]$ with $n_ i$ the multiplicity of $W'_ i$ in $W'$ as per definition. So $f^*[W] = \sum n_ i[W'_ i]$ in $Z_{k + r}(X)$. Since each $W'_ i \to W$ is dominant we see that $s_ i = s|_{W'_ i}$ is a nonzero meromorphic section for each $i$. By Lemma 42.25.1 we have the following equality of cycles

\[ h^*\text{div}_{\mathcal{L}|_ W}(s) = \sum n_ i\text{div}_{f^*\mathcal{L}|_{W'_ i}}(s_ i) \]

in $Z_{k + r - 1}(W')$. This finishes the proof since the left hand side is a cycle on $W'$ which pushes to $f^*(c_1(\mathcal{L}) \cap [W])$ in $\mathop{\mathrm{CH}}\nolimits _{k + r - 1}(X)$ and the right hand side is a cycle on $W'$ which pushes to $c_1(f^*\mathcal{L}) \cap f^*[W]$ in $\mathop{\mathrm{CH}}\nolimits _{k + r - 1}(X)$. $\square$

Lemma 42.25.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a proper morphism. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Let $s$ be a nonzero meromorphic section $s$ of $\mathcal{L}$ on $Y$. Assume $X$, $Y$ integral, $f$ dominant, and $\dim _\delta (X) = \dim _\delta (Y)$. Then

\[ f_*\left(\text{div}_{f^*\mathcal{L}}(f^*s)\right) = [R(X) : R(Y)]\text{div}_\mathcal {L}(s). \]

as cycles on $Y$. In particular

\[ f_*(c_1(f^*\mathcal{L}) \cap [X]) = c_1(\mathcal{L}) \cap [Y]. \]

Proof. The last equation follows from the first since $f_*[X] = [R(X) : R(Y)][Y]$ by definition. It turns out that we can re-use Lemma 42.18.1 to prove this. Namely, since we are trying to prove an equality of cycles, we may work locally on $Y$. Hence we may assume that $\mathcal{L} = \mathcal{O}_ Y$. In this case $s$ corresponds to a rational function $g \in R(Y)$, and we are simply trying to prove

\[ f_*\left(\text{div}_ X(g)\right) = [R(X) : R(Y)]\text{div}_ Y(g). \]

Comparing with the result of the aforementioned Lemma 42.18.1 we see this true since $\text{Nm}_{R(X)/R(Y)}(g) = g^{[R(X) : R(Y)]}$ as $g \in R(Y)^*$. $\square$

Lemma 42.25.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $p : X \to Y$ be a proper morphism. Let $\alpha \in Z_{k + 1}(X)$. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Then

\[ p_*(c_1(p^*\mathcal{L}) \cap \alpha ) = c_1(\mathcal{L}) \cap p_*\alpha \]

in $\mathop{\mathrm{CH}}\nolimits _ k(Y)$.

Proof. Suppose that $p$ has the property that for every integral closed subscheme $W \subset X$ the map $p|_ W : W \to Y$ is a closed immersion. Then, by definition of capping with $c_1(\mathcal{L})$ the lemma holds.

We will use this remark to reduce to a special case. Namely, write $\alpha = \sum n_ i[W_ i]$ with $n_ i \not= 0$ and $W_ i$ pairwise distinct. Let $W'_ i \subset Y$ be the image of $W_ i$ (as an integral closed subscheme). Consider the diagram

\[ \xymatrix{ X' = \coprod W_ i \ar[r]_-q \ar[d]_{p'} & X \ar[d]^ p \\ Y' = \coprod W'_ i \ar[r]^-{q'} & Y. } \]

Since $\{ W_ i\} $ is locally finite on $X$, and $p$ is proper we see that $\{ W'_ i\} $ is locally finite on $Y$ and that $q, q', p'$ are also proper morphisms. We may think of $\sum n_ i[W_ i]$ also as a $k$-cycle $\alpha ' \in Z_ k(X')$. Clearly $q_*\alpha ' = \alpha $. We have $q_*(c_1(q^*p^*\mathcal{L}) \cap \alpha ') = c_1(p^*\mathcal{L}) \cap q_*\alpha '$ and $(q')_*(c_1((q')^*\mathcal{L}) \cap p'_*\alpha ') = c_1(\mathcal{L}) \cap q'_*p'_*\alpha '$ by the initial remark of the proof. Hence it suffices to prove the lemma for the morphism $p'$ and the cycle $\sum n_ i[W_ i]$. Clearly, this means we may assume $X$, $Y$ integral, $f : X \to Y$ dominant and $\alpha = [X]$. In this case the result follows from Lemma 42.25.3. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AYA. Beware of the difference between the letter 'O' and the digit '0'.