The Stacks project

80.22 Intersecting with effective Cartier divisors

This section is the analogue of Chow Homology, Section 42.28. Please read the introduction of that section we motivation.

Recall that effective Cartier divisors correspond $1$-to-$1$ to isomorphism classes of pairs $(\mathcal{L}, s)$ where $\mathcal{L}$ is an invertible sheaf and $s$ is a global section, see Divisors on Spaces, Lemma 69.7.8. If $D$ corresponds to $(\mathcal{L}, s)$, then $\mathcal{L} = \mathcal{O}_ X(D)$. Please keep this in mind while reading this section.

Definition 80.22.1. In Situation 80.2.1 let $X/B$ be good. Let $(\mathcal{L}, s)$ be a pair consisting of an invertible sheaf and a global section $s \in \Gamma (X, \mathcal{L})$. Let $D = Z(s)$ be the vanishing locus of $s$, and denote $i : D \to X$ the closed immersion. We define, for every integer $k$, a (refined) Gysin homomorphism

\[ i^* : Z_{k + 1}(X) \to \mathop{\mathrm{CH}}\nolimits _ k(D). \]

by the following rules:

  1. Given a integral closed subspace $W \subset X$ with $\dim _\delta (W) = k + 1$ we define

    1. if $W \not\subset D$, then $i^*[W] = [D \cap W]_ k$ as a $k$-cycle on $D$, and

    2. if $W \subset D$, then $i^*[W] = i'_*(c_1(\mathcal{L}|_ W) \cap [W])$, where $i' : W \to D$ is the induced closed immersion.

  2. For a general $(k + 1)$-cycle $\alpha = \sum n_ j[W_ j]$ we set

    \[ i^*\alpha = \sum n_ j i^*[W_ j] \]
  3. If $D$ is an effective Cartier divisor, then we denote $D \cdot \alpha = i_*i^*\alpha $ the pushforward of the class to a class on $X$.

In fact, as we will see later, this Gysin homomorphism $i^*$ can be viewed as an example of a non-flat pullback. Thus we will sometimes informally call the class $i^*\alpha $ the pullback of the class $\alpha $.

Remark 80.22.2. Let $S$, $B$, $X$, $\mathcal{L}$, $s$, $i : D \to X$ be as in Definition 80.22.1 and assume that $\mathcal{L}|_ D \cong \mathcal{O}_ D$. In this case we can define a canonical map $i^* : Z_{k + 1}(X) \to Z_ k(D)$ on cycles, by requiring that $i^*[W] = 0$ whenever $W \subset D$. The possibility to do this will be useful later on.

Remark 80.22.3. Let $f : X' \to X$ be a morphism of good algebraic spaces over $B$ as in Situation 80.2.1. Let $(\mathcal{L}, s, i : D \to X)$ be a triple as in Definition 80.22.1. Then we can set $\mathcal{L}' = f^*\mathcal{L}$, $s' = f^*s$, and $D' = X' \times _ X D = Z(s')$. This gives a commutative diagram

\[ \xymatrix{ D' \ar[d]_ g \ar[r]_{i'} & X' \ar[d]^ f \\ D \ar[r]^ i & X } \]

and we can ask for various compatibilities between $i^*$ and $(i')^*$.

Lemma 80.22.4. In Situation 80.2.1 let $X/B$ be good. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 80.22.1. Let $\alpha $ be a $(k + 1)$-cycle on $X$. Then $i_*i^*\alpha = c_1(\mathcal{L}) \cap \alpha $ in $\mathop{\mathrm{CH}}\nolimits _ k(X)$. In particular, if $D$ is an effective Cartier divisor, then $D \cdot \alpha = c_1(\mathcal{O}_ X(D)) \cap \alpha $.

Proof. Write $\alpha = \sum n_ j[W_ j]$ where $i_ j : W_ j \to X$ are integral closed subspaces with $\dim _\delta (W_ j) = k$. Since $D$ is the vanishing locus of $s$ we see that $D \cap W_ j$ is the vanishing locus of the restriction $s|_{W_ j}$. Hence for each $j$ such that $W_ j \not\subset D$ we have $c_1(\mathcal{L}) \cap [W_ j] = [D \cap W_ j]_ k$ by Lemma 80.18.4. So we have

\[ c_1(\mathcal{L}) \cap \alpha = \sum \nolimits _{W_ j \not\subset D} n_ j[D \cap W_ j]_ k + \sum \nolimits _{W_ j \subset D} n_ j i_{j, *}(c_1(\mathcal{L})|_{W_ j}) \cap [W_ j]) \]

in $\mathop{\mathrm{CH}}\nolimits _ k(X)$ by Definition 80.18.1. The right hand side matches (termwise) the pushforward of the class $i^*\alpha $ on $D$ from Definition 80.22.1. Hence we win. $\square$

Lemma 80.22.5. In Situation 80.2.1. Let $f : X' \to X$ be a proper morphism of good algebraic spaces over $B$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 80.22.1. Form the diagram

\[ \xymatrix{ D' \ar[d]_ g \ar[r]_{i'} & X' \ar[d]^ f \\ D \ar[r]^ i & X } \]

as in Remark 80.22.3. For any $(k + 1)$-cycle $\alpha '$ on $X'$ we have $i^*f_*\alpha ' = g_*(i')^*\alpha '$ in $\mathop{\mathrm{CH}}\nolimits _ k(D)$ (this makes sense as $f_*$ is defined on the level of cycles).

Proof. Suppose $\alpha = [W']$ for some integral closed subspace $W' \subset X'$. Let $W \subset X$ be the “image” of $W'$ as in Lemma 80.7.1. In case $W' \not\subset D'$, then $W \not\subset D$ and we see that

\[ [W' \cap D']_ k = \text{div}_{\mathcal{L}'|_{W'}}({s'|_{W'}}) \quad \text{and}\quad [W \cap D]_ k = \text{div}_{\mathcal{L}|_ W}(s|_ W) \]

and hence $f_*$ of the first cycle equals the second cycle by Lemma 80.19.3. Hence the equality holds as cycles. In case $W' \subset D'$, then $W \subset D$ and $f_*(c_1(\mathcal{L}|_{W'}) \cap [W'])$ is equal to $c_1(\mathcal{L}|_ W) \cap [W]$ in $\mathop{\mathrm{CH}}\nolimits _ k(W)$ by the second assertion of Lemma 80.19.3. By Remark 80.15.3 the result follows for general $\alpha '$. $\square$

Lemma 80.22.6. In Situation 80.2.1. Let $f : X' \to X$ be a flat morphism of relative dimension $r$ of good algebraic spaces over $B$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 80.22.1. Form the diagram

\[ \xymatrix{ D' \ar[d]_ g \ar[r]_{i'} & X' \ar[d]^ f \\ D \ar[r]^ i & X } \]

as in Remark 80.22.3. For any $(k + 1)$-cycle $\alpha $ on $X$ we have $(i')^*f^*\alpha = g^*i^*\alpha '$ in $\mathop{\mathrm{CH}}\nolimits _{k + r}(D)$ (this makes sense as $f^*$ is defined on the level of cycles).

Proof. Suppose $\alpha = [W]$ for some integral closed subspace $W \subset X$. Let $W' = f^{-1}(W) \subset X'$. In case $W \not\subset D$, then $W' \not\subset D'$ and we see that

\[ W' \cap D' = g^{-1}(W \cap D) \]

as closed subspaces of $D'$. Hence the equality holds as cycles, see Lemma 80.10.5. In case $W \subset D$, then $W' \subset D'$ and $W' = g^{-1}(W)$ with $[W']_{k + 1 + r} = g^*[W]$ and equality holds in $\mathop{\mathrm{CH}}\nolimits _{k + r}(D')$ by Lemma 80.19.2. By Remark 80.15.3 the result follows for general $\alpha '$. $\square$

Lemma 80.22.7. In Situation 80.2.1 let $X/B$ be good. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 80.22.1. Let $Z \subset X$ be a closed subscheme such that $\dim _\delta (Z) \leq k + 1$ and such that $D \cap Z$ is an effective Cartier divisor on $Z$. Then $i^*([Z]_{k + 1}) = [D \cap Z]_ k$.

Proof. The assumption means that $s|_ Z$ is a regular section of $\mathcal{L}|_ Z$. Thus $D \cap Z = Z(s)$ and we get

\[ [D \cap Z]_ k = \sum n_ i [Z(s_ i)]_ k \]

as cycles where $s_ i = s|_{Z_ i}$, the $Z_ i$ are the irreducible components of $\delta $-dimension $k + 1$, and $[Z]_{k + 1} = \sum n_ i[Z_ i]$. See Lemma 80.18.3. We have $D \cap Z_ i = Z(s_ i)$. Comparing with the definition of the gysin map we conclude. $\square$


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 0ER0. Beware of the difference between the letter 'O' and the digit '0'.