81.22 Intersecting with effective Cartier divisors
This section is the analogue of Chow Homology, Section 42.29. 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 70.7.8. If $D$ corresponds to $(\mathcal{L}, s)$, then $\mathcal{L} = \mathcal{O}_ X(D)$. Please keep this in mind while reading this section.
Definition 81.22.1. In Situation 81.2.1 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:
Given a integral closed subspace $W \subset X$ with $\dim _\delta (W) = k + 1$ we define
if $W \not\subset D$, then $i^*[W] = [D \cap W]_ k$ as a $k$-cycle on $D$, and
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.
For a general $(k + 1)$-cycle $\alpha = \sum n_ j[W_ j]$ we set
\[ i^*\alpha = \sum n_ j i^*[W_ j] \]
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 $.
Lemma 81.22.4. In Situation 81.2.1 let $X/B$ be good. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 81.22.1. 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 81.18.4. 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 81.18.1. The right hand side matches (termwise) the pushforward of the class $i^*\alpha $ on $D$ from Definition 81.22.1. Hence we win.
$\square$
Lemma 81.22.5. In Situation 81.2.1. 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 81.22.1. Form the diagram
\[ \xymatrix{ D' \ar[d]_ g \ar[r]_{i'} & X' \ar[d]^ f \\ D \ar[r]^ i & X } \]
as in Remark 81.22.3. 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 81.7.1. 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 81.19.3. 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 81.19.3. By Remark 81.15.3 the result follows for general $\alpha '$.
$\square$
Lemma 81.22.6. In Situation 81.2.1. 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 81.22.1. Form the diagram
\[ \xymatrix{ D' \ar[d]_ g \ar[r]_{i'} & X' \ar[d]^ f \\ D \ar[r]^ i & X } \]
as in Remark 81.22.3. 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 81.10.5. 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 81.19.2. By Remark 81.15.3 the result follows for general $\alpha '$.
$\square$
Lemma 81.22.7. In Situation 81.2.1 let $X/B$ be good. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 81.22.1. 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 81.18.3. We have $D \cap Z_ i = Z(s_ i)$. Comparing with the definition of the gysin map we conclude.
$\square$
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