82.22 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 71.7.8. If D corresponds to (\mathcal{L}, s), then \mathcal{L} = \mathcal{O}_ X(D). Please keep this in mind while reading this section.
Definition 82.22.1. In Situation 82.2.1 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 82.22.4. In Situation 82.2.1 let X/B be good. Let (\mathcal{L}, s, i : D \to X) be as in Definition 82.22.1. 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 82.18.4. 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 82.18.1. The right hand side matches (termwise) the pushforward of the class i^*\alpha on D from Definition 82.22.1. Hence we win.
\square
Lemma 82.22.5. In Situation 82.2.1. 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 82.22.1. Form the diagram
\xymatrix{ D' \ar[d]_ g \ar[r]_{i'} & X' \ar[d]^ f \\ D \ar[r]^ i & X }
as in Remark 82.22.3. 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 82.7.1. 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 82.19.3. 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 82.19.3. By Remark 82.15.3 the result follows for general \alpha '.
\square
Lemma 82.22.6. In Situation 82.2.1. 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 82.22.1. Form the diagram
\xymatrix{ D' \ar[d]_ g \ar[r]_{i'} & X' \ar[d]^ f \\ D \ar[r]^ i & X }
as in Remark 82.22.3. 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 82.10.5. 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 82.19.2. By Remark 82.15.3 the result follows for general \alpha '.
\square
Lemma 82.22.7. In Situation 82.2.1 let X/B be good. Let (\mathcal{L}, s, i : D \to X) be as in Definition 82.22.1. 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 82.18.3. We have D \cap Z_ i = Z(s_ i). Comparing with the definition of the gysin map we conclude.
\square
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