Section 42.29 (02T7): Gysin homomorphisms

42.29 Gysin homomorphisms

In this section we define the gysin map for the zero locus $D$ of a section of an invertible sheaf. An interesting case occurs when $D$ is an effective Cartier divisor, but the generalization to arbitrary $D$ allows us a flexibility to formulate various compatibilities, see Remark 42.29.7 and Lemmas 42.29.8, 42.29.9, and 42.30.5. These results can be generalized to locally principal closed subschemes endowed with a virtual normal bundle (Remark 42.29.2) or to pseudo-divisors (Remark 42.29.3).

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 regular global section, see Divisors, Lemma 31.14.10. If $D$ corresponds to $(\mathcal{L}, s)$ , then $\mathcal{L} = \mathcal{O}_ X(D)$ . Please keep this in mind while reading this section.

Definition 42.29.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$ . 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 zero scheme of $s$ , and denote $i : D \to X$ the closed immersion. We define, for every integer $k$ , a Gysin homomorphism

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

by the following rules:

Given a integral closed subscheme $W \subset X$ 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.
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 $i^*\alpha$ 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 42.29.2. Let $X$ be a scheme locally of finite type over $S$ as in Situation 42.7.1. Let $(D, \mathcal{N}, \sigma )$ be a triple consisting of a locally principal (Divisors, Definition 31.13.1) closed subscheme $i : D \to X$ , an invertible $\mathcal{O}_ D$ -module $\mathcal{N}$ , and a surjection $\sigma : \mathcal{N}^{\otimes -1} \to i^*\mathcal{I}_ D$ of $\mathcal{O}_ D$ -modules ¹. Here $\mathcal{N}$ should be thought of as a virtual normal bundle of $D$ in $X$ . The construction of $i^* : Z_{k + 1}(X) \to \mathop{\mathrm{CH}}\nolimits _ k(D)$ in Definition 42.29.1 generalizes to such triples, see Section 42.54.

Remark 42.29.3. Let $X$ be a scheme locally of finite type over $S$ as in Situation 42.7.1. In a pseudo-divisor on $X$ is defined as a triple $D = (\mathcal{L}, Z, s)$ where $\mathcal{L}$ is an invertible $\mathcal{O}_ X$ -module, $Z \subset X$ is a closed subset, and $s \in \Gamma (X \setminus Z, \mathcal{L})$ is a nowhere vanishing section. Similarly to the above, one can define for every $\alpha$ in $\mathop{\mathrm{CH}}\nolimits _{k + 1}(X)$ a product $D \cdot \alpha$ in $\mathop{\mathrm{CH}}\nolimits _ k(Z \cap |\alpha |)$ where $|\alpha |$ is the support of $\alpha$ .

Lemma 42.29.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$ . Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 42.29.1. 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 subschemes with $\dim _\delta (W_ j) = k$ . Since $D$ is the zero scheme of $s$ we see that $D \cap W_ j$ is the zero scheme 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 42.25.4. 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 42.25.1. The right hand side matches (termwise) the pushforward of the class $i^*\alpha$ on $D$ from Definition 42.29.1. Hence we win. $\square$

Lemma 42.29.5. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$ . Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 42.29.1.

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 $i^*[Z]_{k + 1} = [D \cap Z]_ k$ .
Let $\mathcal{F}$ be a coherent sheaf on $X$ such that $\dim _\delta (\text{Supp}(\mathcal{F})) \leq k + 1$ and $s : \mathcal{F} \to \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}$ is injective. Then

$i^*[\mathcal{F}]_{k + 1} = [i^*\mathcal{F}]_ k$

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

Proof. Assume $Z \subset X$ as in (1). Then set $\mathcal{F} = \mathcal{O}_ Z$ . The assumption that $D \cap Z$ is an effective Cartier divisor is equivalent to the assumption that $s : \mathcal{F} \to \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}$ is injective. Moreover $[Z]_{k + 1} = [\mathcal{F}]_{k + 1}]$ and $[D \cap Z]_ k = [\mathcal{O}_{D \cap Z}]_ k = [i^*\mathcal{F}]_ k$ . See Lemma 42.10.3. Hence part (1) follows from part (2).

Write $[\mathcal{F}]_{k + 1} = \sum m_ j[W_ j]$ with $m_ j > 0$ and pairwise distinct integral closed subschemes $W_ j \subset X$ of $\delta$ -dimension $k + 1$ . The assumption that $s : \mathcal{F} \to \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}$ is injective implies that $W_ j \not\subset D$ for all $j$ . By definition we see that

$i^*[\mathcal{F}]_{k + 1} = \sum m_ j [D \cap W_ j]_ k.$

We claim that

$\sum [D \cap W_ j]_ k = [i^*\mathcal{F}]_ k$

as cycles. Let $Z \subset D$ be an integral closed subscheme of $\delta$ -dimension $k$ . Let $\xi \in Z$ be its generic point. Let $A = \mathcal{O}_{X, \xi }$ . Let $M = \mathcal{F}_\xi$ . Let $f \in A$ be an element generating the ideal of $D$ , i.e., such that $\mathcal{O}_{D, \xi } = A/fA$ . By assumption $\dim (\text{Supp}(M)) = 1$ , the map $f : M \to M$ is injective, and $\text{length}_ A(M/fM) < \infty$ . Moreover, $\text{length}_ A(M/fM)$ is the coefficient of $[Z]$ in $[i^*\mathcal{F}]_ k$ . On the other hand, let $\mathfrak q_1, \ldots , \mathfrak q_ t$ be the minimal primes in the support of $M$ . Then

$\sum \text{length}_{A_{\mathfrak q_ i}}(M_{\mathfrak q_ i}) \text{ord}_{A/\mathfrak q_ i}(f)$

is the coefficient of $[Z]$ in $\sum [D \cap W_ j]_ k$ . Hence we see the equality by Lemma 42.3.2. $\square$

Remark 42.29.6. Let $X \to S$ , $\mathcal{L}$ , $s$ , $i : D \to X$ be as in Definition 42.29.1 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 $W \subset D$ is an integral closed subscheme. The possibility to do this will be useful later on.

Remark 42.29.7. Let $f : X' \to X$ be a morphism of schemes locally of finite type over $S$ as in Situation 42.7.1. Let $(\mathcal{L}, s, i : D \to X)$ be a triple as in Definition 42.29.1. 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 42.29.8. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X' \to X$ be a proper morphism of schemes locally of finite type over $S$ . Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 42.29.1. Form the diagram

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

as in Remark 42.29.7. 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 subscheme $W' \subset X'$ . Let $W = f(W') \subset X$ . 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 42.26.3. 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 42.26.3. By Remark 42.19.6 the result follows for general $\alpha '$ . $\square$

Lemma 42.29.9. Let $(S, \delta )$ be as in Situation 42.7.1. Let $f : X' \to X$ be a flat morphism of relative dimension $r$ of schemes locally of finite type over $S$ . Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 42.29.1. Form the diagram

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

as in Remark 42.29.7. 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 subscheme $W \subset X$ . 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 subschemes of $D'$ . Hence the equality holds as cycles, see Lemma 42.14.4. 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 42.26.2. By Remark 42.19.6 the result follows for general $\alpha '$ . $\square$

[1] This condition assures us that if

$D$ is an effective Cartier divisor, then

$\mathcal{N} = \mathcal{O}_ X(D)|_ D$ .

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42.29 Gysin homomorphisms

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