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
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 $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 $.
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.
$\square$
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 $\alpha '$.
$\square$
Comments (0)