## 50.23 Gysin maps for closed immersions

In this section we define the gysin map for closed immersions.

Remark 50.23.1. Let $X \to S$ be a morphism of schemes. Let $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$. Let $Z \subset X$ be the closed subscheme cut out by $f_1, \ldots , f_ c$. Below we will study the gysin map

50.23.1.1
$$\label{derham-equation-gysin} \gamma ^ p_{f_1, \ldots , f_ c} : \Omega ^ p_{Z/S} \longrightarrow \mathcal{H}_ Z^ c(\Omega ^{p + c}_{X/S})$$

defined as follows. Given a local section $\omega$ of $\Omega ^ p_{Z/S}$ which is the restriction of a section $\tilde\omega$ of $\Omega ^ p_{X/S}$ we set

$\gamma ^ p_{f_1, \ldots , f_ c}(\omega ) = c_{f_1, \ldots , f_ c}(\tilde\omega |_ Z) \wedge \text{d}f_1 \wedge \ldots \wedge \text{d}f_ c$

where $c_{f_1, \ldots , f_ c} : \Omega ^ p_{X/S} \otimes \mathcal{O}_ Z \to \mathcal{H}_ Z^ c(\Omega ^ p_{X/S})$ is the map constructed in Derived Categories of Schemes, Remark 36.6.10. This is well defined: given $\omega$ we can change our choice of $\tilde\omega$ by elements of the form $\sum f_ i \omega '_ i + \sum \text{d}(f_ i) \wedge \omega ''_ i$ which are mapped to zero by the construction.

Lemma 50.23.2. The gysin map (50.23.1.1) is compatible with the de Rham differentials on $\Omega ^\bullet _{X/S}$ and $\Omega ^\bullet _{Z/S}$.

Proof. This follows from an almost trivial calculation once we correctly interpret this. First, we recall that the functor $\mathcal{H}^ c_ Z$ computed on the category of $\mathcal{O}_ X$-modules agrees with the similarly defined functor on the category of abelian sheaves on $X$, see Cohomology, Lemma 20.34.8. Hence, the differential $\text{d} : \Omega ^ p_{X/S} \to \Omega ^{p + 1}_{X/S}$ induces a map $\mathcal{H}^ c_ Z(\Omega ^ p_{X/S}) \to \mathcal{H}^ c_ Z(\Omega ^{p + 1}_{X/S})$. Moreover, the formation of the extended alternating Čech complex in Derived Categories of Schemes, Remark 36.6.4 works on the category of abelian sheaves. The map

$\mathop{\mathrm{Coker}}\left(\bigoplus \mathcal{F}_{1 \ldots \hat i \ldots c} \to \mathcal{F}_{1 \ldots c}\right) \longrightarrow i_*\mathcal{H}^ c_ Z(\mathcal{F})$

used in the construction of $c_{f_1, \ldots , f_ c}$ in Derived Categories of Schemes, Remark 36.6.10 is well defined and functorial on the category of all abelian sheaves on $X$. Hence we see that the lemma follows from the equality

$\text{d}\left( \frac{\tilde\omega \wedge \text{d}f_1 \wedge \ldots \wedge \text{d}f_ c}{f_1 \ldots f_ c}\right) = \frac{\text{d}(\tilde\omega ) \wedge \text{d}f_1 \wedge \ldots \wedge \text{d}f_ c}{f_1 \ldots f_ c}$

which is clear. $\square$

Lemma 50.23.3. Let $X \to S$ be a morphism of schemes. Let $Z \to X$ be a closed immersion of finite presentation whose conormal sheaf $\mathcal{C}_{Z/X}$ is locally free of rank $c$. Then there is a canonical map

$\gamma ^ p : \Omega ^ p_{Z/S} \to \mathcal{H}^ c_ Z(\Omega ^{p + c}_{X/S})$

which is locally given by the maps $\gamma ^ p_{f_1, \ldots , f_ c}$ of Remark 50.23.1.

Proof. The assumptions imply that given $x \in Z \subset X$ there exists an open neighbourhood $U$ of $x$ such that $Z$ is cut out by $c$ elements $f_1, \ldots , f_ c \in \mathcal{O}_ X(U)$. Thus it suffices to show that given $f_1, \ldots , f_ c$ and $g_1, \ldots , g_ c$ in $\mathcal{O}_ X(U)$ cutting out $Z \cap U$, the maps $\gamma ^ p_{f_1, \ldots , f_ c}$ and $\gamma ^ p_{g_1, \ldots , g_ c}$ are the same. To do this, after shrinking $U$ we may assume $g_ j = \sum a_{ji} f_ i$ for some $a_{ji} \in \mathcal{O}_ X(U)$. Then we have $c_{f_1, \ldots , f_ c} = \det (a_{ji}) c_{g_1, \ldots , g_ c}$ by Derived Categories of Schemes, Lemma 36.6.12. On the other hand we have

$\text{d}(g_1) \wedge \ldots \wedge \text{d}(g_ c) \equiv \det (a_{ji}) \text{d}(f_1) \wedge \ldots \wedge \text{d}(f_ c) \bmod (f_1, \ldots , f_ c)\Omega ^ c_{X/S}$

Combining these relations, a straightforward calculation gives the desired equality. $\square$

Lemma 50.23.4. Let $X \to S$ and $i : Z \to X$ be as in Lemma 50.23.3. The gysin map $\gamma ^ p$ is compatible with the de Rham differentials on $\Omega ^\bullet _{X/S}$ and $\Omega ^\bullet _{Z/S}$.

Proof. We may check this locally and then it follows from Lemma 50.23.2. $\square$

Lemma 50.23.5. Let $X \to S$ and $i : Z \to X$ be as in Lemma 50.23.3. Given $\alpha \in H^ q(X, \Omega ^ p_{X/S})$ we have $\gamma ^ p(\alpha |_ Z) = i^{-1}\alpha \wedge \gamma ^0(1)$ in $H^ q(Z, \mathcal{H}^ c_ Z(\Omega ^{p + c}_{X/S}))$. Please see proof for notation.

Proof. The restriction $\alpha |_ Z$ is the element of $H^ q(Z, \Omega ^ p_{Z/S})$ given by functoriality for Hodge cohomology. Applying functoriality for cohomology using $\gamma ^ p : \Omega ^ p_{Z/S} \to \mathcal{H}^ c_ Z(\Omega ^{p + c}_{X/S})$ we get get $\gamma ^ p(\alpha |_ Z)$ in $H^ q(Z, \mathcal{H}^ c_ Z(\Omega ^{p + c}_{X/S}))$. This explains the left hand side of the formula.

To explain the right hand side, we first pullback by the map of ringed spaces $i : (Z, i^{-1}\mathcal{O}_ X) \to (X, \mathcal{O}_ X)$ to get the element $i^{-1}\alpha \in H^ q(Z, i^{-1}\Omega ^ p_{X/S})$. Let $\gamma ^0(1) \in H^0(Z, \mathcal{H}_ Z^ c(\Omega ^ c_{X/S}))$ be the image of $1 \in H^0(Z, \mathcal{O}_ Z) = H^0(Z, \Omega ^0_{Z/S})$ by $\gamma ^0$. Using cup product we obtain an element

$i^{-1}\alpha \cup \gamma ^0(1) \in H^{q + c}(Z, i^{-1}\Omega ^ p_{X/S} \otimes _{i^{-1}\mathcal{O}_ X} \mathcal{H}^ c_ Z(\Omega ^ c_{X/S}))$

Using Cohomology, Remark 20.34.9 and wedge product there are canonical maps

$i^{-1}\Omega ^ p_{X/S} \otimes _{i^{-1}\mathcal{O}_ X}^\mathbf {L} R\mathcal{H}_ Z(\Omega ^ c_{X/S}) \to R\mathcal{H}_ Z(\Omega ^ p_{X/S} \otimes _{\mathcal{O}_ X}^\mathbf {L} \Omega ^ c_{X/S}) \to R\mathcal{H}_ Z(\Omega ^{p + c}_{X/S})$

By Derived Categories of Schemes, Lemma 36.6.8 the objects $R\mathcal{H}_ Z(\Omega ^ j_{X/S})$ have vanishing cohomology sheaves in degrees $> c$. Hence on cohomology sheaves in degree $c$ we obtain a map

$i^{-1}\Omega ^ p_{X/S} \otimes _{i^{-1}\mathcal{O}_ X} \mathcal{H}^ c_ Z(\Omega ^ c_{X/S}) \longrightarrow \mathcal{H}^ c_ Z(\Omega ^{p + c}_{X/S})$

The expression $i^{-1}\alpha \wedge \gamma ^0(1)$ is the image of the cup product $i^{-1}\alpha \cup \gamma ^0(1)$ by the functoriality of cohomology.

Having explained the content of the formula in this manner, by general properties of cup products (Cohomology, Section 20.31), it now suffices to prove that the diagram

$\xymatrix{ i^{-1}\Omega ^ p_ X \otimes \Omega ^0_ Z \ar[rr]_{\text{id} \otimes \gamma ^0} \ar[d] & & i^{-1}\Omega ^ p_ X \otimes \mathcal{H}^ c_ Z(\Omega ^ c_ X) \ar[d]^\wedge \\ \Omega ^ p_ Z \otimes \Omega ^0_ Z \ar[r]^\wedge & \Omega ^ p_ Z \ar[r]^{\gamma ^ p} & \mathcal{H}^ c_ Z(\Omega ^{p + c}_ X) }$

is commutative in the category of sheaves on $Z$ (with obvious abuse of notation). This boils down to a simple computation for the maps $\gamma ^ j_{f_1, \ldots , f_ c}$ which we omit; in fact these maps are chosen exactly such that this works and such that $1$ maps to $\frac{\text{d}f_1 \wedge \ldots \wedge \text{d}f_ c}{f_1 \ldots f_ c}$. $\square$

Lemma 50.23.6. Let $c \geq 0$ be a integer. Let

$\xymatrix{ Z' \ar[d]_ h \ar[r] & X' \ar[d]_ g \ar[r] & S' \ar[d] \\ Z \ar[r] & X \ar[r] & S }$

be a commutative diagram of schemes. Assume

1. $Z \to X$ and $Z' \to X'$ satisfy the assumptions of Lemma 50.23.3,

2. the left square in the diagram is cartesian, and

3. $h^*\mathcal{C}_{Z/X} \to \mathcal{C}_{Z'/X'}$ (Morphisms, Lemma 29.31.3) is an isomorphism.

Then the diagram

$\xymatrix{ h^*\Omega ^ p_{Z/S} \ar[rr]_-{h^{-1}\gamma ^ p} \ar[d] & & \mathcal{O}_{X'}|_{Z'} \otimes _{h^{-1}\mathcal{O}_ X|_ Z} h^{-1}\mathcal{H}^ c_ Z(\Omega ^{p + c}_{X/S}) \ar[d] \\ \Omega ^ p_{Z'/S'} \ar[rr]^{\gamma ^ p} & & \mathcal{H}^ c_{Z'}(\Omega ^{p + c}_{X'/S'}) }$

is commutative. The left vertical arrow is functoriality of modules of differentials and the right vertical arrow uses Cohomology, Remark 20.34.12.

Proof. More precisely, consider the composition

\begin{align*} \mathcal{O}_{X'}|_{Z'} \otimes _{h^{-1}\mathcal{O}_ X|_ Z}^\mathbf {L} h^{-1}R\mathcal{H}_ Z(\Omega ^{p + c}_{X/S}) & \to R\mathcal{H}_{Z'}(Lg^*\Omega ^{p + c}_{X/S}) \\ & \to R\mathcal{H}_{Z'}(g^*\Omega ^{p + c}_{X/S}) \\ & \to R\mathcal{H}_{Z'}(\Omega ^{p + c}_{X'/S'}) \end{align*}

where the first arrow is given by Cohomology, Remark 20.34.12 and the last one by functoriality of differentials. Since we have the vanishing of cohomology sheaves in degrees $> c$ by Derived Categories of Schemes, Lemma 36.6.8 this induces the right vertical arrow. We can check the commutativity locally. Thus we may assume $Z$ is cut out by $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$. Then $Z'$ is cut out by $f'_ i = g^\sharp (f_ i)$. The maps $c_{f_1, \ldots , f_ c}$ and $c_{f'_1, \ldots , f'_ c}$ fit into the commutative diagram

$\xymatrix{ h^*i^*\Omega ^ p_{X/S} \ar[rr]_-{h^{-1}c_{f_1, \ldots , f_ c}} \ar[d] & & \mathcal{O}_{X'}|_{Z'} \otimes _{h^{-1}\mathcal{O}_ X|_ Z} h^{-1}\mathcal{H}^ c_ Z(\Omega ^ p_{X/S}) \ar[d] \\ (i')^*\Omega ^ p_{X'/S'} \ar[rr]^{c_{f'_1, \ldots , f'_ c}} & & \mathcal{H}^ c_{Z'}(\Omega ^ p_{X'/S'}) }$

See Derived Categories of Schemes, Remark 36.6.14. Recall given a $p$-form $\omega$ on $Z$ we define $\gamma ^ p(\omega )$ by choosing (locally on $X$ and $Z$) a $p$-form $\tilde\omega$ on $X$ lifting $\omega$ and taking $\gamma ^ p(\omega ) = c_{f_1, \ldots , f_ c}(\tilde\omega ) \wedge \text{d}f_1 \wedge \ldots \wedge \text{d}f_ c$. Since the form $\text{d}f_1 \wedge \ldots \wedge \text{d}f_ c$ pulls back to $\text{d}f'_1 \wedge \ldots \wedge \text{d}f'_ c$ we conclude. $\square$

Remark 50.23.7. Let $X \to S$, $i : Z \to X$, and $c \geq 0$ be as in Lemma 50.23.3. Let $p \geq 0$ and assume that $\mathcal{H}^ i_ Z(\Omega ^{p + c}_{X/S}) = 0$ for $i = 0, \ldots , c - 1$. This vanishing holds if $X \to S$ is smooth and $Z \to X$ is a Koszul regular immersion, see Derived Categories of Schemes, Lemma 36.6.9. Then we obtain a map

$\gamma ^{p, q} : H^ q(Z, \Omega ^ p_{Z/S}) \longrightarrow H^{q + c}(X, \Omega ^{p + c}_{X/S})$

by first using $\gamma ^ p : \Omega ^ p_{Z/S} \to \mathcal{H}^ c_ Z(\Omega ^{p + c}_{X/S})$ to map into

$H^ q(Z, \mathcal{H}^ c_ Z(\Omega ^{p + c}_{X/S})) = H^ q(Z, R\mathcal{H}_ Z(\Omega ^{p + c}_{X/S})[c]) = H^ q(X, i_*R\mathcal{H}_ Z(\Omega ^{p + c}_{X/S})[c])$

and then using the adjunction map $i_*R\mathcal{H}_ Z(\Omega ^{p + c}_{X/S}) \to \Omega ^{p + c}_{X/S}$ to continue on to the desired Hodge cohomology module.

Lemma 50.23.8. Let $X \to S$ and $i : Z \to X$ be as in Lemma 50.23.3. Assume $X \to S$ is smooth and $Z \to X$ Koszul regular. The gysin maps $\gamma ^{p, q}$ are compatible with the de Rham differentials on $\Omega ^\bullet _{X/S}$ and $\Omega ^\bullet _{Z/S}$.

Proof. This follows immediately from Lemma 50.23.4. $\square$

Lemma 50.23.9. Let $X \to S$, $i : Z \to X$, and $c \geq 0$ be as in Lemma 50.23.3. Assume $X \to S$ smooth and $Z \to X$ Koszul regular. Given $\alpha \in H^ q(X, \Omega ^ p_{X/S})$ we have $\gamma ^{p, q}(\alpha |_ Z) = \alpha \cup \gamma ^{0, 0}(1)$ in $H^{q + c}(X, \Omega ^{p + c}_{X/S})$ with $\gamma ^{a, b}$ as in Remark 50.23.7.

Proof. This lemma follows from Lemma 50.23.5 and Cohomology, Lemma 20.34.11. We suggest the reader skip over the more detailed discussion below.

We will use without further mention that $R\mathcal{H}_ Z(\Omega ^ j_{X/S}) = \mathcal{H}^ c_ Z(\Omega ^ j_{X/S})[-c]$ for all $j$ as pointed out in Remark 50.23.7. We will also silently use the identifications $H^{q + c}_ Z(X, \Omega ^ j_{X/S}) = H^{q + c}(Z, R\mathcal{H}_ Z(\Omega ^ j_{X/S}) = H^ q(Z, \mathcal{H}^ c_ Z(\Omega ^ j_{X/S}))$, see Cohomology, Lemma 20.34.4 for the first one. With these identifications

1. $\gamma ^0(1) \in H^ c_ Z(X, \Omega ^ c_{X/S})$ maps to $\gamma ^{0, 0}(1)$ in $H^ c(X, \Omega ^ c_{X/S})$,

2. the right hand side $i^{-1}\alpha \wedge \gamma ^0(1)$ of the equality in Lemma 50.23.5 is the (image by wedge product of the) cup product of Cohomology, Remark 20.34.10 of the elements $\alpha$ and $\gamma ^0(1)$, in other words, the constructions in the proof of Lemma 50.23.5 and in Cohomology, Remark 20.34.10 match,

3. by Cohomology, Lemma 20.34.11 this maps to $\alpha \cup \gamma ^{0, 0}(1)$ in $H^{q + c}(X, \Omega ^ p_{X/S} \otimes \Omega ^ c_{X/S})$, and

4. the left hand side $\gamma ^ p(\alpha |_ Z)$ of the equality in Lemma 50.23.5 maps to $\gamma ^{p, q}(\alpha |_ Z)$.

This finishes the proof. $\square$

Lemma 50.23.10. Let $c \geq 0$ and

$\xymatrix{ Z' \ar[d]_ h \ar[r] & X' \ar[d]_ g \ar[r] & S' \ar[d] \\ Z \ar[r] & X \ar[r] & S }$

satisfy the assumptions of Lemma 50.23.6 and assume in addition that $X \to S$ and $X' \to S'$ are smooth and that $Z \to X$ and $Z' \to X'$ are Koszul regular immersions. Then the diagram

$\xymatrix{ H^ q(Z, \Omega ^ p_{Z/S}) \ar[rr]_-{\gamma ^{p, q}} \ar[d] & & H^{q + c}(X, \Omega ^{p + c}_{X/S}) \ar[d] \\ H^ q(Z', \Omega ^ p_{Z'/S'}) \ar[rr]^{\gamma ^{p, q}} & & H^{q + c}(X', \Omega ^{p + c}_{X'/S'}) }$

is commutative where $\gamma ^{p, q}$ is as in Remark 50.23.7.

Proof. This follows on combining Lemma 50.23.6 and Cohomology, Lemma 20.34.13. $\square$

Lemma 50.23.11. Let $k$ be a field. Let $X$ be an irreducible smooth proper scheme over $k$ of dimension $d$. Let $Z \subset X$ be the reduced closed subscheme consisting of a single $k$-rational point $x$. Then the image of $1 \in k = H^0(Z, \mathcal{O}_ Z) = H^0(Z, \Omega ^0_{Z/k})$ by the map $H^0(Z, \Omega ^0_{Z/k}) \to H^ d(X, \Omega ^ d_{X/k})$ of Remark 50.23.7 is nonzero.

Proof. The map $\gamma ^0 : \mathcal{O}_ Z \to \mathcal{H}^ d_ Z(\Omega ^ d_{X/k}) = R\mathcal{H}_ Z(\Omega ^ d_{X/k})[d]$ is adjoint to a map

$g^0 : i_*\mathcal{O}_ Z \longrightarrow \Omega ^ d_{X/k}[d]$

in $D(\mathcal{O}_ X)$. Recall that $\Omega ^ d_{X/k} = \omega _ X$ is a dualizing sheaf for $X/k$, see Duality for Schemes, Lemma 48.27.1. Hence the $k$-linear dual of the map in the statement of the lemma is the map

$H^0(X, \mathcal{O}_ X) \to \mathop{\mathrm{Ext}}\nolimits ^ d_ X(i_*\mathcal{O}_ Z, \omega _ X)$

which sends $1$ to $g^0$. Thus it suffices to show that $g^0$ is nonzero. This we may do in any neighbourhood $U$ of the point $x$. Choose $U$ such that there exist $f_1, \ldots , f_ d \in \mathcal{O}_ X(U)$ vanishing only at $x$ and generating the maximal ideal $\mathfrak m_ x \subset \mathcal{O}_{X, x}$. We may assume assume $U = \mathop{\mathrm{Spec}}(R)$ is affine. Looking over the construction of $\gamma ^0$ we find that our extension is given by

$k \to (R \to \bigoplus \nolimits _{i_0} R_{f_{i_0}} \to \bigoplus \nolimits _{i_0 < i_1} R_{f_{i_0}f_{i_1}} \to \ldots \to R_{f_1\ldots f_ r})[d] \to R[d]$

where $1$ maps to $1/f_1 \ldots f_ c$ under the first map. This is nonzero because $1/f_1 \ldots f_ c$ is a nonzero element of local cohomology group $H^ d_{(f_1, \ldots , f_ d)}(R)$ in this case, $\square$

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