The Stacks project

42.48 Gysin at infinity

This section is about the bivariant class constructed in the next lemma. We urge the reader to skip the rest of the section.

Lemma 42.48.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $b : W \to \mathbf{P}^1_ X$ be a proper morphism of schemes which is an isomorphism over $\mathbf{A}^1_ X$. Denote $i_\infty : W_\infty \to W$ the inverse image of the divisor $D_\infty \subset \mathbf{P}^1_ X$ with complement $\mathbf{A}^1_ X$. Then there is a canonical bivariant class

\[ C \in A^0(W_\infty \to X) \]

with the property that $i_{\infty , *}(C \cap \alpha ) = i_{0, *}\alpha $ for $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ and similarly after any base change by $X' \to X$ locally of finite type.

Proof. Given $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ there exists a $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(W)$ restricting to the flat pullback of $\alpha $ on $b^{-1}(\mathbf{A}^1_ X)$, see Lemma 42.14.2. A second choice of $\beta $ differs from $\beta $ by a cycle supported on $W_\infty $, see Lemma 42.19.3. Since the normal bundle of the effective Cartier divisor $D_\infty \subset \mathbf{P}^1_ X$ of ( is trivial, the gysin homomorphism $i_\infty ^*$ kills cycle classes supported on $W_\infty $, see Remark 42.29.6. Hence setting $C \cap \alpha = i_\infty ^*\beta $ is well defined.

Since $W_\infty $ and $W_0 = X \times \{ 0\} $ are the pullbacks of the rationally equivalent effective Cartier divisors $D_0, D_\infty $ in $\mathbf{P}^1_ X$, we see that $i_\infty ^*\beta $ and $i_0^*\beta $ map to the same cycle class on $W$; namely, both represent the class $c_1(\mathcal{O}_{\mathbf{P}^1_ X}(1)) \cap \beta $ by Lemma 42.29.4. By our choice of $\beta $ we have $i_0^*\beta = \alpha $ as cycles on $W_0 = X \times \{ 0\} $, see for example Lemma 42.31.1. Thus we see that $i_{\infty , *}(C \cap \alpha ) = i_{0, *}\alpha $ as stated in the lemma.

Observe that the assumptions on $b$ are preserved by any base change by $X' \to X$ locally of finite type. Hence we get an operation $C \cap - : \mathop{\mathrm{CH}}\nolimits _ k(X') \to \mathop{\mathrm{CH}}\nolimits _ k(W'_\infty )$ by the same construction as above. To see that this family of operations defines a bivariant class, we consider the diagram

\[ \xymatrix{ & & & \mathop{\mathrm{CH}}\nolimits _*(X) \ar[d]^{\text{flat pullback}} \\ \mathop{\mathrm{CH}}\nolimits _{* + 1}(W_\infty ) \ar[r] \ar[rd]^0 & \mathop{\mathrm{CH}}\nolimits _{* + 1}(W) \ar[d]^{i_\infty ^*} \ar[rr]^{\text{flat pullback}} & & \mathop{\mathrm{CH}}\nolimits _{* + 1}(\mathbf{A}^1_ X) \ar[r] \ar@{..>}[lld]^{C \cap -} & 0 \\ & \mathop{\mathrm{CH}}\nolimits _*(W_\infty ) } \]

for $X$ as indicated and the base change of this diagram for any $X' \to X$. We know that flat pullback and $i_\infty ^*$ are bivariant operations, see Lemmas 42.33.2 and 42.33.3. Then a formal argument (involving huge diagrams of schemes and their chow groups) shows that the dotted arrow is a bivariant operation. $\square$

Lemma 42.48.2. In Lemma 42.48.1 let $X' \to X$ be a morphism which is locally of finite type. Denote $b' : W' \to \mathbf{P}^1_{X'}$ and $i'_\infty : W'_\infty \to W'$ the base changes of $b$ and $i_\infty $. Then the class $C' \in A^0(W'_\infty \to X')$ constructed as in Lemma 42.48.1 using $b'$ is the restriction (Remark 42.33.5) of $C$.

Proof. Immediate from the construction and the fact that a similar statement holds for flat pullback and $i_\infty ^*$. $\square$

Lemma 42.48.3. In Lemma 42.48.1 let $g : W' \to W$ be a proper morphism which is an isomorphism over $\mathbf{A}^1_ X$. Let $C' \in A^0(W'_\infty \to X)$ and $C \in A^0(W_\infty \to X)$ be the classes constructed in Lemma 42.48.1. Then $g_{\infty , *} \circ C' = C$ in $A^0(W_\infty \to X)$.

Proof. Set $b' = b \circ g : W' \to \mathbf{P}^1_ X$. Denote $i'_\infty : W'_\infty \to W'$ the inclusion morphism. Denote $g_\infty : W'_\infty \to W_\infty $ the restriction of $g$. Given $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ choose $\beta ' \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(W')$ restricting to the flat pullback of $\alpha $ on $(b')^{-1}\mathbf{A}^1_ X$. Then $\beta = g_*\beta ' \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(W)$ restricts to the flat pullback of $\alpha $ on $b^{-1}\mathbf{A}^1_ X$. Then $i_\infty ^*\beta = g_{\infty , *}(i'_\infty )^*\beta '$ by Lemma 42.29.8. This and the corresponding fact after base change by morphisms $X' \to X$ locally of finite type, corresponds to the assertion made in the lemma. $\square$

Proof. Let $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(W)$. Denote $i_0 : X = X \times \{ 0\} \to W$ the closed immersion of the fibre over $0$ in $\mathbf{P}^1$. Then $(W_\infty \to X)_* i_\infty ^* \beta = i_0^*\beta $ in $\mathop{\mathrm{CH}}\nolimits _ k(X)$ because $i_{\infty , *}i_\infty ^*\beta $ and $i_{0, *}i_0^*\beta $ represent the same class on $W$ (for example by Lemma 42.29.4) and hence pushforward to the same class on $X$. The restriction of $\beta $ to $b^{-1}(\mathbf{A}^1_ X)$ restricts to the flat pullback of $i_0^*\beta = (W_\infty \to X)_* i_\infty ^* \beta $ because we can check this after pullback by $i_0$, see Lemmas 42.32.2 and 42.32.4. Hence we may use $\beta $ when computing the image of $(W_\infty \to X)_*i_\infty ^*\beta $ under $C$ and we get the desired result. $\square$

Lemma 42.48.5. In Lemma 42.48.1 let $f : Y \to X$ be a morphism locally of finite type and $c \in A^*(Y \to X)$. Then $C \circ c = c \circ C$ in $A^*(W_\infty \times _ X Y \to X)$.

Proof. Consider the commutative diagram

\[ \xymatrix{ W_\infty \times _ X Y \ar@{=}[r] & W_{Y, \infty } \ar[r]_{i_{Y, \infty }} \ar[d] & W_ Y \ar[r]_{b_ Y} \ar[d] & \mathbf{P}^1_ Y \ar[r]_{p_ Y} \ar[d] & Y \ar[d]^ f \\ & W_\infty \ar[r]^{i_\infty } & W \ar[r]^ b & \mathbf{P}^1_ X \ar[r]^ p & X } \]

with cartesian squares. For an elemnent $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ choose $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(W)$ whose restriction to $b^{-1}(\mathbf{A}^1_ X)$ is the flat pullback of $\alpha $. Then $c \cap \beta $ is a class in $\mathop{\mathrm{CH}}\nolimits _*(W_ Y)$ whose restriction to $b_ Y^{-1}(\mathbf{A}^1_ Y)$ is the flat pullback of $c \cap \alpha $. Next, we have

\[ i_{Y, \infty }^*(c \cap \beta ) = c \cap i_\infty ^*\beta \]

because $c$ is a bivariant class. This exactly says that $C \cap c \cap \alpha = c \cap C \cap \alpha $. The same argument works after any base change by $X' \to X$ locally of finite type. This proves the lemma. $\square$

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