42.51 Two technical lemmas

In this section we develop some additional tools to allow us to work more comfortably with localized Chern classes. The following lemma is a more precise version of something we've already encountered in the proofs of Lemmas 42.49.6 and 42.49.7.

Lemma 42.51.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $b : W \longrightarrow \mathbf{P}^1_ X$ be a proper morphism of schemes. Let $n \geq 1$. For $i = 1, \ldots , n$ let $Z_ i \subset X$ be a closed subscheme, let $Q_ i \in D(\mathcal{O}_ W)$ be a perfect object, let $p_ i \geq 0$ be an integer, and let $T_ i \subset W_\infty$, $i = 1, \ldots , n$ be closed. Denote $W_ i = b^{-1}(\mathbf{P}^1_{Z_ i})$. Assume

1. for $i = 1, \ldots , n$ the assumption of Lemma 42.49.1 hold for $b, Z_ i, Q_ i, T_ i, p_ i$,

2. $Q_ i|_{W \setminus W_ i}$ is zero, resp. isomorphic to a finite locally free module of rank $< p_ i$ placed in cohomological degree $0$,

3. $Q_ i$ on $W$ satisfies assumption (3) of Situation 42.50.1.

Then $P'_{p_ n}(Q_ n) \circ \ldots \circ P'_{p_1}(Q_1)$ is equal to

$(W_{n, \infty } \cap \ldots \cap W_{1, \infty } \to Z_ n \cap \ldots \cap Z_1)_* \circ P'_{p_ n}(Q_ n|_{W_{n, \infty }}) \circ \ldots \circ P'_{p_1}(Q_1|_{W_{1, \infty }}) \circ C$

in $A^{p_ n + \ldots + p_1}(Z_ n \cap \ldots \cap Z_1 \to X)$, resp. $c'_{p_ n}(Q_ n) \circ \ldots \circ c'_{p_1}(Q_1)$ is equal to

$(W_{n, \infty } \cap \ldots \cap W_{1, \infty } \to Z_ n \cap \ldots \cap Z_1)_* \circ c'_{p_ n}(Q_ n|_{W_{n, \infty }}) \circ \ldots \circ c'_{p_1}(Q_1|_{W_{1, \infty }}) \circ C$

in $A^{p_ n + \ldots + p_1}(Z_ n \cap \ldots \cap Z_1 \to X)$.

Proof. Let us prove the statement on Chern classes by induction on $n$; the statement on $P_ p(-)$ is proved in the exact same manner. The case $n = 1$ is the construction of $c'_{p_1}(Q_1)$ because $W_{1, \infty }$ is the inverse image of $Z_1$ in $W_\infty$. For $n > 1$ we have by induction that $c'_{p_ n}(Q_ n) \circ \ldots \circ c'_{p_1}(Q_1)$ is equal to

$c'_{p_ n}(Q_ n) \circ (W_{n - 1, \infty } \cap \ldots \cap W_{1, \infty } \to Z_{n - 1} \cap \ldots \cap Z_1)_* \circ c'_{p_{n - 1}}(Q_{n - 1}|_{W_{n - 1}, \infty }) \circ \ldots \circ c'_{p_1}(Q_1|_{W_{1, \infty }}) \circ C$

By Lemma 42.49.2 the restriction of $c'_{p_ n}(Q_ n)$ to $Z_{n - 1} \cap \ldots \cap Z_1$ is computed by the closed subset $Z_ n \cap \ldots \cap Z_1$, the morphism $b' : W_{n - 1} \cap \ldots \cap W_1 \to \mathbf{P}^1_{Z_{n - 1} \cap \ldots \cap Z_1}$ and the restriction of $Q_ n$ to $W_{n - 1} \cap \ldots \cap W_1$. Observe that $(b')^{-1}(Z_ n) = W_ n \cap \ldots \cap W_1$ and that $(W_ n \cap \ldots \cap W_1)_\infty = W_{n, \infty } \cap \ldots \cap W_{1, \infty }$. Denote $C_{n - 1} \in A^0(W_{n - 1, \infty } \cap \ldots \cap W_{1, \infty } \to Z_{n - 1} \cap \ldots \cap Z_1)$ the class of Lemma 42.48.1. We conclude the restriction of $c'_{p_ n}(Q_ n)$ to $Z_{n - 1} \cap \ldots \cap Z_1$ is

\begin{align*} & (W_{n, \infty } \cap \ldots \cap W_{1, \infty } \to Z_ n \cap \ldots \cap Z_1)_* \circ c'_{p_ n}(Q_ n|_{(W_ n \cap \ldots \cap W_1)_\infty }) \circ C_{n - 1} \\ & = (W_{n, \infty } \cap \ldots \cap W_{1, \infty } \to Z_ n \cap \ldots \cap Z_1)_* \circ c'_{p_ n}(Q_ n|_{W_{n, \infty }}) \circ C_{n - 1} \end{align*}

where the equality follows from Lemma 42.47.3 (we omit writing the restriction on the right). Hence the above becomes

\begin{align*} (W_{n, \infty } \cap \ldots \cap W_{1, \infty } \to Z_ n \cap \ldots \cap Z_1)_* \circ c'_{p_ n}(Q_ n|_{W_ n, \infty }) \circ \\ C_{n - 1} \circ (W_{n - 1, \infty } \cap \ldots \cap W_{1, \infty } \to Z_{n - 1} \cap \ldots \cap Z_1)_* \\ \circ c'_{p_{n - 1}}(Q_{n - 1}|_{W_{n - 1}, \infty }) \circ \ldots \circ c'_{p_1}(Q_1|_{W_{1, \infty }}) \circ C \end{align*}

By Lemma 42.48.4 we know that the composition $C_{n - 1} \circ (W_{n - 1, \infty } \cap \ldots \cap W_{1, \infty } \to Z_{n - 1} \cap \ldots \cap Z_1)_*$ is the identity on elements in the image of the gysin map

$(W_{n - 1, \infty } \cap \ldots \cap W_{1, \infty } \to W_{n - 1} \cap \ldots \cap W_1)^*$

Thus it suffices to show that any element in the image of $c'_{p_{n - 1}}(Q_{n - 1}|_{W_{n - 1}, \infty }) \circ \ldots \circ c'_{p_1}(Q_1|_{W_{1, \infty }}) \circ C$ is in the image of the gysin map. We may write

$c'_{p_ i}(Q_ i|_{W_{i, \infty }}) = \text{restriction of } c_{p_ i}(W_ i \to W, Q_ i) \text{ to } W_{i, \infty }$

by Lemma 42.50.9 and assumptions (2) and (3) on $Q_ i$ in the statement of the lemma. Thus, if $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(W)$ restricts to the flat pullback of $\alpha$ on $b^{-1}(\mathbf{A}^1_ X)$, then

\begin{align*} & c'_{p_{n - 1}}(Q_{n - 1}|_{W_{n - 1}, \infty }) \cap \ldots \cap c'_{p_1}(Q_1|_{W_{1, \infty }}) \cap C \cap \alpha \\ & = c'_{p_{n - 1}}(Q_{n - 1}|_{W_{n - 1}, \infty }) \cap \ldots \cap c'_{p_1}(Q_1|_{W_{1, \infty }}) \cap i_\infty ^* \beta \\ & = c_{p_{n - 1}}(W_{n - 1} \to W, Q_{n - 1}) \cap \ldots \cap c_{p_{n - 1}}(W_1 \to W, Q_1) \cap i_\infty ^* \beta \\ & = (W_{n - 1, \infty } \cap \ldots \cap W_{1, \infty } \to W_{n - 1} \cap \ldots \cap W_1)^* \left(c_{p_{n - 1}}(W_{n - 1} \to W, Q_{n - 1}) \cap \ldots \cap c_{p_1}(W_1 \to W, Q_1) \cap \beta \right) \end{align*}

as desired. Namely, for the last equality we use that $c_{p_ i}(W_ i \to W, Q_ i)$ is a bivariant class and hence commutes with $i_\infty ^*$ by definition. $\square$

The following lemma gives us a tremendous amount of flexibility if we want to compute the localized Chern classes of a complex.

Lemma 42.51.2. Assume $(S, \delta ), X, Z, b : W \to \mathbf{P}^1_ X, Q, T, p$ satisfy the assumptions of Lemma 42.49.1. Let $F \in D(\mathcal{O}_ X)$ be a perfect object such that

1. the restriction of $Q$ to $b^{-1}(\mathbf{A}^1_ X)$ is isomorphic to the pullback of $F$,

2. $F|_{X \setminus Z}$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z}$-module of rank $< p$ sitting in cohomological degree $0$, and

3. $Q$ on $W$ and $F$ on $X$ satisfy assumption (3) of Situation 42.50.1.

Then the class $P'_ p(Q)$, resp. $c'_ p(Q)$ in $A^ p(Z \to X)$ constructed in Lemma 42.49.1 is equal to $P_ p(Z \to X, F)$, resp. $c_ p(Z \to X, F)$ from Definition 42.50.3.

Proof. The assumptions are preserved by base change with a morphism $X' \to X$ locally of finite type. Hence it suffices to show that $P_ p(Z \to X, F) \cap \alpha = P'_ p(Q) \cap \alpha$, resp. $c_ p(Z \to X, F) \cap \alpha = c'_ p(Q) \cap \alpha$ for any $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$. Choose $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(W)$ whose restriction to $b^{-1}(\mathbf{A}^1_ X)$ is equal to the flat pullback of $\alpha$ as in the construction of $C$ in Lemma 42.48.1. Denote $W' = b^{-1}(Z)$ and denote $E = W'_\infty \subset W_\infty$ the inverse image of $Z$ by $W_\infty \to X$. The lemma follows from the following sequence of equalities (the case of $P_ p$ is similar)

\begin{align*} c'_ p(Q) \cap \alpha & = (E \to Z)_*(c'_ p(Q|_ E) \cap i_\infty ^*\beta ) \\ & = (E \to Z)_*(c_ p(E \to W_\infty , Q|_{W_\infty }) \cap i_\infty ^*\beta ) \\ & = (W'_\infty \to Z)_*(c_ p(W' \to W, Q) \cap i_\infty ^*\beta ) \\ & = (W'_\infty \to Z)_*((i'_\infty )^*(c_ p(W' \to W, Q) \cap \beta )) \\ & = (W'_\infty \to Z)_*((i'_\infty )^*(c_ p(Z' \to X, F) \cap \beta )) \\ & = (W'_0 \to Z)_*((i'_0)^*(c_ p(Z' \to X, F) \cap \beta )) \\ & = (W'_0 \to Z)_*(c_ p(Z' \to X, F) \cap i_0^*\beta )) \\ & = c_ p(Z \to X, F) \cap \alpha \end{align*}

The first equality is the construction of $c'_ p(Q)$ in Lemma 42.49.1. The second is Lemma 42.50.9. The base change of $W' \to W$ by $W_\infty \to W$ is the morphism $E = W'_\infty \to W_\infty$. Hence the third equality holds by Lemma 42.50.4. The fourth equality, in which $i'_\infty : W'_\infty \to W'$ is the inclusion morphism, follows from the fact that $c_ p(W' \to W, Q)$ is a bivariant class. For the fith equality, observe that $c_ p(W' \to W, Q)$ and $c_ p(Z' \to X, F)$ restrict to the same bivariant class in $A^ p((b')^{-1} \to b^{-1}(\mathbf{A}^1_ X))$ by assumption (1) of the lemma which says that $Q$ and $F$ restrict to the same object of $D(\mathcal{O}_{b^{-1}(\mathbf{A}^1_ X)})$; use Lemma 42.50.4. Since $(i'_\infty )^*$ annihilates cycles supported on $W'_\infty$ (see Remark 42.29.6) we conclude the fifth equality is true. The sixth equality holds because $W'_\infty$ and $W'_0$ are the pullbacks of the rationally equivalent effective Cartier divisors $D_0, D_\infty$ in $\mathbf{P}^1_ Z$ and hence $i_\infty ^*\beta$ and $i_0^*\beta$ map to the same cycle class on $W'$; namely, both represent the class $c_1(\mathcal{O}_{\mathbf{P}^1_ Z}(1)) \cap c_ p(Z \to X, F_) \cap \beta$ by Lemma 42.29.4. The seventh equality holds because $c_ p(Z \to X, F)$ is a bivariant class. By construction $W'_0 = Z$ and $i_0^*\beta = \alpha$ which explains why the final equality holds. $\square$

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