## 42.49 Preparation for localized Chern classes

In this section we discuss some properties of the bivariant classes constructed in the following lemma. We urge the reader to skip the rest of the section.

Lemma 42.49.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme. Let

$b : W \longrightarrow \mathbf{P}^1_ X$

be a proper morphism of schemes. Let $Q \in D(\mathcal{O}_ W)$ be a perfect object. Denote $W_\infty \subset W$ the inverse image of the divisor $D_\infty \subset \mathbf{P}^1_ X$ with complement $\mathbf{A}^1_ X$. We assume

1. Chern classes of $Q$ are defined (Section 42.46),

2. $b$ is an isomorphism over $\mathbf{A}^1_ X$,

3. there exists a closed subscheme $T \subset W_\infty$ containing all points of $W_\infty$ lying over $X \setminus Z$ such that $Q|_ T$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_ T$-module of rank $< p$ sitting in cohomological degree $0$.

Then there exists a canonical bivariant class

$P'_ p(Q),\text{ resp. }c'_ p(Q) \in A^ p(Z \to X)$

with $(Z \to X)_* \circ P'_ p(Q) = P_ p(Q|_{X \times \{ 0\} })$, resp. $(Z \to X)_* \circ c'_ p(Q) = c_ p(Q|_{X \times \{ 0\} })$.

Proof. Denote $E \subset W_\infty$ the inverse image of $Z$. Then $W_\infty = T \cup E$ and $b$ induces a proper morphism $E \to Z$. Denote $C \in A^0(W_\infty \to X)$ the bivariant class constructed in Lemma 42.48.1. Denote $P'_ p(Q|_ E)$, resp. $c'_ p(Q|_ E)$ in $A^ p(E \to W_\infty )$ the bivariant class constructed in Lemma 42.47.1. This makes sense because $(Q|_ E)|_{T \cap E}$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{E \cap T}$-module of rank $< p$ sitting in cohomological degree $0$ by assumption (A2). Then we define

$P'_ p(Q) = (E \to Z)_* \circ P'_ p(Q|_ E) \circ C,\text{ resp. } c'_ p(Q) = (E \to Z)_* \circ c'_ p(Q|_ E) \circ C$

This is a bivariant class, see Lemma 42.33.4. Since $E \to Z \to X$ is equal to $E \to W_\infty \to W \to X$ we see that

\begin{align*} (Z \to X)_* \circ c'_ p(Q) & = (W \to X)_* \circ i_{\infty , *} \circ (E \to W_\infty )_* \circ c'_ p(Q|_ E) \circ C \\ & = (W \to X)_* \circ i_{\infty , *} \circ c_ p(Q|_{W_\infty }) \circ C \\ & = (W \to X)_* \circ c_ p(Q) \circ i_{\infty , *} \circ C \\ & = (W \to X)_*\circ c_ p(Q) \circ i_{0, *} \\ & = (W \to X)_* \circ i_{0, *} \circ c_ p(Q|_{X \times \{ 0\} }) \\ & = c_ p(Q|_{X \times \{ 0\} }) \end{align*}

The second equality holds by Lemma 42.47.4. The third equality because $c_ p(Q)$ is a bivariant class. The fourth equality by Lemma 42.48.1. The fifth equality because $c_ p(Q)$ is a bivariant class. The final equality because $(W_0 \to W) \circ (W \to X)$ is the identity on $X$ if we identify $W_0$ with $X$ as we've done above. The exact same sequence of equations works to prove the property for $P'_ p(Q)$. $\square$

Lemma 42.49.2. In Lemma 42.49.1 let $X' \to X$ be a morphism which is locally of finite type. Denote $Z'$, $b' : W' \to \mathbf{P}^1_{X'}$, and $T' \subset W'_\infty$ the base changes of $Z$, $b : W \to \mathbf{P}^1_ X$, and $T \subset W_\infty$. Set $Q' = (W' \to W)^*Q$. Then the class $P'_ p(Q')$, resp. $c'_ p(Q')$ in $A^ p(Z' \to X')$ constructed as in Lemma 42.49.1 using $b'$, $Q'$, and $T'$ is the restriction (Remark 42.33.5) of the class $P'_ p(Q)$, resp. $c'_ p(Q)$ in $A^ p(Z \to X)$.

Proof. Recall that the construction is as follows

$P'_ p(Q) = (E \to Z)_* \circ P'_ p(Q|_ E) \circ C,\text{ resp. } c'_ p(Q) = (E \to Z)_* \circ c'_ p(Q|_ E) \circ C$

Thus the lemma follows from the corresponding base change property for $C$ (Lemma 42.48.2) and the fact that the same base change property holds for the classes constructed in Lemma 42.47.1 (small detail omitted). $\square$

Lemma 42.49.3. In Lemma 42.49.1 the bivariant class $P'_ p(Q)$, resp. $c'_ p(Q)$ is independent of the choice of the closed subscheme $T$. Moreover, given a proper morphism $g : W' \to W$ which is an isomorphism over $\mathbf{A}^1_ X$, then setting $Q' = g^*Q$ we have $P'_ p(Q) = P'_ p(Q')$, resp. $c'_ p(Q) = c'_ p(Q')$.

Proof. The independence of $T$ follows immediately from Lemma 42.47.2.

Let $g : W' \to W$ be a proper morphism which is an isomorphism over $\mathbf{A}^1_ X$. Observe that taking $T' = g^{-1}(T) \subset W'_\infty$ is a closed subscheme satisfying (A2) hence the operator $P'_ p(Q')$, resp. $c'_ p(Q')$ in $A^ p(Z \to X)$ corresponding to $b' = b \circ g : W' \to \mathbf{P}^1_ X$ and $Q'$ is defined. Denote $E' \subset W'_\infty$ the inverse image of $Z$ in $W'_\infty$. Recall that

$c'_ p(Q') = (E' \to Z)_* \circ c'_ p(Q'|_{E'}) \circ C'$

with $C' \in A^0(W'_\infty \to X)$ and $c'_ p(Q'|_{E'}) \in A^ p(E' \to W'_\infty )$. By Lemma 42.48.3 we have $g_{\infty , *} \circ C' = C$. Observe that $E'$ is also the inverse image of $E$ in $W'_\infty$ by $g_\infty$. Since moreover $Q' = g^*Q$ we find that $c'_ p(Q'|_{E'})$ is simply the restriction of $c'_ p(Q|_ E)$ to schemes lying over $W'_\infty$, see Remark 42.33.5. Thus we obtain

\begin{align*} c'_ p(Q') & = (E' \to Z)_* \circ c'_ p(Q'|_{E'}) \circ C' \\ & = (E \to Z)_* \circ (E' \to E)_* \circ c'_ p(Q|_ E) \circ C' \\ & = (E \to Z)_* \circ c'_ p(Q|_ E) \circ g_{\infty , *} \circ C' \\ & = (E \to Z)_* \circ c'_ p(Q|_ E) \circ C \\ & = c'_ p(Q) \end{align*}

In the third equality we used that $c'_ p(Q|_ E)$ commutes with proper pushforward as it is a bivariant class. The equality $P'_ p(Q) = P'_ p(Q')$ is proved in exactly the same way. $\square$

Lemma 42.49.4. In Lemma 42.49.1 assume $Q|_ T$ is isomorphic to a finite locally free $\mathcal{O}_ T$-module of rank $< p$. Denote $C \in A^0(W_\infty \to X)$ the class of Lemma 42.48.1. Then

$C \circ c_ p(Q|_{X \times \{ 0\} }) = C \circ (Z \to X)_* \circ c'_ p(Q) = c_ p(Q|_{W_\infty }) \circ C$

Proof. The first equality holds because $c_ p(Q|_{X \times \{ 0\} }) = (Z \to X)_* \circ c'_ p(Q)$ by Lemma 42.49.1. We may prove the second equality one cycle class at a time (see Lemma 42.35.3). Since the construction of the bivariant classes in the lemma is compatible with base change, we may assume we have some $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ and we have to show that $C \cap (Z \to X)_*(c'_ p(Q) \cap \alpha ) = c_ p(Q|_{W_\infty }) \cap C \cap \alpha$. Observe that

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

as desired. For the first equality we used that $c'_ p(Q) = (E \to Z)_* \circ c'_ p(Q|_ E) \circ C$ where $E \subset W_\infty$ is the inverse image of $Z$ and $c'_ p(Q|_ E)$ is the class constructed in Lemma 42.47.1. The second equality is just the statement that $E \to Z \to X$ is equal to $E \to W_\infty \to X$. For the third equality we choose $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + 1}(W)$ whose restriction to $b^{-1}(\mathbf{A}^1_ X)$ is the flat pullback of $\alpha$ so that $C \cap \alpha = i_\infty ^*\beta$ by construction. The fourth equality is Lemma 42.47.4. The fifth equality is the fact that $c_ p(Q)$ is a bivariant class and hence commutes with $i_\infty ^*$. The sixth equality is Lemma 42.48.4. The seventh uses again that $c_ p(Q)$ is a bivariant class. The final holds as $C \cap \alpha = i_\infty ^*\beta$. $\square$

Lemma 42.49.5. In Lemma 42.49.1 let $Y \to X$ be a morphism locally of finite type and let $c \in A^*(Y \to X)$ be a bivariant class. Then

$P'_ p(Q) \circ c = c \circ P'_ p(Q) \quad \text{resp.}\quad c'_ p(Q) \circ c = c \circ c'_ p(Q)$

in $A^*(Y \times _ X Z \to X)$.

Proof. Let $E \subset W_\infty$ be the inverse image of $Z$. Recall that $P'_ p(Q) = (E \to Z)_* \circ P'_ p(Q|_ E) \circ C$, resp. $c'_ p(Q) = (E \to Z)_* \circ c'_ p(Q|_ E) \circ C$ where $C$ is as in Lemma 42.48.1 and $P'_ p(Q|_ E)$, resp. $c'_ p(Q|_ E)$ are as in Lemma 42.47.1. By Lemma 42.48.5 we see that $C$ commutes with $c$ and by Lemma 42.47.6 we see that $P'_ p(Q|_ E)$, resp. $c'_ p(Q|_ E)$ commutes with $c$. Since $c$ is a bivariant class it commutes with proper pushforward by $E \to Z$ by definition. This finishes the proof. $\square$

Lemma 42.49.6. In Lemma 42.49.1 assume $Q|_ T$ is zero. In $A^*(Z \to X)$ we have

\begin{align*} P'_1(Q) & = c'_1(Q), \\ P'_2(Q) & = c'_1(Q)^2 - 2c'_2(Q), \\ P'_3(Q) & = c'_1(Q)^3 - 3c'_1(Q)c'_2(Q) + 3c'_3(Q), \\ P'_4(Q) & = c'_1(Q)^4 - 4c'_1(Q)^2c'_2(Q) + 4c'_1(Q)c'_3(Q) + 2c'_2(Q)^2 - 4c'_4(Q), \end{align*}

and so on with multiplication as in Remark 42.34.7.

Proof. The statement makes sense because the zero sheaf has rank $< 1$ and hence the classes $c'_ p(Q)$ are defined for all $p \geq 1$. In the proof of Lemma 42.49.1 we have constructed the classes $P'_ p(Q)$ and $c'_ p(Q)$ using the bivariant class $C \in A^0(W_\infty \to X)$ of Lemma 42.48.1 and the bivariant classes $P'_ p(Q|_ E)$ and $c'_ p(Q|_ E)$ of Lemma 42.47.1 for the restriction $Q|_ E$ of $Q$ to the inverse image $E$ of $Z$ in $W_\infty$. Observe that by Lemma 42.47.7 we have the desired relationship between $P'_ p(Q|_ E)$ and $c'_ p(Q|_ E)$. Recall that

$P'_ p(Q) = (E \to Z)_* \circ P'_ p(Q|_ E) \circ C \quad \text{and}\quad c'_ p(Q) = (E \to Z)_* \circ c'_ p(Q|_ E) \circ C$

To finish the proof it suffices to show the multiplications defined in Remark 42.34.7 on the classes $a_ p = c'_ p(Q)$ and on the classes $b_ p = c'_ p(Q|_ E)$ agree:

$a_{p_1}a_{p_2} \ldots a_{p_ r} = (E \to Z)_* \circ b_{p_1}b_{p_2} \ldots b_{p_ r} \circ C$

Some details omitted. If $r = 1$, then this is true. For $r > 1$ note that by Remark 42.34.8 the multiplication in Remark 42.34.7 proceeds by inserting $(Z \to X)_*$, resp. $(E \to W_\infty )_*$ in between the factors of the product $a_{p_1}a_{p_2} \ldots a_{p_ r}$, resp. $b_{p_1}b_{p_2} \ldots b_{p_ r}$ and taking compositions as bivariant classes. Now by Lemma 42.47.1 we have

$(E \to W_\infty )_* \circ b_{p_ i} = c_{p_ i}(Q|_{W_\infty })$

and by Lemma 42.49.4 we have

$C \circ (Z \to X)_* \circ a_{p_ i} = c_{p_ i}(Q|_{W_\infty }) \circ C$

for $i = 2, \ldots , r$. A calculation shows that the left and right hand side of the desired equality both simplify to

$(E \to Z)_* \circ c'_{p_1}(Q|_ E) \circ c_{p_2}(Q|_{W_\infty }) \circ \ldots \circ c_{p_ r}(Q|_{W_\infty }) \circ C$

and the proof is complete. $\square$

Lemma 42.49.7. In Lemma 42.49.1 assume $Q|_ T$ is isomorphic to a finite locally free $\mathcal{O}_ T$-module of rank $< p$. Assume we have another perfect object $Q' \in D(\mathcal{O}_ W)$ whose Chern classes are defined with $Q'|_ T$ isomorphic to a finite locally free $\mathcal{O}_ T$-module of rank $< p'$ placed in cohomological degree $0$. With notation as in Remark 42.34.7 set

$c^{(p)}(Q) = 1 + c_1(Q|_{X \times \{ 0\} }) + \ldots + c_{p - 1}(Q|_{X \times \{ 0\} }) + c'_{p}(Q) + c'_{p + 1}(Q) + \ldots$

in $A^{(p)}(Z \to X)$ with $c'_ i(Q)$ for $i \geq p$ as in Lemma 42.49.1. Similarly for $c^{(p')}(Q')$ and $c^{(p + p')}(Q \oplus Q')$. Then $c^{(p + p')}(Q \oplus Q') = c^{(p)}(Q)c^{(p')}(Q')$ in $A^{(p + p')}(Z \to X)$.

Proof. Recall that the image of $c'_ i(Q)$ in $A^ p(X)$ is equal to $c_ i(Q|_{X \times \{ 0\} })$ for $i \geq p$ and similarly for $Q'$ and $Q \oplus Q'$, see Lemma 42.49.1. Hence the equality in degrees $< p + p'$ follows from the additivity of Lemma 42.46.7.

Let's take $n \geq p + p'$. As in the proof of Lemma 42.49.1 let $E \subset W_\infty$ denote the inverse image of $Z$. Observe that we have the equality

$c^{(p + p')}(Q|_ E \oplus Q'|_ E) = c^{(p)}(Q|_ E)c^{(p')}(Q'|_ E)$

in $A^{(p + p')}(E \to W_\infty )$ by Lemma 42.47.8. Since by construction

$c'_ p(Q \oplus Q') = (E \to Z)_* \circ c'_ p(Q|_ E \oplus Q'|_ E) \circ C$

we conclude that suffices to show for all $i + j = n$ we have

$(E \to Z)_* \circ c^{(p)}_ i(Q|_ E)c^{(p')}_ j(Q'|_ E) \circ C = c^{(p)}_ i(Q)c^{(p')}_ j(Q')$

in $A^ n(Z \to X)$ where the multiplication is the one from Remark 42.34.7 on both sides. There are three cases, depending on whether $i \geq p$, $j \geq p'$, or both.

Assume $i \geq p$ and $j \geq p'$. In this case the products are defined by inserting $(E \to W_\infty )_*$, resp. $(Z \to X)_*$ in between the two factors and taking compositions as bivariant classes, see Remark 42.34.8. In other words, we have to show

$(E \to Z)_* \circ c'_ i(Q|_ E) \circ (E \to W_\infty )_* \circ c'_ j(Q'|_ E) \circ C = c'_ i(Q) \circ (Z \to X)_* \circ c'_ j(Q')$

By Lemma 42.47.1 the left hand side is equal to

$(E \to Z)_* \circ c'_ i(Q|_ E) \circ c_ j(Q'|_{W_\infty }) \circ C$

Since $c'_ i(Q) = (E \to Z)_* \circ c'_ i(Q|_ E) \circ C$ the right hand side is equal to

$(E \to Z)_* \circ c'_ i(Q|_ E) \circ C \circ (Z \to X)_* \circ c'_ j(Q')$

which is immediately seen to be equal to the above by Lemma 42.49.4.

Assume $i \geq p$ and $j < p$. Unwinding the products in this case we have to show

$(E \to Z)_* \circ c'_ i(Q|_ E) \circ c_ j(Q'|_{W_\infty }) \circ C = c'_ i(Q) \circ c_ j(Q'|_{X \times \{ 0\} })$

Again using that $c'_ i(Q) = (E \to Z)_* \circ c'_ i(Q|_ E) \circ C$ we see that it suffices to show $c_ j(Q'|_{W_\infty }) \circ C = C \circ c_ j(Q'|_{X \times \{ 0\} })$ which is part of Lemma 42.49.4.

Assume $i < p$ and $j \geq p'$. Unwinding the products in this case we have to show

$(E \to Z)_* \circ c_ i(Q|_ E) \circ c'_ j(Q'|_ E) \circ C = c_ i(Q|_{Z \times \{ 0\} }) \circ c'_ j(Q')$

However, since $c'_ j(Q|_ E)$ and $c'_ j(Q')$ are bivariant classes, they commute with capping with Chern classes (Lemma 42.38.9). Hence it suffices to prove

$(E \to Z)_* \circ c'_ j(Q'|_ E) \circ c_ i(Q|_{W_\infty }) \circ C = c'_ j(Q') \circ c_ i(Q|_{X \times \{ 0\} })$

which we reduces us to the case discussed in the preceding paragraph. $\square$

Lemma 42.49.8. In Lemma 42.49.1 assume $Q|_ T$ is zero. Assume we have another perfect object $Q' \in D(\mathcal{O}_ W)$ whose Chern classes are defined such that the restriction $Q'|_ T$ is zero. In this case the classes $P'_ p(Q), P'_ p(Q'), P'_ p(Q \oplus Q') \in A^ p(Z \to X)$ constructed in Lemma 42.49.1 satisfy $P'_ p(Q \oplus Q') = P'_ p(Q) + P'_ p(Q')$.

Proof. This follows immediately from the construction of these classes and Lemma 42.47.9. $\square$

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