The Stacks project

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

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

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 $P'_ p(Q)$. $\square$

Proof. Recall that the construction is as follows

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). $\square$

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

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

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. $\square$

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

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 $. $\square$

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. $\square$

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

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:

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

and by Lemma 42.49.4 we have

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

and the proof is complete. $\square$

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

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

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

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

By Lemma 42.47.1 the left hand side is equal to

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

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

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

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

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

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