hcju/stacksqa huggingface dataset:
The assumption on rank implies that $E_2|_{X_1 \cap X_2}$ is zero. Hence $c'_i(E_2)$ is defined for all $i \geq 1$ and the statement makes sense. The actual equality follows immediately from Lemma \ref{lemma-chern-class-perfect-tensor-invertible} and the characterization of $c'_i$ in Lemma \ref{lemma-silly}.
First we observe that the statement makes sense. Namely, we have $X = (X_2 \cap X'_2) \cup Y$ where $Y = (X_1 \cap X'_1) \cup (X_1 \cap X'_2) \cup (X_2 \cap X'_1)$ and the object $E \otimes_{\mathcal{O}_X}^\mathbf{L} E'$ restricts to zero on $Y$. The actual equalities follow from the characterization of our classes in Lemma \ref{lemma-silly} and the equalities of Lemma \ref{lemma-chern-classes-perfect-tensor-product}. We omit the details.
Given $\alpha \in \CH_k(X)$ there exists a $\beta \in \CH_{k + 1}(W)$ restricting to the flat pullback of $\alpha$ on $b^{-1}(\mathbf{A}^1_X)$, see Lemma \ref{lemma-exact-sequence-open}. A second choice of $\beta$ differs from $\beta$ by a cycle supported on $W_\infty$, see Lemma \ref{lemma-restrict-to-open}. Since the normal bundle of the effective Cartier divisor $D_\infty \subset \mathbf{P}^1_X$ of (\ref{equation-zero-infty}) is trivial, the gysin homomorphism $i_\infty^*$ kills cycle classes supported on $W_\infty$, see Remark \ref{remark-gysin-on-cycles}. Hence setting $C \cap \alpha = i_\infty^*\beta$ is well defined. \medskip\noindent 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 \ref{lemma-support-cap-effective-Cartier}. By our choice of $\beta$ we have $i_0^*\beta = \alpha$ as cycles on $W_0 = X \times \{0\}$, see for example Lemma \ref{lemma-relative-effective-cartier}. Thus we see that $i_{\infty, *}(C \cap \alpha) = i_{0, *}\alpha$ as stated in the lemma. \medskip\noindent 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 $C \cap - : \CH_k(X') \to \CH_k(W'_\infty)$ by the same construction as above. To see that this family of operations defines a bivariant class, we consider the diagram $$ \xymatrix{ & & & \CH_*(X) \ar[d]^{\text{flat pullback}} \\ \CH_{* + 1}(W_\infty) \ar[r] \ar[rd]^0 & \CH_{* + 1}(W) \ar[d]^{i_\infty^*} \ar[rr]^{\text{flat pullback}} & & \CH_{* + 1}(\mathbf{A}^1_X) \ar[r] \ar@{..>}[lld]^{C \cap -} & 0 \\ & \CH_*(W_\infty) } $$ 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 \ref{lemma-flat-pullback-bivariant} and \ref{lemma-gysin-bivariant}. Then a formal argument (involving huge diagrams of schemes and their chow groups) shows that the dotted arrow is a bivariant operation.
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 $\alpha \in \CH_k(X)$ choose $\beta' \in \CH_{k + 1}(W')$ restricting to the flat pullback of $\alpha$ on $(b')^{-1}\mathbf{A}^1_X$. Then $\beta = g_*\beta' \in \CH_{k + 1}(W)$ 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 \ref{lemma-closed-in-X-gysin}. 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.
Let $\beta \in \CH_{k + 1}(W)$. 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 $\CH_k(X)$ because $i_{\infty, *}i_\infty^*\beta$ and $i_{0, *}i_0^*\beta$ represent the same class on $W$ (for example by Lemma \ref{lemma-support-cap-effective-Cartier}) 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 \ref{lemma-linebundle} and \ref{lemma-linebundle-formulae}. 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.
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 \ref{lemma-gysin-at-infty}. Denote $P'_p(Q|_E)$, resp.\ $c'_p(Q|_E)$ in $A^p(E \to W_\infty)$ the bivariant class constructed in Lemma \ref{lemma-silly}. 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 \ref{lemma-push-proper-bivariant}. 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 \ref{lemma-silly-silly}. The third equality because $c_p(Q)$ is a bivariant class. The fourth equality by Lemma \ref{lemma-gysin-at-infty}. 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 $P'_p(Q)$.
The independence of $T$ follows immediately from Lemma \ref{lemma-silly-independent}. \medskip\noindent 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 \ref{lemma-gysin-at-infty-independent} 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 \ref{remark-restriction-bivariant}. 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.
The first equality holds because $c_p(Q|_{X \times \{0\}}) = (Z \to X)_* \circ c'_p(Q)$ by Lemma \ref{lemma-localized-chern-pre}. We may prove the second equality one cycle class at a time (see Lemma \ref{lemma-bivariant-zero}). Since the construction of the bivariant classes in the lemma is compatible with base change, we may assume we have some $\alpha \in \CH_k(X)$ 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 \ref{lemma-silly}. 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 $\beta \in \CH_{k + 1}(W)$ 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 \ref{lemma-silly-silly}. 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 \ref{lemma-homomorphism-pre}. The seventh uses again that $c_p(Q)$ is a bivariant class. The final holds as $C \cap \alpha = i_\infty^*\beta$.
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 \ref{lemma-gysin-at-infty} and $P'_p(Q|_E)$, resp.\ $c'_p(Q|_E)$ are as in Lemma \ref{lemma-silly}. By Lemma \ref{lemma-gysin-at-infty-commutes} we see that $C$ commutes with $c$ and by Lemma \ref{lemma-silly-commutes} 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.
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 \ref{lemma-localized-chern-pre} 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 \ref{lemma-gysin-at-infty} and the bivariant classes $P'_p(Q|_E)$ and $c'_p(Q|_E)$ of Lemma \ref{lemma-silly} for the restriction $Q|_E$ of $Q$ to the inverse image $E$ of $Z$ in $W_\infty$. Observe that by Lemma \ref{lemma-silly-compose} 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 \ref{remark-ring-loc-classes} 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 \ref{remark-res-push} the multiplication in Remark \ref{remark-ring-loc-classes} 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 \ref{lemma-silly} we have $$ (E \to W_\infty)_* \circ b_{p_i} = c_{p_i}(Q|_{W_\infty}) $$ and by Lemma \ref{lemma-homomorphism} 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.
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 \ref{lemma-localized-chern-pre}. Hence the equality in degrees $< p + p'$ follows from the additivity of Lemma \ref{lemma-additivity-on-perfect}. \medskip\noindent Let's take $n \geq p + p'$. As in the proof of Lemma \ref{lemma-localized-chern-pre} 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 \ref{lemma-silly-sum-c}. 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 \ref{remark-ring-loc-classes} on both sides. There are three cases, depending on whether $i \geq p$, $j \geq p'$, or both. \medskip\noindent 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 \ref{remark-res-push}. 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 \ref{lemma-silly} 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 \ref{lemma-homomorphism}. \medskip\noindent 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 \ref{lemma-homomorphism}. \medskip\noindent 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 \ref{lemma-cap-commutative-chern}). 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.
Here are the choices we made above: the bounded complex $\mathcal{E}^\bullet$  of finite locally free $\mathcal{O}_X$-modules representing $E$, the blowup $b : W \to \mathbf{P}^1_X$, the choice of $\mathcal{Q}^\bullet$, and the closed subscheme $T'$. In Lemma \ref{lemma-localized-chern-pre-independent} we have seen that the class is independent of the choice of $T'$. In More on Flatness, Lemma \ref{flat-lemma-complex-and-divisor-derived} we have seen that the blowing up $b : W \to \mathbf{P}^1_X$ and the isomorphism class $Q$ of $\mathcal{Q}^\bullet$ in $D(\mathcal{O}_W)$ only depend on the isomorphism class of $L\text{pr}^*E$ in $D(\mathcal{O}_{\mathbf{P}^1_X})$ where $\text{pr} : \mathbf{P}^1_X \to X$ is the projection morphism. Since the construction of Lemma \ref{lemma-localized-chern-pre} depends only on the isomorphism class $Q$, we conclude.
Choose a bounded complex $\mathcal{E}^\bullet$ of finite locally free $\mathcal{O}_X$-modules representing $E$. Denote $(\mathcal{E}')^\bullet = f^*\mathcal{E}^\bullet$. Observe that $\mathbf{P}^1_{X'} \to \mathbf{P}^1_X$ is a morphism of schemes such that the pullback of the effective Cartier divisor $(\mathbf{P}^1_X)_\infty$ is the effective Cartier divisor $(\mathbf{P}^1_{X'})_\infty$. By More on Flatness, Lemma \ref{flat-lemma-complex-and-divisor-blowup-base-change} we obtain a commutative diagram $$ \xymatrix{ W' \ar[rd]_{b'} \ar[r]_-g & \mathbf{P}^1_{X'} \times_{\mathbf{P}^1_X} W \ar[d]_r \ar[r]_-q & W \ar[d]^b \\ & \mathbf{P}^1_{X'} \ar[r] & \mathbf{P}^1_X } $$ such that $W'$ is the strict transform of $\mathbf{P}^1_{X'}$ with respect to $b$ and such that $(\mathcal{Q}')^\bullet = g^*q^*\mathcal{Q}^\bullet$. The restriction of the bivariant class $P_p(Z \to X, E)$, resp.\ $c_p(Z \to X, E)$ corresponds to the class constructed in Lemma \ref{lemma-localized-chern-pre} using the proper morphism $r$ and the complex $q^*\mathcal{Q}^\bullet$. On the other hand, the bivariant class $P_p(Z' \to X', E')$, resp.\ $c_p(Z' \to X', E')$ corresponds to the proper morphism $b'$ and the complex $(\mathcal{Q}')^\bullet$. Thus we conclude by Lemma \ref{lemma-localized-chern-pre-independent}.
We only prove the second equality and we omit the proof of the first. Since $c_p(Z \to X, E)$ is a bivariant class and since the base change of $Z \to X$ by $Z \to X$ is $\text{id} : Z \to Z$ we have $c_p(Z \to X, E) \cap i_*\alpha = c_p(Z \to X, E) \cap \alpha$. By Lemma \ref{lemma-base-change-loc-chern} the restriction of $c_p(Z \to X, E)$ to $Z$ (!) is the localized Chern class for $\text{id} : Z \to Z$ and $E|_Z$. Thus the result follows from (\ref{equation-defining-property-localized-classes}) with $X = Z$.
This is true because the construction of $P_p(Z' \to X, E)$, resp.\ $c_p(Z' \to X, E)$ uses the exact same morphism $b : W \to \mathbf{P}^1_X$ and $\mathcal{Q}^\bullet$. Then we can use Lemma \ref{lemma-silly-shrink} to conclude. Some details omitted.
The assumptions on $E$ imply that there is an open $U \subset X$ containing $X_1$ such that $E|_U$ is zero, resp.\ isomorphic to a finite locally free $\mathcal{O}_U$-module of rank $< p$. See More on Algebra, Lemma \ref{more-algebra-lemma-lift-perfect-from-residue-field}. Let $Z \subset X$ be the complement of $U$ in $X$ endowed with the reduced induced closed subscheme structure. Then $P_p(X_2 \to X, E) = (Z \to X_2)_* \circ P_p(Z \to X, E)$, resp.\ $c_p(X_2 \to X, E) = (Z \to X_2)_* \circ c_p(Z \to X, E)$ by Lemma \ref{lemma-loc-chern-shrink-Z}. Now we can prove that $P_p(X_2 \to X, E)$, resp.\ $c_p(X_2 \to X, E)$ satisfies the characterization of $P'_p(E_2)$, resp.\ $c'_p(E_2)$ given in Lemma \ref{lemma-silly}. Namely, by the relation $P_p(X_2 \to X, E) = (Z \to X_2)_* \circ P_p(Z \to X, E)$, resp.\ $c_p(X_2 \to X, E) = (Z \to X_2)_* \circ c_p(Z \to X, E)$ just proven and the fact that $X_1 \cap Z = \emptyset$, the composition $P_p(X_2 \to X, E) \circ i_{1, *}$, resp.\ $c_p(X_2 \to X, E) \circ i_{1, *}$ is zero by Lemma \ref{lemma-loc-chern-disjoint}. On the other hand, $P_p(X_2 \to X, E) \circ i_{2, *} = P_p(E_2)$, resp.\ $c_p(X_2 \to X, E) \circ i_{2, *} = c_p(E_2)$ by Lemma \ref{lemma-loc-chern-after-pushforward}.
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 $c'_{p_1}(Q_1)$. For $n > 1$ we have by induction that $c'_{p_n}(Q_n) \circ \ldots \circ c'_{p_1}(Q_1)$ is equal to $$ c'_{p_n}(Q_n) \circ (W_{n - 1, \infty} \cap \ldots \cap W_{1, \infty} \to Z_{n - 1} \cap \ldots \cap Z_1)_* \circ c'_{p_{n - 1}}(Q_{n - 1}|_{W_{n - 1}, \infty}) \circ \ldots \circ c'_{p_1}(Q_1|_{W_{1, \infty}}) \circ C $$ Observe that the restriction of $c'_{p_n}(Q_n)$ to $Z_{n - 1} \cap \ldots \cap Z_1$ is computed by $b' : W_{n - 1} \cap \ldots \cap W_1 \to \mathbf{P}^1_{Z_{n - 1} \cap \ldots \cap Z_1}$ and the restriction of $Q_n$ to $W_{n - 1} \cap \ldots \cap W_1$. 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 \ref{lemma-gysin-at-infty}. 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 \ref{lemma-homomorphism-pre} 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 \ref{lemma-loc-chern-agree} and assumption (2) on $Q_i$ in the statement of the lemma. Thus, if $\beta \in \CH_{k + 1}(W)$ 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.
The statement makes sense because the zero sheaf has rank $< 1$ and hence the classes $c_p(Z \to X, E)$ are defined for all $p \geq 1$. The result itself follows immediately from the more general Lemma \ref{lemma-localized-chern-pre-compose} as the localized Chern classes where defined using the procedure of Lemma \ref{lemma-localized-chern-pre} in Section \ref{section-localized-chern}.
This follows from Lemma \ref{lemma-homomorphism-commute}. Namely, our assumptions say $E$ is represented to a bounded complex $\mathcal{E}^\bullet$ of finite locally free $\mathcal{O}_X$-modules. Let $$ b : W \to \mathbf{P}^1_X \quad\text{and}\quad \mathcal{Q}^\bullet $$ be the blowing up and complex of $\mathcal{O}_W$-modules constructed in More on Flatness, Section \ref{flat-section-blowup-complexes-III}. Let $T \subset W_\infty$ be the closed subscheme whose existence is averted in More on Flatness, Lemma \ref{flat-lemma-graph-construction}. Let $T' \subset T$ be the open and closed subscheme such that $\mathcal{Q}^\bullet|_{T'}$ is zero, resp.\ isomorphic to a finite locally free sheaf of rank $< p$ placed in degree $0$. By definition $$ c_p(Z \to X, E) = c'_p(\mathcal{Q}^\bullet) $$ as bivariant operations (and not just on cycles over $X$) where the right hand side is the bivariant class constructed in Lemma \ref{lemma-localized-chern-pre} using $W, b, \mathcal{Q}^\bullet, T'$. By Lemma \ref{lemma-homomorphism-commute} we have $$ P'_p(\mathcal{Q}^\bullet) \circ c = c \circ P'_p(\mathcal{Q}^\bullet) \quad\text{resp.}\quad c'_p(\mathcal{Q}^\bullet) \circ c = c \circ c'_p(\mathcal{Q}^\bullet) $$ in $A^*(Y \times_X Z \to X)$ and we conclude.
The proof is exactly the same as the proof of Lemma \ref{lemma-additivity-loc-chern-c} except it uses Lemma \ref{lemma-localized-chern-pre-sum-P} at the very end. For $p > 0$ we can deduce this lemma from Lemma \ref{lemma-additivity-loc-chern-c} with $p_1 = p_3 = 1$ and the relationship between $P_p(Z \to X, E)$ and $c_p(Z \to X, E)$ given in Lemma \ref{lemma-loc-chern-character}. The case $p = 0$ can be shown directly (it is only interesting if $X$ has a connected component entirely contained in $Z$).