hcju/stacksqa huggingface dataset:
Choose proper morphisms $b_i : W_i \to \mathbf{P}^1_X$ and $Q_i \in D(\mathcal{O}_{W_i})$ as well as closed subschemes $T_i \subset W_{i, \infty}$ as in the construction of the localized Chern classes for $F_i$ or more generally as in Lemma \ref{lemma-independent-loc-chern-bQ}. Choose a commutative diagram $$ \xymatrix{ W \ar[d]^{g_1} \ar[rd]^b \ar[r]_{g_2} & W_2 \ar[d]^{b_2} \\ W_1 \ar[r]^{b_1} & \mathbf{P}^1_X } $$ where all morphisms are proper and isomorphisms over $\mathbf{A}^1_X$. For example, we can take $W$ to be the closure of the graph of the isomorphism between $b_1^{-1}(\mathbf{A}^1_X)$ and $b_2^{-1}(\mathbf{A}^1_X)$. By Lemma \ref{lemma-independent-loc-chern-bQ} we may work with $W$, $b = b_i \circ g_i$, $Lg_i^*Q_i$, and $g_i^{-1}(T_i)$ to construct the localized Chern classes $c_p(Z_i \to X, F_i)$. Thus we reduce to the situation described in the next paragraph. \medskip\noindent Assume we have \begin{enumerate} \item a proper morphism $b : W \to \mathbf{P}^1_X$ which is an isomorphism over $\mathbf{A}^1_X$, \item $E_i \subset W_\infty$ is the inverse image of $Z_i$, \item perfect objects $Q_i \in D(\mathcal{O}_W)$ whose Chern classes are defined, such that \begin{enumerate} \item the restriction of $Q_i$ to $b^{-1}(\mathbf{A}^1_X)$ is the pullback of $F_i$, and \item there exists a closed subscheme $T_i \subset W_\infty$ containing all points of $W_\infty$ lying over $X \setminus Z_i$ such that $Q_i|_{T_i}$ is zero. \end{enumerate} \end{enumerate} By Lemma \ref{lemma-independent-loc-chern-bQ} we have $$ c_p(Z_i \to X, F_i) = c'_p(Q_i) = (E_i \to Z_i)_* \circ c'_p(Q_i|_{E_i}) \circ C $$ and $$ P_p(Z_i \to X, F_i) = P'_p(Q_i) = (E_i \to Z_i)_* \circ P'_p(Q_i|_{E_i}) \circ C $$ for $i = 1, 2$. Next, we observe that $Q = Q_1 \otimes_{\mathcal{O}_W}^\mathbf{L} Q_2$ satisfies (3)(a) and (3)(b) for $F_1 \otimes_{\mathcal{O}_X}^\mathbf{L} F_2$ and $T_1 \cup T_2$. Hence we see that $$ c_p(Z_1 \cap Z_2 \to X, F_1 \otimes_{\mathcal{O}_X}^\mathbf{L} F_2) = (E_1 \cap E_2 \to Z_1 \cap Z_2)_* \circ c'_p(Q|_{E_1 \cap E_2}) \circ C $$ and $$ P_p(Z_1 \cap Z_2 \to X, F_1 \otimes_{\mathcal{O}_X}^\mathbf{L} F_2) = (E_1 \cap E_2 \to Z_1 \cap Z_2)_* \circ P'_p(Q|_{E_1 \cap E_2}) \circ C $$ by the same lemma. By Lemma \ref{lemma-silly-tensor-product} the classes $c'_p(Q|_{E_1 \cap E_2})$ and $P'_p(Q|_{E_1 \cap E_2})$ can be expanded in the correct manner in terms of the classes $c'_p(Q_i|_{E_i})$ and $P'_p(Q_i|_{E_i})$. Then finally Lemma \ref{lemma-homomorphism-final} tells us that polynomials in $c'_p(Q_i|_{E_i})$ and $P'_p(Q_i|_{E_i})$ agree with the corresponding polynomials in $c'_p(Q_i)$ and $P'_p(Q_i)$ as desired.
Since both $i_* \circ j^* \circ C$ and $p^*$ are bivariant classes (see Lemmas \ref{lemma-flat-pullback-bivariant} and \ref{lemma-push-proper-bivariant}) we can use the equation $$ i_* \circ j^* \circ C  = p^* \circ c(Z \to X, \mathcal{N}) $$ (suitably interpreted) to define $c(Z \to X, \mathcal{N})$ as a bivariant class. This works because $p^*$ is always bijective on chow groups by Lemma \ref{lemma-vectorbundle}. \medskip\noindent Let $X' \to X$, $Z' \to X'$, and $\mathcal{N}'$ be as in Lemma \ref{lemma-pullback-virtual-normal-sheaf}. Write $c = c(Z \to X, \mathcal{N})$ and $c' = c(Z' \to X', \mathcal{N}')$. The second statement of the lemma means that $c'$ is the restriction of $c$ as in Remark \ref{remark-restriction-bivariant}. Since we claim this is true for all $X'/X$ locally of finite type, a formal argument shows that it suffices to check that $c' \cap \alpha' = c \cap \alpha'$ for $\alpha' \in \CH_k(X')$. To see this, note that we have a commutative diagram $$ \xymatrix{ C_{Z'}X' \ar[d] \ar[r] & W'_\infty \ar[d] \ar[r] & W' \ar[d] \ar[r] & \mathbf{P}^1_{X'} \ar[d] \\ C_ZX \ar[r] & W_\infty \ar[r] & W \ar[r] & \mathbf{P}^1_X } $$ which induces closed immersions: $$ W' \to W \times_{\mathbf{P}^1_X} \mathbf{P}^1_{X'},\quad W'_\infty \to W_\infty \times_X X',\quad C_{Z'}X' \to C_ZX \times_Z Z' $$ To get $c \cap \alpha'$ we use the class $C \cap \alpha'$ defined using the morphism $W \times_{\mathbf{P}^1_X} \mathbf{P}^1_{X'} \to \mathbf{P}^1_{X'}$ in Lemma \ref{lemma-gysin-at-infty}. To get $c' \cap \alpha'$ on the other hand, we use the class $C' \cap \alpha'$ defined using the morphism $W' \to \mathbf{P}^1_{X'}$. By Lemma \ref{lemma-gysin-at-infty-independent} the pushforward of $C' \cap \alpha'$ by the closed immersion $W'_\infty \to (W \times_{\mathbf{P}^1_X} \mathbf{P}^1_{X'})_\infty$, is equal to $C \cap \alpha'$. Hence the same is true for the pullbacks to the opens $$ C_{Z'}X' \subset W'_\infty,\quad C_ZX \times_Z Z' \subset (W \times_{\mathbf{P}^1_X} \mathbf{P}^1_{X'})_\infty $$ by Lemma \ref{lemma-flat-pullback-proper-pushforward}. Since we have a commutative diagram $$ \xymatrix{ C_{Z'} X' \ar[d] \ar[r] & N' \ar@{=}[d] \\ C_ZX \times_Z Z' \ar[r] & N \times_Z Z' } $$ these classes pushforward to the same class on $N'$ which proves that we obtain the same element $c \cap \alpha' = c' \cap \alpha'$ in $\CH_*(Z')$.
Denote $N' \to N$ the closed immersion of vector bundles corresponding to the surjection $\mathcal{N}^\vee \to (\mathcal{N}')^\vee$. Then we have closed immersions $$ C_ZX \to N' \to N $$ Thus the desired relationship between the bivariant classes follows immediately from Lemma \ref{lemma-easy-virtual-class}.
The bivariant class $c_{top}(\mathcal{E}) \in A^*(Z \times_X Y)$ was defined in Remark \ref{remark-top-chern-class}. By Lemma \ref{lemma-construction-gysin} we may replace $X$ by $Y$. Thus we may assume $Z \to X$ is a regular closed immersion of codimension $r$, we have $\dim_\delta(X) = n$, and we have to show that $c(Z \to X, \mathcal{N}) \cap [X]_n = c_{top}(\mathcal{E}) \cap [Z]_{n - r}$ in $\CH_*(Z)$. By Lemma \ref{lemma-gysin-decompose} we may even assume $\mathcal{N}^\vee \to \mathcal{C}_{Z/X}$ is an isomorphism. In other words, we have to show $c(Z \to X, \mathcal{C}_{Z/X}^\vee) \cap [X]_n = [Z]_{n - r}$ in $\CH_*(Z)$. \medskip\noindent Let us trace through the steps in the definition of $c(Z \to X, \mathcal{C}_{Z/X}^\vee) \cap [X]_n$. Let $b : W \to \mathbf{P}^1_X$ be the blowing up of $\infty(Z)$. We first have to compute $C \cap [X]_n$ where $C \in A^0(W_\infty \to X)$ is the class of Lemma \ref{lemma-gysin-at-infty}. To do this, note that $[W]_{n + 1}$ is a cycle on $W$ whose restriction to $\mathbf{A}^1_X$ is equal to the flat pullback of $[X]_n$. Hence $C \cap [X]_n$ is equal to $i_\infty^*[W]_{n + 1}$. Since $W_\infty$ is an effective Cartier divisor on $W$ we have $i_\infty^*[W]_{n + 1} = [W_\infty]_n$, see Lemma \ref{lemma-easy-gysin}. The restriction of this class to the open $C_ZX \subset W_\infty$ is of course just $[C_ZX]_n$. Because $Z \subset X$ is regularly embedded we have $$ \mathcal{C}_{Z/X, *} = \text{Sym}(\mathcal{C}_{Z/X}) $$ as graded $\mathcal{O}_Z$-algebras, see Divisors, Lemma \ref{divisors-lemma-quasi-regular-immersion}. Hence $p : N = C_ZX \to Z$ is the structure morphism of the vector bundle associated to the finite locally free module $\mathcal{C}_{Z/X}$ of rank $r$. Then it is clear that $p^*[Z]_{n - r} = [C_ZX]_n$ and the proof is complete.
The statement makes sense as $Z \times_X Y$ is a closed subscheme of $Y$. Because $\mathcal{N}$ has rank $r$ we know that $c(Z \to X, \mathcal{N}) \cap [Y]_n$ is in $\CH_{n - r}(Z \times_X Y)$. Since $\dim_\delta(Z \cap Y) \leq n - r$ the chow group $\CH_{n - r}(Z \times_X Y)$ is freely generated by the cycle classes of the irreducible components $W \subset Z \times_X Y$ of $\delta$-dimension $n - r$. Let $\xi \in W$ be the generic point. By assumption (2) we see that $\dim(\mathcal{O}_{Y, \xi}) = r$. On the other hand, since $\mathcal{N}$ has rank $r$ and since $\mathcal{N}^\vee \to \mathcal{C}_{Z/X}$ is surjective, we see that the ideal sheaf of $Z$ is locally cut out by $r$ equations. Hence the quasi-coherent ideal sheaf $\mathcal{I} \subset \mathcal{O}_Y$ of $Z \times_X Y$ in $Y$ is locally generated by $r$ elements. Since $\mathcal{O}_{Y, \xi}$ is Cohen-Macaulay of dimension $r$ and since $\mathcal{I}_\xi$ is an ideal of definition (as $\xi$ is a generic point of $Z \times_X Y$) it follows that $\mathcal{I}_\xi$ is generated by a regular sequence (Algebra, Lemma \ref{algebra-lemma-reformulate-CM}). By Divisors, Lemma \ref{divisors-lemma-Noetherian-scheme-regular-ideal} we see that $\mathcal{I}$ is generated by a regular sequence over an open neighbourhood $V \subset Y$ of $\xi$. By our description of $\CH_{n - r}(Z \times_X Y)$ it suffices to show that $c(Z \to X, \mathcal{N}) \cap [V]_n = [Z \times_X V]_{n - r}$ in $\CH_{n - r}(Z \times_X V)$. This follows from Lemma \ref{lemma-gysin-fundamental} because the excess normal sheaf is $0$ over $V$.
We will use the criterion of Lemma \ref{lemma-bivariant-zero}. Thus we may assume that $X$ is an integral scheme and we have to show that $i^*[X]$ is equal to $c \cap [X]$. Let $n = \dim_\delta(X)$. As usual, there are two cases. \medskip\noindent If $X = D$, then we see that both classes are represented by $c_1(\mathcal{N}) \cap [X]_n$. See Lemma \ref{lemma-gysin-fundamental} and Definition \ref{definition-gysin-homomorphism}. \medskip\noindent If $D \not = X$, then $D \to X$ is an effective Cartier divisor and in particular a regular closed immersion of codimension $1$. Again by Lemma \ref{lemma-gysin-fundamental} we conclude $c(D \to X, \mathcal{N}) \cap [X]_n = [D]_{n - 1}$. The same is true by definition for the gysin homomorphism and we conclude once again.
To check this we may use Lemma \ref{lemma-bivariant-zero}. Thus we may assume $X$ is an integral scheme and we have to show $c \cap c(Z \to X, \mathcal{N}) \cap [X] = c(Z \to X, \mathcal{N}) \cap c \cap [X]$ in $\CH_*(Z \times_X Y)$. \medskip\noindent If $Z = X$, then $c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{N})$ by Lemma \ref{lemma-gysin-fundamental} which commutes with the bivariant class $c$, see Lemma \ref{lemma-cap-commutative-chern}. \medskip\noindent Assume that $Z$ is not equal to $X$. By Lemma \ref{lemma-bivariant-zero} it even suffices to prove the result after blowing up $X$ (in a nonzero ideal). Let us blowup $X$ in the ideal sheaf of $Z$. This reduces us to the case where $Z$ is an effective Cartier divisor, see Divisors, Lemma \ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}, \medskip\noindent If $Z$ is an effective Cartier divisor, then we have $$ c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{E}) \circ i^* $$ where $i^* \in A^1(Z \to X)$ is the gysin homomorphism associated to $i : Z \to X$ (Lemma \ref{lemma-gysin-bivariant}) and $\mathcal{E}$ is the dual of the kernel of $\mathcal{N}^\vee \to \mathcal{C}_{Z/X}$, see Lemmas \ref{lemma-gysin-decompose} and \ref{lemma-gysin-agrees}. Then we conclude because Chern classes are in the center of the bivariant ring (in the strong sense formulated in Lemma \ref{lemma-cap-commutative-chern}) and $c$ commutes with the gysin homomorphism $i^*$ by definition of bivariant classes.
Denote $W \to \mathbf{P}^1_X$ the blowing up of $\infty(Z)$ as in Section \ref{section-blowup-Z-first}. Similarly, denote $W' \to \mathbf{P}^1_X$ the blowing up of $\infty(Y)$. Since $\infty(Z) \subset \infty(Y)$ we get an opposite inclusion of ideal sheaves and hence a map of the graded algebras defining these blowups. This produces a rational morphism from $W$ to $W'$ which in fact has a canonical representative $$ W \supset U \longrightarrow W' $$ See Constructions, Lemma \ref{constructions-lemma-morphism-relative-proj}. A local calculation (omitted) shows that $U$ contains at least all points of $W$ not lying over $\infty$ and the open subscheme $C_Z X$ of the special fibre. After shrinking $U$ we may assume $U_\infty = C_Z X$ and $\mathbf{A}^1_X \subset U$. Another local calculation (omitted) shows that the morphism $U_\infty \to W'_\infty$ induces the canonical morphism $C_Z X \to C_Y X \subset W'_\infty$ of normal cones induced by the inclusion of ideals sheaves coming from $Z \subset Y$. Denote $W'' \subset W$ the strict transform of $\mathbf{P}^1_Y \subset \mathbf{P}^1_X$ in $W$. Then $W''$ is the blowing up of $\mathbf{P}^1_Y$ in $\infty(Z)$ by Divisors, Lemma \ref{divisors-lemma-strict-transform} and hence $(W'' \cap U)_\infty = C_ZY$. \medskip\noindent Consider the effective Cartier divisor $i : \mathbf{P}^1_Y \to W'$ from (\ref{item-find-Z-in-blowup}) and its associated bivariant class $i^* \in A^1(\mathbf{P}^1_Y \to W')$ from Lemma \ref{lemma-gysin-bivariant}. We similarly denote $(i'_\infty)^* \in A^1(W'_\infty \to W')$ the gysin map at infinity. Observe that the restriction of $i'_\infty$ (Remark \ref{remark-restriction-bivariant}) to $U$ is the restriction of $i_\infty^* \in A^1(W_\infty \to W)$ to $U$. On the one hand we have $$ (i'_\infty)^* i^* [U]_{n + 1} = i_\infty^* i^* [U]_{n + 1} = i_\infty^* [(W'' \cap U)_\infty]_{n + 1} = [C_ZY]_n $$ because $i_\infty^*$ kills all classes supported over $\infty$, because $i^*[U]$ and $[W'']$ agree as cycles over $\mathbf{A}^1$, and because $C_ZY$ is the fibre of $W'' \cap U$ over $\infty$. On the other hand, we have $$ (i'_\infty)^* i^* [U]_{n + 1} = i^* i_\infty^*[U]_{n + 1} = i^* [U_\infty] = o^*[C_YX]_n $$ because $(i'_\infty)^*$ and $i^*$ commute (Lemma \ref{lemma-gysin-commutes-gysin}) and because the fibre of $i : \mathbf{P}^1_Y \to W'$ over $\infty$ factors as $o : Y \to C_YX$ and the open immersion $C_YX \to W'_\infty$. The lemma follows.
The canonical map $\pi^*\mathcal{C} \to \mathcal{O}_E(1)$ vanishes exactly along $i(X)$. Hence the Koszul complex on the map $$ \pi^*\mathcal{C} \otimes \mathcal{O}_E(-1) \to \mathcal{O}_E $$ is a resolution of $i_*\mathcal{O}_X$. In particular we see that $i_*\mathcal{O}_X$ is a perfect object of $D(\mathcal{O}_E)$ whose Chern classes are defined. The vanishing of $c_t(i_*\mathcal{O}_X)$ for $t = 1, \ldots, t - 1$ follows from Lemma \ref{lemma-compute-koszul}. This lemma also gives $$ c_r(i_*\mathcal{O}_X) = - (r - 1)! c_r(\pi^*\mathcal{C} \otimes \mathcal{O}_E(-1)) $$ On the other hand, by Lemma \ref{lemma-chern-classes-dual} we have $$ c_r(\pi^*\mathcal{C} \otimes \mathcal{O}_E(-1)) = (-1)^r c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)) $$ and $\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)$ has a section $s$ vanishing exactly along $i(X)$. \medskip\noindent After replacing $X$ by a scheme locally of finite type over $X$, it suffices to prove that both sides of the equality have the same effect on an element $\alpha \in \CH_*(E)$. Since $C \to X$ is a vector bundle, every cycle class on $C$ is of the form $p^*\beta$ for some $\beta \in \CH_*(X)$ (Lemma \ref{lemma-vectorbundle}). Hence by Lemma \ref{lemma-restrict-to-open} we can write $\alpha = \pi^*\beta + \gamma$ where $\gamma$ is supported on $E \setminus C$. Using the equalities above it suffices to show that $$ p^*(\pi_*(c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)) \cap [W])) = j^*[W] $$ when $W \subset E$ is an integral closed subscheme which is either (a) disjoint from $C$ or (b) is of the form $W = \pi^{-1}Y$ for some integral closed subscheme $Y \subset X$. Using the section $s$ and Lemma \ref{lemma-top-chern-class} we find in case (a) $c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)) \cap [W] = 0$ and in case (b) $c_r(\pi^*\mathcal{C}^\vee \otimes \mathcal{O}_E(1)) \cap [W] = [i(Y)]$. The result follows easily from this; details omitted.
Parts (1) and (2) follow immediately from Lemma \ref{lemma-agreement-with-loc-chern} combined with Lemma \ref{lemma-gysin-fundamental}. Then we deduce parts (3) and (4) using the relationship between $ch_p = (1/p!)P_p$ and $c_p$ given in Lemma \ref{lemma-loc-chern-character}. (Namely, $(-1)^{r - 1}(r - 1)!ch_r = c_r$ provided $c_1 = c_2 = \ldots = c_{r - 1} = 0$.)
Observe that the map $\prod_{i \geq 0} A^i(X) \to \prod_{i \geq 0} A^i(X)$ multiplying by $2^i$ on $A^i(X)$ is a ring map. Hence, since $\psi^2$ is also a ring map, it suffices to prove the formulas for additive generators of $K_0(\textit{Vect}(X))$. Thus we may assume $\alpha = [\mathcal{E}]$ for some finite locally free $\mathcal{O}_X$-module $\mathcal{E}$. By construction of the Chern classes of $\mathcal{E}$ we immediately reduce to the case where $\mathcal{E}$ has constant rank $r$, see Remark \ref{remark-extend-to-finite-locally-free}. In this case, we can choose a projective smooth morphism $p : P \to X$ such that restriction $A^*(X) \to A^*(P)$ is injective and such that $p^*\mathcal{E}$ has a finite filtration whose graded parts are invertible $\mathcal{O}_P$-modules $\mathcal{L}_j$, see Lemma \ref{lemma-splitting-principle}. Then $[p^*\mathcal{E}] = \sum [\mathcal{L}_j]$ and hence $\psi^2([p^\mathcal{E}]) = \sum [\mathcal{L}_j^{\otimes 2}]$ by definition of $\psi^2$. Setting $x_j  = c_1(\mathcal{L}_j)$ we have $$ c(\alpha) = \prod (1 + x_j) \quad\text{and}\quad c(\psi^2(\alpha)) = \prod (1 + 2 x_j) $$ in $\prod A^i(P)$ and we have $$ ch(\alpha) = \sum \exp(x_j) \quad\text{and}\quad ch(\psi^2(\alpha)) = \sum \exp(2 x_j) $$ in $\prod A^i(P)$. From these formulas the desired result follows.
We have $K'_0(X) = K_0(X) = K_0(\textit{Vect}(X))$ by Derived Categories of Schemes, Lemmas \ref{perfect-lemma-Kprime-K} and \ref{perfect-lemma-K-is-old-K}. By Remark \ref{remark-chern-classes-agree} the composition given agrees with the map of Proposition \ref{proposition-K-tensor-Q} for $X = Y$. Thus the result follows from the proposition.
The first statement is Divisors, Lemma \ref{divisors-lemma-composition-regular-immersion}. By Divisors, Lemma \ref{divisors-lemma-transitivity-conormal-quasi-regular} there is a short exact sequence $$ 0 \to i^*(\mathcal{C}_{Y/Z}) \to \mathcal{C}_{X/Z} \to \mathcal{C}_{X/Y} \to 0 $$ Thus the result by the more general Lemma \ref{lemma-gysin-composition}.
The first statement is Divisors, Lemma \ref{divisors-lemma-section-smooth-regular-immersion}. It suffices to show that $s^! \cap p^*[Z] = [Z]$ in $\CH_*(X)$ for any integral closed subscheme $Z \subset X$ as the assumptions are preserved by base change by $X' \to X$ locally of finite type. After replacing $P$ by an open neighbourhood of $s(Z)$ we may assume $P \to X$ is smooth of fixed relative dimension $r$. Say $\dim_\delta(Z) = n$. Then every irreducible component of $p^{-1}(Z)$ has dimension $r + n$ and $p^*[Z]$ is given by $[p^{-1}(Z)]_{n + r}$. Observe that $s(X) \cap p^{-1}(Z) = s(Z)$ scheme theoretically. Hence by the same reference as used above $s(X) \cap p^{-1}(Z)$ is a closed subscheme regularly embedded in $\overline{p}^{-1}(Z)$ of codimension $r$. We conclude by Lemma \ref{lemma-gysin-fundamental}.
Given a second such factorization $f = g' \circ i'$ we can consider the smooth morphism $g'' : P \times_Y P' \to Y$, the immersion $i'' : X \to P \times_Y P'$ and the factorization $f = g'' \circ i''$. Thus we may assume that we have a diagram $$ \xymatrix{ & P' \ar[d]^p \ar[rd]^{g'} \\ X \ar[r]^i \ar[ru]^{i'} & P \ar[r]^g & Y } $$ where $p$ is a smooth morphism. Then $(g')^* = p^* \circ g^*$ (Lemma \ref{lemma-compose-flat-pullback}) and hence it suffices to show that $i^! = (i')^! \circ p^*$ in $A^*(X \to P)$. Consider the commutative diagram $$ \xymatrix{ & X \times_P P' \ar[d]^{\overline{p}} \ar[r]_j & P' \ar[d]^p \\ X \ar[ru]^s \ar[r]^1 & X \ar[r]^i & P } $$ where $s =(1, i')$. Then $s$ and $j$ are regular immersions (by Divisors, Lemma \ref{divisors-lemma-section-smooth-regular-immersion} and Divisors, Lemma \ref{divisors-lemma-flat-base-change-regular-immersion}) and $i' = j \circ s$. By Lemma \ref{lemma-composition-regular-immersion} we have $(i')^! = s^! \circ j^!$. Since the square is cartesian, the bivariant class $j^!$ is the restriction (Remark \ref{remark-restriction-bivariant}) of $i^!$ to $P'$, see Lemma \ref{lemma-construction-gysin}. Since bivariant classes commute with flat pullbacks we find $j^! \circ p^* = \overline{p}^* \circ i^!$. Thus it suffices to show that $s^! \circ \overline{p}^* = \text{id}$ which is done in Lemma \ref{lemma-section-smooth}.
Choose a factorization $f = g \circ i$ with $i : X \to P$ an immersion and $g : P \to Y$ smooth. Observe that for any morphism $Y' \to Y$ which is locally of finite type, the base changes of $f'$, $g'$, $i'$ satisfy the same assumptions (see Morphisms, Lemmas \ref{morphisms-lemma-base-change-smooth} and \ref{morphisms-lemma-base-change-syntomic} and More on Morphisms, Lemma \ref{more-morphisms-lemma-flat-lci}). Thus we reduce to proving that $f^*[Y] = i^!(g^*[Y])$ in case $Y$ is integral, see Lemma \ref{lemma-bivariant-zero}. Set $n = \dim_\delta(Y)$. After decomposing $X$ and $P$ into connected components we may assume $f$ is flat of relative dimension $r$ and $g$ is smooth of relative dimension $t$. Then $f^*[Y] = [X]_{n + s}$ and $g^*[Y] = [P]_{n + t}$. On the other hand $i$ is a regular immersion of codimension $t - s$. Thus $i^![P]_{n + t} = [X]_{n + s}$ (Lemma \ref{lemma-gysin-fundamental}) and the proof is complete.
Observe that $g \circ f$ is a local complete intersection morphism by More on Morphisms, Lemma \ref{more-morphisms-lemma-composition-lci} and hence the statement of the lemma makes sense. If $X \to P$ is an immersion of $X$ into a scheme $P$ smooth over $Z$ then $X \to P \times_Z Y$ is an immersion of $X$ into a scheme smooth over $Y$. This prove the first assertion of the lemma. Let $Y \to P'$ be an immersion of $Y$ into a scheme $P'$ smooth over $Z$. Consider the commutative diagram $$ \xymatrix{ X \ar[r] \ar[d] & P \times_Z Y \ar[r]_a \ar[ld]^p & P \times_Z P' \ar[ld]^q \\ Y \ar[r]_b \ar[d] & P' \ar[ld] \\ Z } $$ Here the horizontal arrows are regular immersions, the south-west arrows are smooth, and the square is cartesian. Whence $a^! \circ q^* = p^* \circ b^!$ as bivariant classes commute with flat pullback. Combining this fact with Lemmas \ref{lemma-composition-regular-immersion} and \ref{lemma-compose-flat-pullback} the reader finds the statement of the lemma holds true. Small detail omitted.
Choose a factorization $f = g \circ i$ with $g$ smooth and $i$ an immersion. Since $f^! = i^! \circ g^!$ it suffices to prove the lemma for $g^!$ (which is given by flat pullback) and for $i^!$. The result for flat pullback is part of the definition of a bivariant class. The case of $i^!$ follows immediately from Lemma \ref{lemma-gysin-commutes}.