hcju/stacksqa huggingface dataset:
The last equation follows from the first since $f_*[X] = [R(X) : R(Y)][Y]$ by definition. Proof of the first equaltion. Let $q : T \to Y$ be the morphism of Lemma \ref{lemma-Gm-torsor} constructed using $\mathcal{L}$ on $Y$. We will use all the properties of $T$ stated in this lemma. Consider the fibre product diagram $$ \xymatrix{ T' \ar[r]_{q'} \ar[d]_h & X \ar[d]^f \\ T \ar[r]^q & Y } $$ Then $q' : T' \to X$ is the morphism constructed using $f^*\mathcal{L}$ on $X$. It suffices to prove the equality after pulling back to $T'$. The left hand side pulls back to \begin{align*} q^*f_*\left(\text{div}_{f^*\mathcal{L}}(f^*s)\right) & = h_*(q')^*\text{div}_{f^*\mathcal{L}}(f^*s) \\ & = h_*\text{div}_{(q')^*f^*\mathcal{L}}((q')^*f^*s) \\ & = h_*\text{div}_{h^*q^*\mathcal{L}}(h^*q^*s) \end{align*} The first equality by Lemma \ref{lemma-flat-pullback-proper-pushforward}. The second by Lemma \ref{lemma-prepare-flat-pullback-cap-c1} using that $T'$ is integral. The third because the displayed diagram commutes. The right hand side pulls back to $$ [R(X) : R(Y)]q^*\text{div}_\mathcal{L}(s) = [R(T') : R(T)]\text{div}_{q^*\mathcal{L}}(q^*s) $$ This follows from Lemma \ref{lemma-prepare-flat-pullback-cap-c1}, the fact that $T$ is integral, and the equality $[R(T') : R(T)] = [R(X) : R(Y)]$ whose proof we omit (it follows from Lemma \ref{lemma-flat-pullback-proper-pushforward} but that would be a silly way to prove the equality). Thus it suffices to prove the lemma for $h : T' \to T$, the invertible module $q^\mathcal{L}$ and the section $q^*s$. Since $q^*\mathcal{L} = \mathcal{O}_T$ we reduce to the case where $\mathcal{L} \cong \mathcal{O}$ discussed in the next paragraph. \medskip\noindent Assume that $\mathcal{L} = \mathcal{O}_Y$. In this case $s$ corresponds to a rational function $g \in R(Y)$. Using the embedding $R(Y) \subset R(X)$ we may think of $g$ as a rational on $X$ and we are simply trying to prove $$ f_*\left(\text{div}_X(g)\right) = [R(X) : R(Y)]\text{div}_Y(g). $$ Comparing with the result of Lemma \ref{lemma-proper-pushforward-alteration} we see this true since $\text{Nm}_{R(X)/R(Y)}(g) = g^{[R(X) : R(Y)]}$ as $g \in R(Y)^*$.
Suppose that $p$ has the property that for every integral closed subspace $W \subset X$ the map $p|_W : W \to Y$ is a closed immersion. Then, by definition of capping with $c_1(\mathcal{L})$ the lemma holds. \medskip\noindent We will use this remark to reduce to a special case. Namely, write $\alpha = \sum n_i[W_i]$ with $n_i \not = 0$ and $W_i$ pairwise distinct. Let $W'_i \subset Y$ be the ``image'' of $W_i$ as in Lemma \ref{lemma-proper-image}. Consider the diagram $$ \xymatrix{ X' = \coprod W_i \ar[r]_-q \ar[d]_{p'} & X \ar[d]^p \\ Y' = \coprod W'_i \ar[r]^-{q'} & Y. } $$ Since $\{W_i\}$ is locally finite on $X$, and $p$ is proper we see that $\{W'_i\}$ is locally finite on $Y$ and that $q, q', p'$ are also proper morphisms. We may think of $\sum n_i[W_i]$ also as a $k$-cycle $\alpha' \in Z_k(X')$. Clearly $q_*\alpha' = \alpha$. We have $q_*(c_1(q^*p^*\mathcal{L}) \cap \alpha') = c_1(p^*\mathcal{L}) \cap q_*\alpha'$ and $(q')_*(c_1((q')^*\mathcal{L}) \cap p'_*\alpha') = c_1(\mathcal{L}) \cap q'_*p'_*\alpha'$ by the initial remark of the proof. Hence it suffices to prove the lemma for the morphism $p'$ and the cycle $\sum n_i[W_i]$. Clearly, this means we may assume $X$, $Y$ integral, $f : X \to Y$ dominant and $\alpha = [X]$. In this case the result follows from Lemma \ref{lemma-equal-c1-as-cycles}.
Let $\alpha \in Z_{k + 1}(X)$, and $\alpha \sim_{rat} 0$. We have to show that $c_1(\mathcal{L}) \cap \alpha$ as defined in Definition \ref{definition-cap-c1} is zero. By Definition \ref{definition-rational-equivalence} there exists a locally finite family $\{W_j\}$ of integral closed subspaces with $\dim_\delta(W_j) = k + 2$ and rational functions $f_j \in R(W_j)^*$ such that $$ \alpha = \sum (i_j)_*\text{div}_{W_j}(f_j) $$ Note that $p : \coprod W_j \to X$ is a proper morphism, and hence $\alpha = p_*\alpha'$ where $\alpha' \in Z_{k + 1}(\coprod W_j)$ is the sum of the principal divisors $\text{div}_{W_j}(f_j)$. By Lemma \ref{lemma-pushforward-cap-c1} we have $c_1(\mathcal{L}) \cap \alpha = p_*(c_1(p^*\mathcal{L}) \cap \alpha')$. Hence it suffices to show that each $c_1(\mathcal{L}|_{W_j}) \cap \text{div}_{W_j}(f_j)$ is zero. In other words we may assume that $X$ is integral and $\alpha = \text{div}_X(f)$ for some $f \in R(X)^*$. \medskip\noindent Assume $X$ is integral and $\alpha = \text{div}_X(f)$ for some $f \in R(X)^*$. We can think of $f$ as a regular meromorphic section of the invertible sheaf $\mathcal{N} = \mathcal{O}_X$. Choose a meromorphic section $s$ of $\mathcal{L}$ and denote $\beta = \text{div}_\mathcal{L}(s)$. By Lemma \ref{lemma-commutativity-on-integral} we conclude that $$ c_1(\mathcal{L}) \cap \alpha = c_1(\mathcal{O}_X) \cap \beta. $$ However, by Lemma \ref{lemma-c1-cap-additive} we see that the right hand side is zero in $\CH_k(X)$ as desired.
Write $\alpha = \sum m_j[Z_j]$ for some locally finite collection of integral closed subspaces $Z_j \subset X$ with $\dim_\delta(Z_j) = k + 2$. Consider the proper morphism $p : \coprod Z_j \to X$. Set $\alpha' = \sum m_j[Z_j]$ as a $(k + 2)$-cycle on $\coprod Z_j$. By several applications of Lemma \ref{lemma-pushforward-cap-c1} we see that $c_1(\mathcal{L}) \cap c_1(\mathcal{N}) \cap \alpha = p_*(c_1(p^*\mathcal{L}) \cap c_1(p^*\mathcal{N}) \cap \alpha')$ and $c_1(\mathcal{N}) \cap c_1(\mathcal{L}) \cap \alpha = p_*(c_1(p^*\mathcal{N}) \cap c_1(p^*\mathcal{L}) \cap \alpha')$. Hence it suffices to prove the formula in case $X$ is integral and $\alpha = [X]$. In this case the result follows from Lemma \ref{lemma-commutativity-on-integral} and the definitions.
Write $\alpha = \sum n_j[W_j]$ where $i_j : W_j \to X$ are integral closed subspaces with $\dim_\delta(W_j) = k$. Since $D$ is the vanishing locus of $s$ we see that $D \cap W_j$ is the vanishing locus of the restriction $s|_{W_j}$. Hence for each $j$ such that $W_j \not \subset D$ we have $c_1(\mathcal{L}) \cap [W_j] = [D \cap W_j]_k$ by Lemma \ref{lemma-geometric-cap}. So we have $$ c_1(\mathcal{L}) \cap \alpha = \sum\nolimits_{W_j \not \subset D} n_j[D \cap W_j]_k + \sum\nolimits_{W_j \subset D} n_j i_{j, *}(c_1(\mathcal{L})|_{W_j}) \cap [W_j]) $$ in $\CH_k(X)$ by Definition \ref{definition-cap-c1}. The right hand side matches (termwise) the pushforward of the class $i^*\alpha$ on $D$ from Definition \ref{definition-gysin-homomorphism}. Hence we win.
Suppose $\alpha = [W']$ for some integral closed subspace $W' \subset X'$. Let $W \subset X$ be the ``image'' of $W'$ as in Lemma \ref{lemma-proper-image}. In case $W' \not \subset D'$, then $W \not \subset D$ and we see that $$ [W' \cap D']_k = \text{div}_{\mathcal{L}'|_{W'}}({s'|_{W'}}) \quad\text{and}\quad [W \cap D]_k = \text{div}_{\mathcal{L}|_W}(s|_W) $$ and hence $f_*$ of the first cycle equals the second cycle by Lemma \ref{lemma-equal-c1-as-cycles}. Hence the equality holds as cycles. In case $W' \subset D'$, then $W \subset D$ and $f_*(c_1(\mathcal{L}|_{W'}) \cap [W'])$ is equal to $c_1(\mathcal{L}|_W) \cap [W]$ in $\CH_k(W)$ by the second assertion of Lemma \ref{lemma-equal-c1-as-cycles}. By Remark \ref{remark-infinite-sums-rational-equivalences} the result follows for general $\alpha'$.
Suppose $\alpha = [W]$ for some integral closed subspace $W \subset X$. Let $W' = f^{-1}(W) \subset X'$. In case $W \not \subset D$, then $W' \not \subset D'$ and we see that $$ W' \cap D' = g^{-1}(W \cap D) $$ as closed subspaces of $D'$. Hence the equality holds as cycles, see Lemma \ref{lemma-pullback-coherent}. In case $W \subset D$, then $W' \subset D'$ and $W' = g^{-1}(W)$ with $[W']_{k + 1 + r} = g^*[W]$ and equality holds in $\CH_{k + r}(D')$ by Lemma \ref{lemma-flat-pullback-cap-c1}. By Remark \ref{remark-infinite-sums-rational-equivalences} the result follows for general $\alpha'$.
The assumption means that $s|_Z$ is a regular section of $\mathcal{L}|_Z$. Thus $D \cap Z = Z(s)$ and we get $$ [D \cap Z]_k = \sum n_i [Z(s_i)]_k $$ as cycles where $s_i = s|_{Z_i}$, the $Z_i$ are the irreducible components of $\delta$-dimension $k + 1$, and $[Z]_{k + 1} = \sum n_i[Z_i]$. See Lemma \ref{lemma-prepare-geometric-cap}. We have $D \cap Z_i = Z(s_i)$. Comparing with the definition of the gysin map we conclude.
Write $\text{div}_\mathcal{N}(t) = \sum \text{ord}_{Z_i, \mathcal{N}}(t)[Z_i]$ for some integral closed subspaces $Z_i \subset X$ of $\delta$-dimension $n - 1$. We may assume that the family $\{Z_i\}$ is locally finite, that $t \in \Gamma(U, \mathcal{N}|_U)$ is a generator where $U = X \setminus \bigcup Z_i$, and that every irreducible component of $D$ is one of the $Z_i$, see Spaces over Fields, Lemmas \ref{spaces-over-fields-lemma-components-locally-finite}, \ref{spaces-over-fields-lemma-divisor-locally-finite}, and \ref{spaces-over-fields-lemma-divisor-meromorphic-locally-finite}. \medskip\noindent Set $\mathcal{L} = \mathcal{O}_X(D)$. Denote $s \in \Gamma(X, \mathcal{O}_X(D)) = \Gamma(X, \mathcal{L})$ the canonical section. We will apply the discussion of Section \ref{section-key} to our current situation. For each $i$ let $\xi_i \in |Z_i|$ be its generic point. Let $B_i = \mathcal{O}_{X, \xi_i}^h$. For each $i$ we pick generators $s_i$ of $\mathcal{L}_{\xi_i}$ and $t_i$ of $\mathcal{N}_{\xi_i}$ over $B_i$ but we insist that we pick $s_i = s$ if $Z_i \not \subset D$. Write $s = f_i s_i$ and $t = g_i t_i$ with $f_i, g_i \in B_i$. Then $\text{ord}_{Z_i, \mathcal{N}}(t) = \text{ord}_{B_i}(g_i)$. On the other hand, we have $f_i \in B_i$ and $$ [D]_{n - 1} = \sum \text{ord}_{B_i}(f_i)[Z_i] $$ because of our choices of $s_i$. We claim that $$ i^*\text{div}_\mathcal{N}(t) = \sum \text{ord}_{B_i}(g_i) \text{div}_{\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) $$ as cycles. More precisely, the right hand side is a cycle representing the left hand side. Namely, this is clear by our formula for $\text{div}_\mathcal{N}(t)$ and the fact that $\text{div}_{\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) = [Z(s_i|_{Z_i})]_{n - 2} = [Z_i \cap D]_{n - 2}$ when $Z_i \not \subset D$ because in that case $s_i|_{Z_i} = s|_{Z_i}$ is a regular section, see Lemma \ref{lemma-compute-c1}. Similarly, $$ c_1(\mathcal{N}) \cap [D]_{n - 1} = \sum \text{ord}_{B_i}(f_i) \text{div}_{\mathcal{N}|_{Z_i}}(t_i|_{Z_i}) $$ The key formula (Lemma \ref{lemma-key-formula}) gives the equality $$ \sum \left( \text{ord}_{B_i}(f_i) \text{div}_{\mathcal{N}|_{Z_i}}(t_i|_{Z_i}) - \text{ord}_{B_i}(g_i) \text{div}_{\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) \right) = \sum \text{div}_{Z_i}(\partial_{B_i}(f_i, g_i)) $$ of cycles. If $Z_i \not \subset D$, then $f_i = 1$ and hence $\text{div}_{Z_i}(\partial_{B_i}(f_i, g_i)) = 0$. Thus we get a rational equivalence between our specific cycles representing $i^*\text{div}_\mathcal{N}(t)$ and $c_1(\mathcal{N}) \cap [D]_{n - 1}$ on $D$. This finishes the proof.
Let $\alpha \in Z_{k + 1}(X)$ and assume that $\alpha \sim_{rat} 0$. This means there exists a locally finite collection of integral closed subspaces $W_j \subset X$ of $\delta$-dimension $k + 2$ and $f_j \in R(W_j)^*$ such that $\alpha = \sum i_{j, *}\text{div}_{W_j}(f_j)$. Set $X' = \coprod W_i$ and consider the diagram $$ \xymatrix{ D' \ar[d]_q \ar[r]_{i'} & X' \ar[d]^p \\ D \ar[r]^i & X } $$ of Remark \ref{remark-pullback-pairs}. Since $X' \to X$ is proper we see that $i^*p_* = q_*(i')^*$ by Lemma \ref{lemma-closed-in-X-gysin}. As we know that $q_*$ factors through rational equivalence (Lemma \ref{lemma-proper-pushforward-rational-equivalence}), it suffices to prove the result for $\alpha' = \sum \text{div}_{W_j}(f_j)$ on $X'$. Clearly this reduces us to the case where $X$ is integral and $\alpha = \text{div}(f)$ for some $f \in R(X)^*$. \medskip\noindent Assume $X$ is integral and $\alpha = \text{div}(f)$ for some $f \in R(X)^*$. If $X = D$, then we see that $i^*\alpha$ is equal to $c_1(\mathcal{L}) \cap \alpha$. This is rationally equivalent to zero by Lemma \ref{lemma-factors}. If $D \not = X$, then we see that $i^*\text{div}_X(f)$ is equal to $c_1(\mathcal{O}_D) \cap [D]_{n - 1}$ in $\CH_k(D)$ by Lemma \ref{lemma-gysin-factors-general}. Of course capping with $c_1(\mathcal{O}_D)$ is the zero map.
Denote $j : D \cap D' \to D$ and $j' : D \cap D' \to D'$ the closed immersions corresponding to $(\mathcal{L}|_{D'}, s|_{D'}$ and $(\mathcal{L}'_D, s|_D)$. We have to show that $(j')^*i^*\alpha = j^* (i')^*\alpha$ for all $\alpha \in \CH_k(X)$. Let $W \subset X$ be an integral closed subscheme of dimension $k$. We will prove the equality in case $\alpha = [W]$. The general case will then follow from the observation in Remark \ref{remark-infinite-sums-rational-equivalences} (and the specific shape of our rational equivalence produced below). We will deduce the equality for $\alpha = [W]$ from the key formula. \medskip\noindent We let $\sigma$ be a nonzero meromorphic section of $\mathcal{L}|_W$ which we require to be equal to $s|_W$ if $W \not \subset D$. We let $\sigma'$ be a nonzero meromorphic section of $\mathcal{L}'|_W$ which we require to be equal to $s'|_W$ if $W \not \subset D'$. Write $$ \text{div}_{\mathcal{L}|_W}(\sigma) = \sum \text{ord}_{Z_i, \mathcal{L}|_W}(\sigma)[Z_i] = \sum n_i[Z_i] $$ and similarly $$ \text{div}_{\mathcal{L}'|_W}(\sigma') = \sum \text{ord}_{Z_i, \mathcal{L}'|_W}(\sigma')[Z_i] = \sum n'_i[Z_i] $$ as in the discussion in Section \ref{section-key}. Then we see that $Z_i \subset D$ if $n_i \not = 0$ and $Z'_i \subset D'$ if $n'_i \not = 0$. For each $i$, let $\xi_i \in |Z_i|$ be the generic point. As in Section \ref{section-key} we choose for each $i$ an element $\sigma_i \in \mathcal{L}_{\xi_i}$, resp.\ $\sigma'_i \in \mathcal{L}'_{\xi_i}$ which generates over $B_i = \mathcal{O}_{W, \xi_i}^h$ and which is equal to the image of $s$, resp.\ $s'$ if $Z_i \not \subset D$, resp.\ $Z_i \not \subset D'$. Write $\sigma = f_i \sigma_i$ and $\sigma' = f'_i\sigma'_i$ so that $n_i = \text{ord}_{B_i}(f_i)$ and $n'_i = \text{ord}_{B_i}(f'_i)$. From our definitions it follows that $$ (j')^*i^*[W] = \sum \text{ord}_{B_i}(f_i) \text{div}_{\mathcal{L}'|_{Z_i}}(\sigma'_i|_{Z_i}) $$ as cycles and $$ j^*(i')^*[W] = \sum \text{ord}_{B_i}(f'_i) \text{div}_{\mathcal{L}|_{Z_i}}(\sigma_i|_{Z_i}) $$ The key formula (Lemma \ref{lemma-key-formula}) now gives the equality $$ \sum \left( \text{ord}_{B_i}(f_i) \text{div}_{\mathcal{L}'|_{Z_i}}(\sigma'_i|_{Z_i}) - \text{ord}_{B_i}(f'_i) \text{div}_{\mathcal{L}|_{Z_i}}(\sigma_i|_{Z_i}) \right) = \sum \text{div}_{Z_i}(\partial_{B_i}(f_i, f'_i)) $$ of cycles. Note that $\text{div}_{Z_i}(\partial_{B_i}(f_i, f'_i)) = 0$ if $Z_i \not \subset D \cap D'$ because in this case either $f_i = 1$ or $f'_i = 1$. Thus we get a rational equivalence between our specific cycles representing $(j')^*i^*[W]$ and $j^*(i')^*[W]$ on $D \cap D' \cap W$.
Let $W \subset Y$ be an integral closed subspace of $\delta$-dimension $k + 1$. By Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-relative-Cartier} we see that $D \cap p^{-1}W$ is an effective Cartier divisor on $p^{-1}W$. By Lemma \ref{lemma-easy-gysin} we get the first equality in $$ i^*[p^{-1}W]_{k + r + 1} = [D \cap p^{-1}W]_{k + r} = [(p|_D)^{-1}(W)]_{k + r}. $$ and the second because $D \cap p^{-1}(W) = (p|_D)^{-1}(W)$ as algebraic spaces. Since by definition $p^*[W] = [p^{-1}W]_{k + r + 1}$ we see that $i^*p^*[W] = (p|_D)^*[W]$ as cycles. If $\alpha = \sum m_j[W_j]$ is a general $k + 1$ cycle, then we get $i^*\alpha = \sum m_j i^*p^*[W_j] = \sum m_j(p|_D)^*[W_j]$ as cycles. This proves then first equality. To deduce the second from the first apply Lemma \ref{lemma-support-cap-effective-Cartier}.
Let $\alpha \in \CH_{k + r}(X)$. Write $\alpha = \sum m_j[W_j]$ with $m_j \not = 0$ and $W_j$ pairwise distinct integral closed subspaces of $\delta$-dimension $k + r$. Then the family $\{W_j\}$ is locally finite in $X$. Let $Z_j \subset Y$ be the integral closed subspace such that we obtain a dominant morphism $W_j \to Z_j$ as in Lemma \ref{lemma-proper-image}. For any quasi-compact open $V \subset Y$ we see that $f^{-1}(V) \cap W_j$ is nonempty only for finitely many $j$. Hence the collection $Z_j$ of closures of images is a locally finite collection of integral closed subspaces of $Y$. \medskip\noindent Consider the fibre product diagrams $$ \xymatrix{ f^{-1}(Z_j) \ar[r] \ar[d]_{f_j} & X \ar[d]^f \\ Z_j \ar[r] & Y } $$ Suppose that $[W_j] \in Z_{k + r}(f^{-1}(Z_j))$ is rationally equivalent to $f_j^*\beta_j$ for some $k$-cycle $\beta_j \in \CH_k(Z_j)$. Then $\beta = \sum m_j \beta_j$ will be a $k$-cycle on $Y$ and $f^*\beta = \sum m_j f_j^*\beta_j$ will be rationally equivalent to $\alpha$ (see Remark \ref{remark-infinite-sums-rational-equivalences}). This reduces us to the case $Y$ integral, and $\alpha = [W]$ for some integral closed subscheme of $X$ dominating $Y$. In particular we may assume that $d = \dim_\delta(Y) < \infty$. \medskip\noindent Hence we can use induction on $d = \dim_\delta(Y)$. If $d < k$, then $\CH_{k + r}(X) = 0$ and the lemma holds; this is the base case of the induction. Consider a nonempty open $V \subset Y$. Suppose that we can show that $\alpha|_{f^{-1}(V)} = f^*\beta$ for some $\beta \in Z_k(V)$. By Lemma \ref{lemma-exact-sequence-open} we see that $\beta = \beta'|_V$ for some $\beta' \in Z_k(Y)$. By the exact sequence $\CH_k(f^{-1}(Y \setminus V)) \to \CH_k(X) \to \CH_k(f^{-1}(V))$ of Lemma \ref{lemma-restrict-to-open} we see that $\alpha - f^*\beta'$ comes from a cycle $\alpha' \in \CH_{k + r}(f^{-1}(Y \setminus V))$. Since $\dim_\delta(Y \setminus V) < d$ we win by induction on $d$. \medskip\noindent In particular, by replacing $Y$ by a suitable open we may assume $Y$ is a scheme with generic point $\eta$. The isomorphism $Y_\eta \cong \mathbf{A}^r_\eta$ extends to an isomorphism over a nonempty open $V \subset Y$, see Limits of Spaces, Lemma \ref{spaces-limits-lemma-descend-finite-presentation}. This reduces us to the case of schemes which is Chow Homology, Lemma \ref{chow-lemma-pullback-affine-fibres-surjective}.
Let $p : L \to X$ be as in Lemma \ref{lemma-linebundle} and let $o : X \to L$ be the zero section. Observe that $p^*\mathcal{L}^{\otimes -1}$ has a canonical section whose vanishing locus is exactly the effective Cartier divisor $o(X)$. Let $\alpha \in \CH_k(X)$. Then we see that $$ p^*(c_1(\mathcal{L}^{\otimes -1}) \cap \alpha) = c_1(p^*\mathcal{L}^{\otimes -1}) \cap p^*\alpha = o_* o^* p^*\alpha $$ by Lemmas \ref{lemma-flat-pullback-cap-c1} and \ref{lemma-relative-effective-cartier}. Since $c$ is a bivariant class we have \begin{align*} p^*(c \cap c_1(\mathcal{L}^{\otimes -1}) \cap \alpha) & = c \cap p^*(c_1(\mathcal{L}^{\otimes -1}) \cap \alpha) \\ & = c \cap o_* o^* p^*\alpha \\ & = o_* o^* p^*(c \cap \alpha) \\ & = p^*(c_1(\mathcal{L}^{\otimes -1}) \cap c \cap \alpha) \end{align*} (last equality by the above applied to $c \cap \alpha$). Since $p^*$ is injective by a lemma cited above we get that $c_1(\mathcal{L}^{\otimes -1})$ is in the center of $A^*(X)$. This proves the lemma.
Let $Z \subset X$ be an integral closed subspace of $\delta$-dimension $k$. We will prove the lemma for $\alpha = [Z]$. We omit the argument deducing the general case from this special case; hint: argue as in Remark \ref{remark-infinite-sums-rational-equivalences}. \medskip\noindent Let $P_Z = P \times_X Z$ be the base change; of course $\pi_Z : P_Z \to Z$ is the projective bundle associated to $\mathcal{E}|_Z$ and $\mathcal{O}_P(1)$ pulls back to the corresponding invertible module on $P_Z$. Since $c_1(\mathcal{O}_P(1) \cap -$, and $\pi^*$ are bivariant classes by Lemmas \ref{lemma-cap-c1-bivariant} and \ref{lemma-flat-pullback-bivariant} we see that $$ \pi_*\left( c_1(\mathcal{O}_P(1))^s \cap \pi^*[Z] \right) = (Z \to X)_*\pi_{Z, *}\left( c_1(\mathcal{O}_{P_Z}(1))^s \cap \pi_Z^*[Z] \right) $$ Hence it suffices to prove the lemma in case $X$ is integral and $\alpha  = [X]$. \medskip\noindent Assume $X$ is integral, $\dim_\delta(X) = k$, and $\alpha = [X]$. Note that $\pi^*[X] = [P]$ as $P$ is integral of $\delta$-dimension $r - 1$. If $s < r - 1$, then by construction $c_1(\mathcal{O}_P(1))^s \cap [P]$ a $(k + r - 1 - s)$-cycle. Hence the pushforward of this cycle is zero for dimension reasons. \medskip\noindent Let $s = r - 1$. By the argument given above we see that $\pi_*(c_1(\mathcal{O}_P(1))^s \cap [P]) = n [X]$ for some $n \in \mathbf{Z}$. We want to show that $n = 1$. For the same dimension reasons as above it suffices to prove this result after replacing $X$ by a dense open. Thus we may assume $X$ is a scheme and the result follows from Chow Homology, Lemma \ref{chow-lemma-cap-projective-bundle}.
(For the case of linebundles, see Lemma \ref{lemma-linebundle}.) For surjectivity see Lemma \ref{lemma-pullback-affine-fibres-surjective}. Let $(\pi  : P \to X, \mathcal{O}_P(1))$ be the projective space bundle associated to the finite locally free sheaf $\mathcal{E} \oplus \mathcal{O}_X$. Let $s \in \Gamma(P, \mathcal{O}_P(1))$ correspond to the global section $(0, 1) \in \Gamma(X, \mathcal{E} \oplus \mathcal{O}_X)$. Let $D = Z(s) \subset P$. Note that $(\pi|_D : D \to X , \mathcal{O}_P(1)|_D)$ is the projective space bundle associated to $\mathcal{E}$. We denote $\pi_D = \pi|_D$ and $\mathcal{O}_D(1) = \mathcal{O}_P(1)|_D$. Moreover, $D$ is an effective Cartier divisor on $P$. Hence $\mathcal{O}_P(D) = \mathcal{O}_P(1)$ (see Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-characterize-OD}). Also there is an isomorphism $E \cong P \setminus D$. Denote $j : E \to P$ the corresponding open immersion. For injectivity we use that the kernel of $$ j^* : \CH_{k + r}(P) \longrightarrow \CH_{k + r}(E) $$ are the cycles supported in the effective Cartier divisor $D$, see Lemma \ref{lemma-restrict-to-open}. So if $p^*\alpha = 0$, then $\pi^*\alpha = i_*\beta$ for some $\beta \in \CH_{k + r}(D)$. By Lemma \ref{lemma-chow-ring-projective-bundle} we may write $$ \beta = \pi_D^*\beta_0 + \ldots + c_1(\mathcal{O}_D(1))^{r - 1} \cap \pi_D^* \beta_{r - 1}. $$ for some $\beta_i \in \CH_{k + i}(X)$. By Lemmas \ref{lemma-relative-effective-cartier} and \ref{lemma-pushforward-cap-c1} this implies $$ \pi^*\alpha = i_*\beta = c_1(\mathcal{O}_P(1)) \cap \pi^*\beta_0 + \ldots + c_1(\mathcal{O}_D(1))^r \cap \pi^*\beta_{r - 1}. $$ Since the rank of $\mathcal{E} \oplus \mathcal{O}_X$ is $r + 1$ this contradicts Lemma \ref{lemma-pushforward-cap-c1} unless all $\alpha$ and all $\beta_i$ are zero.
Uniqueness and existence of the maps $c_i(\mathcal{E}) \cap -$ follows immediately from Lemma \ref{lemma-chow-ring-projective-bundle} and the given description of $c_0(\mathcal{E})$. For every $i \in \mathbf{Z}$ the rule which to every morphism $X' \to X$ of good algebraic spaces over $B$ assigns the map $$ t_i(\mathcal{E}) \cap - : \CH_k(X') \longrightarrow \CH_{k - i}(X'),\quad \alpha \longmapsto \pi'_*(c_1(\mathcal{O}_{P'}(1))^{r - 1 + i} \cap (\pi')^*\alpha) $$ is a bivariant class\footnote{Up to signs these are the Segre classes of $\mathcal{E}$.} by Lemmas \ref{lemma-cap-c1-bivariant}, \ref{lemma-flat-pullback-bivariant}, and \ref{lemma-push-proper-bivariant}. By Lemma \ref{lemma-cap-projective-bundle} we have $t_i(\mathcal{E}) = 0$ for $i < 0$ and $t_0(\mathcal{E}) = 1$. Applying pushforward to the equation in the statement of the lemma we find from Lemma \ref{lemma-cap-projective-bundle} that $$ (-1)^r t_1(\mathcal{E}) + (-1)^{r - 1}c_1(\mathcal{E}) = 0 $$ In particular we find that $c_1(\mathcal{E})$ is a bivariant class. If we multiply the equation in the statement of the lemma by $c_1(\mathcal{O}_{P'}(1))$ and push the result forward to $X'$ we find $$ (-1)^r t_2(\mathcal{E}) + (-1)^{r - 1} t_1(\mathcal{E}) \cap c_1(\mathcal{E}) + (-1)^{r - 2} c_2(\mathcal{E}) = 0 $$ As before we conclude that $c_2(\mathcal{E})$ is a bivariant class. And so on.
Namely, in this case $P = \mathbf{P}(\mathcal{L}) = X$ with $\mathcal{O}_P(1) = \mathcal{L}$ by our normalization of the projective bundle, see Section \ref{section-projective-space-bundle-formula}. Hence the equation in Lemma \ref{lemma-segre-classes} reads $$ (-1)^0 c_1(\mathcal{L})^0 \cap c^{new}_1(\mathcal{L}) \cap \alpha + (-1)^1 c_1(\mathcal{L})^1 \cap c^{new}_0(\mathcal{L}) \cap \alpha = 0 $$ where $c_i^{new}(\mathcal{L})$ is as in Definition \ref{definition-chern-classes}. Since $c_0^{new}(\mathcal{L}) = 1$ and $c_1(\mathcal{L})^0 = 1$ we conclude.