hcju/stacksqa huggingface dataset:
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 $k$. Let us prove the equality in case $\alpha = [W]$. We will deduce it 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 $B_i = \mathcal{O}_{W, \xi_i}$ 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 $D \cap D' \cap W$. By Remark \ref{remark-infinite-sums-rational-equivalences} the result follows for general $\alpha$.
Let $W \subset Y$ be an integral closed subscheme of $\delta$-dimension $k + 1$. By Divisors, Lemma \ref{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 schemes. 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 subschemes of $\delta$-dimension $k + r$. Then the family $\{W_j\}$ is locally finite in $X$. 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 $Z_j = \overline{f(W_j)}$ of closures of images is a locally finite collection of integral closed subschemes 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. By assumption there exists a dense open $V \subset Y$ such that $f^{-1}(V) \cong V \times \mathbf{A}^r$ as schemes over $V$. 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 Thus we may assume that $X = Y \times \mathbf{A}^r$. In this case we can factor $f$ as $$ X = Y \times \mathbf{A}^r \to Y \times \mathbf{A}^{r - 1} \to \ldots \to Y \times \mathbf{A}^1 \to Y. $$ Hence it suffices to do the case $r = 1$. By the argument in the second paragraph of the proof we are reduced to the case $\alpha = [W]$, $Y$ integral, and $W \to Y$ dominant. Again we can do induction on $d = \dim_\delta(Y)$. If $W = Y \times \mathbf{A}^1$, then $[W] = f^*[Y]$. Lastly, $W \subset Y \times \mathbf{A}^1$ is a proper inclusion, then $W \to Y$ induces a finite field extension $R(Y) \subset R(W)$. Let $P(T) \in R(Y)[T]$ be the monic irreducible polynomial such that the generic fibre of $W \to Y$ is cut out by $P$ in $\mathbf{A}^1_{R(Y)}$. Let $V \subset Y$ be a nonempty open such that $P \in \Gamma(V, \mathcal{O}_Y)[T]$, and such that $W \cap f^{-1}(V)$ is still cut out by $P$. Then we see that $\alpha|_{f^{-1}(V)} \sim_{rat} 0$ and hence $\alpha \sim_{rat} \alpha'$ for some cycle $\alpha'$ on $(Y \setminus V) \times \mathbf{A}^1$. By induction on the dimension we win.
For surjectivity see Lemma \ref{lemma-pullback-affine-fibres-surjective}. Let $o : X \to L$ be the zero section of $L \to X$, i.e., the morphism corresponding to the surjection $\text{Sym}^*(\mathcal{L}) \to \mathcal{O}_X$ which maps $\mathcal{L}^{\otimes n}$ to zero for all $n > 0$. Then $p \circ o = \text{id}_X$ and $o(X)$ is an effective Cartier divisor on $L$. Hence by Lemma \ref{lemma-relative-effective-cartier} we see that $o^* \circ p^* = \text{id}$ and we conclude that $p^*$ is injective too.
Since $p_*\mathcal{O}_L = \text{Sym}^*(\mathcal{L})$ we have $p_*(p^*\mathcal{L}^{\otimes -1}) = \text{Sym}^*(\mathcal{L}) \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes -1}$ by the projection formula (Cohomology, Lemma \ref{cohomology-lemma-projection-formula}) and the section mentioned in (1) is the canonical trivialization $\mathcal{O}_X \to \mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes -1}$. We omit the proof that the vanishing locus of this section is precisely $o(X)$. This proves (1). \medskip\noindent Parts (2), (3), and (4) we've seen in the course of the proof of Lemma \ref{lemma-linebundle}. Of course (4) is the first formula in Lemma \ref{lemma-relative-effective-cartier}. \medskip\noindent Part (5) follows from the second formula in Lemma \ref{lemma-relative-effective-cartier}, additivity of capping with $c_1$ (Lemma \ref{lemma-c1-cap-additive}), and the fact that capping with $c_1$ commutes with flat pullback (Lemma \ref{lemma-flat-pullback-cap-c1}). \medskip\noindent Part (6) follows from Lemma \ref{lemma-gysin-back} and the fact that $o^*p^*\mathcal{L} = \mathcal{L}$.
Let $W \subset Y$ be an integral closed subscheme of $\delta$-dimension $k$. We distinguish two cases. \medskip\noindent Assume $W \subset D$. Then $i^*[W] = c_1(\mathcal{L}_1) \cap [W] + c_1(\mathcal{L}_2) \cap [W]$ in $\CH_{k - 1}(D)$ by our definition of gysin homomorphisms and the additivity of Lemma \ref{lemma-c1-cap-additive}. Hence $p_1^*i^*[W] = p_1^*(c_1(\mathcal{L}_1) \cap [W]) + p_1^*(c_1(\mathcal{L}_2) \cap [W])$. On the other hand, we have $p^*[W] = [p^{-1}(W)]_{k + 1}$ by construction of flat pullback. And $p^{-1}(W) = W_1 \cup W_2$ (scheme theoretically) where $W_i = p_i^{-1}(W)$ is a line bundle over $W$ by the lemma (since formation of the diagram commutes with base change). Then $[p^{-1}(W)]_{k + 1} = [W_1] + [W_2]$ as $W_i$ are integral closed subschemes of $L$ of $\delta$-dimension $k + 1$. Hence \begin{align*} i_1^*p^*[W] & = i_1^*[p^{-1}(W)]_{k + 1} \\ & = i_1^*([W_1] + [W_2]) \\ & = c_1(p_1^*\mathcal{L}_1) \cap [W_1] + [W_1 \cap W_2]_k \\ & = c_1(p_1^*\mathcal{L}_1) \cap p_1^*[W] + [W_1 \cap W_2]_k \\ & = p_1^*(c_1(\mathcal{L}_1) \cap [W]) + [W_1 \cap W_2]_k \end{align*} by construction of gysin homomorphisms, the definition of flat pullback (for the second equality), and compatibility of $c_1 \cap -$ with flat pullback (Lemma \ref{lemma-flat-pullback-cap-c1}). Since $W_1 \cap W_2$ is the zero section of the line bundle $W_1 \to W$ we see from Lemma \ref{lemma-linebundle-formulae} that $[W_1 \cap W_2]_k = p_1^*(c_1(\mathcal{L}_2) \cap [W])$. Note that here we use the fact that $D_1$ is the line bundle which is the relative spectrum of the inverse of $\mathcal{L}_2$. Thus we get the same thing as before. \medskip\noindent Assume $W \not \subset D$. In this case, both $i_1^*p^*[W]$ and $p_1^*i^*[W]$ are represented by the $k - 1$ cycle associated to the scheme theoretic inverse image of $W$ in $D_1$.
Observe that $(i')^*$ is defined because we have the triple $(\mathcal{O}_{X'}, s', i' : D' \to X')$ as in Definition \ref{definition-gysin-homomorphism}. Thus the statement makes sense. \medskip\noindent Set $\mathcal{L}_1 = \mathcal{O}_X$, $\mathcal{L}_2 = \mathcal{L}$ and apply Lemma \ref{lemma-decompose-section} with the section $s$ of $\mathcal{L} = \mathcal{L}_1 \otimes_{\mathcal{O}_X} \mathcal{L}_2$. Take $D' = D_1$. The results now follow from the lemma, from Lemma \ref{lemma-decompose-section-formulae} and injectivity by Lemma \ref{lemma-linebundle}.
Of course (2) implies (1). Let $p : L \to X$ be as in Lemma \ref{lemma-linebundle} and let $o : X \to L$ be the zero section. Denote $p' : L' \to X'$ and $o' : X' \to L'$ their base changes. By Lemma \ref{lemma-linebundle-formulae} we have $$ p^*(c_1(\mathcal{L}) \cap \alpha) = - o_* \alpha \quad\text{and}\quad (p')^*(c_1(f^*\mathcal{L}) \cap \alpha') = - o'_* \alpha' $$ Since $c$ is a bivariant class we have \begin{align*} (p')^*(c \cap c_1(\mathcal{L}) \cap \alpha) & = c \cap p^*(c_1(\mathcal{L}) \cap \alpha) \\ & = - c \cap o_* \alpha \\ & = - o'_*(c \cap \alpha) \\ & = (p')^*(c_1(f^*\mathcal{L}) \cap c \cap \alpha) \end{align*} Since $(p')^*$ is injective by one of the lemmas cited above we obtain $c \cap c_1(\mathcal{L}) \cap \alpha = c_1(f^*\mathcal{L}) \cap c \cap \alpha$. The same is true after any base change by $Y \to X$ locally of finite type and hence we have the equality of bivariant classes stated in (2).
We prove this by induction on $d$. The base case $d = 0$ is true because in this case $X$ is a finite set of closed points and hence every invertible module is trivial. Assume $d > 0$. By Divisors, Lemma \ref{divisors-lemma-quasi-projective-Noetherian-pic-effective-Cartier} we can write $\mathcal{L}_{d + 1} \cong \mathcal{O}_X(D) \otimes \mathcal{O}_X(D')^{\otimes -1}$ for some effective Cartier divisors $D, D' \subset X$. Then $c_1(\mathcal{L}_{d + 1})$ is the difference of $c_1(\mathcal{O}_X(D))$ and $c_1(\mathcal{O}_X(D'))$ and hence we may assume $\mathcal{L}_{d + 1} = \mathcal{O}_X(D)$ for some effective Cartier divisor. \medskip\noindent Denote $i : D \to X$ the inclusion morphism and denote $i^* \in A^1(D \to X)$ the bivariant class given by the gysin hommomorphism as in Lemma \ref{lemma-gysin-bivariant}. We have $i_* \circ i^* = c_1(\mathcal{L}_{d + 1})$ in $A^1(X)$ by Lemma \ref{lemma-support-cap-effective-Cartier} (and Lemma \ref{lemma-push-proper-bivariant} to make sense of the left hand side). Since $c_1(\mathcal{L}_i)$ commutes with both $i_*$ and $i^*$ (by definition of bivariant classes) we conclude that $$ c_1(\mathcal{L}_1) \circ \ldots \circ c_1(\mathcal{L}_{d + 1}) = i_* \circ c_1(\mathcal{L}_1) \circ \ldots \circ c_1(\mathcal{L}_d) \circ i^* = i_* \circ c_1(\mathcal{L}_1|_D) \circ \ldots \circ c_1(\mathcal{L}_d|_D) \circ i^* $$ Thus we conclude by induction on $d$. Namely, we have $\dim(D) < d$ as none of the generic points of $X$ are in $D$.
The statement makes sense because given a triple $(\mathcal{L}, s, i : D \to X)$ as in Definition \ref{definition-gysin-homomorphism} such that $\mathcal{L}|_D \cong \mathcal{O}_D$, then the operation $i^*$ is defined on the level of cycles, see Remark \ref{remark-gysin-on-cycles}. Let $\alpha \in Z_k(X')$ be a cycle which is rationally equivalent to zero. We have to show that $c \cap \alpha = 0$. By Lemma \ref{lemma-rational-equivalence-family} there exists a cycle $\beta \in Z_{k + 1}(X' \times \mathbf{P}^1)$ such that $\alpha = i_0^*\beta - i_\infty^*\beta$ where $i_0, i_\infty : X' \to X' \times \mathbf{P}^1$ are the closed immersions of $X'$ over $0, \infty$. Since these are examples of effective Cartier divisors with trivial normal bundles, we see that $c \cap i_0^*\beta = j_0^*(c \cap \beta)$ and $c \cap i_\infty^*\beta = j_\infty^*(c \cap \beta)$ where $j_0, j_\infty : Y' \to Y' \times \mathbf{P}^1$ are closed immersions as before. Since $j_0^*(c \cap \beta) \sim_{rat} j_\infty^*(c \cap \beta)$ (follows from Lemma \ref{lemma-rational-equivalence-family}) we conclude.
Let $Y' \to Y$ be a morphism of schemes which is locally of finite type. Let $(\mathcal{L}', s', i' : D' \to Y')$ be as in Definition \ref{definition-gysin-homomorphism} with pullback $(\mathcal{N}', t', j' : E' \to X')$ to $X'$. We have to show that $c \cap (i')^*\alpha' = (j')^*(c \cap \alpha')$ for all $\alpha' \in \CH_k(Y')$. \medskip\noindent Denote $g : Y'' \to Y'$ the smooth morphism of relative dimension $1$ with $i'' : D'' \to Y''$ and $p : D'' \to D'$ constructed in Lemma \ref{lemma-normal-cone-effective-Cartier}. (Warning: $D''$ isn't the full inverse image of $D'$.) Denote $f : X'' \to X'$ and $E'' \subset X''$ their base changes by $X' \to Y'$. Picture $$ \xymatrix{ & X'' \ar[rr] \ar'[d][dd]_h & & Y'' \ar[dd]^g \\ E'' \ar[rr] \ar[dd]_q \ar[ru]^{j''} & & D'' \ar[dd]^p \ar[ru]^{i''} & \\ & X' \ar'[r][rr] & & Y' \\ E' \ar[rr] \ar[ru]^{j'} & & D' \ar[ru]^{i'} } $$ By the properties given in the lemma we know that $\beta' = (i')^*\alpha'$ is the unique element of $\CH_{k - 1}(D')$ such that $p^*\beta' = (i'')^*g^*\alpha'$. Similarly, we know that $\gamma' = (j')^*(c \cap \alpha')$ is the unique element of $\CH_{k - 1 - p}(E')$ such that $q^*\gamma' = (j'')^*h^*(c \cap \alpha')$. Since we know that $$ (j'')^*h^*(c \cap \alpha') = (j'')^*(c \cap g^*\alpha') = c \cap (i'')^*g^*\alpha' $$ by our assuptions on $c$; note that the modified version of (3) assumed in the statement of the lemma applies to $i''$ and its base change $j''$. We similarly know that $$ q^*(c \cap \beta') = c \cap p^*\beta' $$ We conclude that $\gamma' = c \cap \beta'$ by the uniqueness pointed out above.
Let $Z \subset X$ be an integral closed subscheme of $\delta$-dimension $k$. Note that $\pi^*[Z] = [\pi^{-1}(Z)]$ as $\pi^{-1}(Z)$ is integral of $\delta$-dimension $r - 1$. If $s < r - 1$, then by construction $c_1(\mathcal{O}_P(1))^s \cap \pi^*[Z]$ is represented by a $(k + r - 1 - s)$-cycle supported on $\pi^{-1}(Z)$. 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 $\pi_*(c_1(\mathcal{O}_P(1))^s \cap \pi^*\alpha) = n [Z]$ 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 $X \setminus T$ where $T \subset Z$ is a proper closed subset. Let $\xi$ be the generic point of $Z$. We can choose elements $e_1, \ldots, e_{r - 1} \in \mathcal{E}_\xi$ which form part of a basis of $\mathcal{E}_\xi$. These give rational sections $s_1, \ldots, s_{r - 1}$ of $\mathcal{O}_P(1)|_{\pi^{-1}(Z)}$ whose common zero set is the closure of the image a rational section of $\mathbf{P}(\mathcal{E}|_Z) \to Z$ union a closed subset whose support maps to a proper closed subset $T$ of $Z$. After removing $T$ from $X$ (and correspondingly $\pi^{-1}(T)$ from $P$), we see that $s_1, \ldots, s_n$ form a sequence of global sections $s_i \in \Gamma(\pi^{-1}(Z), \mathcal{O}_{\pi^{-1}(Z)}(1))$ whose common zero set is the image of a section $Z \to \pi^{-1}(Z)$. Hence we see successively that \begin{eqnarray*} \pi^*[Z] & = & [\pi^{-1}(Z)] \\ c_1(\mathcal{O}_P(1)) \cap \pi^*[Z] & = & [Z(s_1)] \\ c_1(\mathcal{O}_P(1))^2 \cap \pi^*[Z] & = & [Z(s_1) \cap Z(s_2)] \\ \ldots & = & \ldots \\ c_1(\mathcal{O}_P(1))^{r - 1} \cap \pi^*[Z] & = & [Z(s_1) \cap \ldots \cap Z(s_{r - 1})] \end{eqnarray*} by repeated applications of Lemma \ref{lemma-geometric-cap}. Since the pushforward by $\pi$ of the image of a section of $\pi$ over $Z$ is clearly $[Z]$ we see the result when $\alpha = [Z]$. We omit the verification that these arguments imply the result for a general cycle $\alpha = \sum n_j [Z_j]$.
(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, Lemma \ref{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.
In this proof we use $c_1(\mathcal{L}) \cap [X]$ to denote the construction of Definition \ref{definition-divisor-invertible-sheaf}. Since $\mathcal{L}$ has rank $1$ we have $\mathbf{P}(\mathcal{L}) = X$ and $\mathcal{O}_{\mathbf{P}(\mathcal{L})}(1) = \mathcal{L}$ by our normalizations. Hence (\ref{equation-chern-classes}) reads $$ (-1)^1 c_1(\mathcal{L}) \cap c_0 + (-1)^0 c_1 = 0 $$ Since $c_0 = [X]$, we conclude $c_1 = c_1(\mathcal{L}) \cap [X]$ as desired.
The uniqueness of $\alpha_0, \ldots, \alpha_r$ such that $\alpha_0 = \alpha$ and such that the displayed equation holds follows from the projective space bundle formula Lemma \ref{lemma-chow-ring-projective-bundle}. The identity holds by definition for $\alpha = [W]$ where $W$ is an integral closed subscheme of $X$. For a general $k$-cycle $\alpha$ on $X$ write $\alpha = \sum n_a[W_a]$ with $n_a \not = 0$, and $i_a : W_a \to X$ pairwise distinct integral closed subschemes. Then the family $\{W_a\}$ is locally finite on $X$. Set $P_a = \pi^{-1}(W_a) = \mathbf{P}(\mathcal{E}|_{W_a})$. Denote $i'_a : P_a \to P$ the corresponding closed immersions. Consider the fibre product diagram $$ \xymatrix{ P' \ar@{=}[r] \ar[d]_{\pi'} & \coprod P_a \ar[d]_{\coprod \pi_a} \ar[r]_{\coprod i'_a} & P \ar[d]^\pi \\ X' \ar@{=}[r] & \coprod W_a \ar[r]^{\coprod i_a} & X } $$ The morphism $p : X' \to X$ is proper. Moreover $\pi' : P' \to X'$ together with the invertible sheaf $\mathcal{O}_{P'}(1) = \coprod \mathcal{O}_{P_a}(1)$ which is also the pullback of $\mathcal{O}_P(1)$ is the projective bundle associated to $\mathcal{E}' = p^*\mathcal{E}$. By definition $$ c_j(\mathcal{E}) \cap [\alpha] = \sum i_{a, *}(c_j(\mathcal{E}|_{W_a}) \cap [W_a]). $$ Write $\beta_{a, j} = c_j(\mathcal{E}|_{W_a}) \cap [W_a]$ which is an element of $\CH_{k - j}(W_a)$. We have $$ \sum\nolimits_{i = 0}^r (-1)^i c_1(\mathcal{O}_{P_a}(1))^i \cap \pi_a^*(\beta_{a, r - i}) = 0 $$ for each $a$ by definition. Thus clearly we have $$ \sum\nolimits_{i = 0}^r (-1)^i c_1(\mathcal{O}_{P'}(1))^i \cap (\pi')^*(\beta_{r - i}) = 0 $$ with $\beta_j = \sum n_a\beta_{a, j} \in \CH_{k - j}(X')$. Denote $p' : P' \to P$ the morphism $\coprod i'_a$. We have $\pi^*p_*\beta_j = p'_*(\pi')^*\beta_j$ by Lemma \ref{lemma-flat-pullback-proper-pushforward}. By the projection formula of Lemma \ref{lemma-pushforward-cap-c1} we conclude that $$ \sum\nolimits_{i = 0}^r (-1)^i c_1(\mathcal{O}_P(1))^i \cap \pi^*(p_*\beta_j) = 0 $$ Since $p_*\beta_j$ is a representative of $c_j(\mathcal{E}) \cap \alpha$ we win.
By Lemma \ref{lemma-determine-intersections} the elements $\alpha_j = c_j(\mathcal{E}) \cap \alpha$, $j \geq 1$ and $\beta_j = c_j(\mathcal{E}) \cap \beta$, $j \geq 1$ are uniquely determined by the {\it same} equation in the chow group of the projective bundle associated to $\mathcal{E}$. (This of course relies on the fact that flat pullback is compatible with rational equivalence, see Lemma \ref{lemma-flat-pullback-rational-equivalence}.) Hence they are equal.
Let $(\pi : P \to Y, \mathcal{O}_P(1))$ be the projective bundle associated to $\mathcal{E}$. Then $P_X = X \times_Y P$ is the projective bundle associated to $p^*\mathcal{E}$ and $\mathcal{O}_{P_X}(1)$ is the pullback of $\mathcal{O}_P(1)$. Write $\alpha_j = c_j(p^*\mathcal{E}) \cap \alpha$, so $\alpha_0 = \alpha$. By Lemma \ref{lemma-determine-intersections} we have $$ \sum\nolimits_{i = 0}^r (-1)^i c_1(\mathcal{O}_P(1))^i \cap \pi_X^*(\alpha_{r - i}) = 0 $$ in the chow group of $P_X$. Consider the fibre product diagram $$ \xymatrix{ P_X \ar[r]_-{p'} \ar[d]_{\pi_X} & P \ar[d]^\pi \\ X \ar[r]^p & Y } $$ Apply proper pushforward $p'_*$ (Lemma \ref{lemma-proper-pushforward-rational-equivalence}) to the displayed equality above. Using Lemmas \ref{lemma-pushforward-cap-c1} and \ref{lemma-flat-pullback-proper-pushforward} we obtain $$ \sum\nolimits_{i = 0}^r (-1)^i c_1(\mathcal{O}_P(1))^i \cap \pi^*(p_*\alpha_{r - i}) = 0 $$ in the chow group of $P$. By the characterization of Lemma \ref{lemma-determine-intersections} we conclude.
Write $\alpha_j = c_j(\mathcal{E}) \cap \alpha$, so $\alpha_0 = \alpha$. By Lemma \ref{lemma-determine-intersections} we have $$ \sum\nolimits_{i = 0}^r (-1)^i c_1(\mathcal{O}_P(1))^i \cap \pi^*(\alpha_{r - i}) = 0 $$ in the chow group of the projective bundle $(\pi : P \to Y, \mathcal{O}_P(1))$ associated to $\mathcal{E}$. Consider the fibre product diagram $$ \xymatrix{ P_X = \mathbf{P}(f^*\mathcal{E}) \ar[r]_-{f'} \ar[d]_{\pi_X} & P \ar[d]^\pi \\ X \ar[r]^f & Y } $$ Note that $\mathcal{O}_{P_X}(1)$ is the pullback of $\mathcal{O}_P(1)$. Apply flat pullback $(f')^*$ (Lemma \ref{lemma-flat-pullback-rational-equivalence}) to the displayed equation above. By Lemmas \ref{lemma-flat-pullback-cap-c1} and \ref{lemma-compose-flat-pullback} we see that $$ \sum\nolimits_{i = 0}^r (-1)^i c_1(\mathcal{O}_{P_X}(1))^i \cap \pi_X^*(f^*\alpha_{r - i}) = 0 $$ holds in the chow group of $P_X$. By the characterization of Lemma \ref{lemma-determine-intersections} we conclude.