hcju/stacksqa huggingface dataset:
Let $\mathcal{F}$ be an object of $\textit{Coh}_Z(X)$. Let $\mathcal{I} \subset \mathcal{O}_X$ be the quasi-coherent ideal sheaf of $Z$. Consider the descending filtration $$ \ldots \subset \mathcal{F}^p = \mathcal{I}^p \mathcal{F} \subset \mathcal{F}^{p - 1} \subset \ldots \subset \mathcal{F}^0 = \mathcal{F} $$ Exactly as in the proof of Lemma \ref{lemma-from-chow-to-K} this filtration is locally finite and hence $\bigoplus_{p \geq 0} \mathcal{F}^p$, $\bigoplus_{p \geq 1} \mathcal{F}^p$, and $\bigoplus_{p \geq 0} \mathcal{F}^p/\mathcal{F}^{p + 1}$ are coherent $\mathcal{O}_X$-modules supported on $Z$. Hence we get $$ [\mathcal{F}] = [\bigoplus\nolimits_{p \geq 0} \mathcal{F}^p/\mathcal{F}^{p + 1}] $$ in $K_0(\textit{Coh}_Z(X))$ exactly as in the proof of Lemma \ref{lemma-from-chow-to-K}. Since the coherent module $\bigoplus_{p \geq 0} \mathcal{F}^p/\mathcal{F}^{p + 1}$ is annihilated by $\mathcal{I}$ we conclude that $[\mathcal{F}]$ is in the image. Actually, we claim that the map $$ \mathcal{F} \longmapsto  c(\mathcal{F}) = [\bigoplus\nolimits_{p \geq 0} \mathcal{F}^p/\mathcal{F}^{p + 1}] $$ factors through $K_0(\textit{Coh}_Z(X))$ and is an inverse to the map in the statement of the lemma. To see this all we have to show is that if $$ 0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0 $$ is a short exact sequence in $\textit{Coh}_Z(X)$, then we get $c(\mathcal{G}) = c(\mathcal{F}) + c(\mathcal{H})$. Observe that for all $q \geq 0$ we have a short exact sequence $$ 0 \to (\mathcal{F} \cap \mathcal{I}^q\mathcal{G})/ (\mathcal{F} \cap \mathcal{I}^{q + 1}\mathcal{G}) \to \mathcal{G}^q/\mathcal{G}^{q + 1} \to \mathcal{H}^q/\mathcal{H}^{q + 1} \to 0 $$ For $p, q \geq 0$ consider the coherent submodule $$ \mathcal{F}^{p, q} = \mathcal{I}^p\mathcal{F} \cap \mathcal{I}^q\mathcal{G} $$ Arguing exactly as above and using that the filtrations $\mathcal{F}^p = \mathcal{I}^p\mathcal{F}$ and $\mathcal{F} \cap \mathcal{I}^q\mathcal{G}$ are locally finite, we find that $$ [\bigoplus\nolimits_{p \geq 0} \mathcal{F}^p/\mathcal{F}^{p + 1}] = [\bigoplus\nolimits_{p, q \geq 0} \mathcal{F}^{p, q}/(\mathcal{F}^{p + 1, q} + \mathcal{F}^{p, q + 1})] = [\bigoplus\nolimits_{q \geq 0} (\mathcal{F} \cap \mathcal{I}^q\mathcal{G})/ (\mathcal{F} \cap \mathcal{I}^{q + 1}\mathcal{G})] $$ in $K_0(\textit{Coh}(Z))$. Combined with the exact sequences above we obtain the desired result. Some details omitted.
Let $Z \subset X$ be an integral closed subscheme of $\delta$-dimension $n - 1$. Let $\xi \in Z$ be its generic point. Choose a generator $s_\xi \in \mathcal{L}_\xi$. Write $s = fs_\xi$ for some $f \in \mathcal{O}_{X, \xi}$. By definition of $Z(s)$, see Divisors, Definition \ref{divisors-definition-zero-scheme-s} we see that $Z(s)$ is cut out by a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ such that $\mathcal{I}_\xi = (f)$. Hence $\text{length}_{\mathcal{O}_{X, x}}(\mathcal{O}_{Z(s), \xi}) = \text{length}_{\mathcal{O}_{X, x}}(\mathcal{O}_{X, \xi}/(f)) = \text{ord}_{\mathcal{O}_{X, x}}(f)$ as desired.
Let $s$ be a nonzero meromorphic section of $\mathcal{L}$. We will show that actually $f^*\text{div}_\mathcal{L}(s) = \text{div}_{f^*\mathcal{L}}(f^*s)$ and hence the lemma holds. To see this let $\xi \in Y$ be a point and let $s_\xi \in \mathcal{L}_\xi$ be a generator. Write $s = gs_\xi$ with $g \in R(X)^*$. Then there is an open neighbourhood $V \subset Y$ of $\xi$ such that $s_\xi \in \mathcal{L}(V)$ and such that $s_\xi$ generates $\mathcal{L}|_V$. Hence we see that $$ \text{div}_\mathcal{L}(s)|_V = \text{div}(g)|_V. $$ In exactly the same way, since $f^*s_\xi$ generates $\mathcal{L}$ over $f^{-1}(V)$ and since $f^*s = g f^*s_\xi$ we also have $$ \text{div}_\mathcal{L}(f^*s)|_{f^{-1}(V)} = \text{div}(g)|_{f^{-1}(V)}. $$ Thus the desired equality of cycles over $f^{-1}(V)$ follows from the corresponding result for pullbacks of principal divisors, see Lemma \ref{lemma-flat-pullback-principal-divisor}.
The additivity follows directly from Divisors, Lemma \ref{divisors-lemma-c1-additive} and the definitions. To see that $c_1(\mathcal{O}_X) \cap \alpha = 0$ consider the section $1 \in \Gamma(X, \mathcal{O}_X)$. This restricts to an everywhere nonzero section on any integral closed subscheme $W \subset X$. Hence $c_1(\mathcal{O}_X) \cap [W] = 0$ as desired.
Let $Z \subset Y$ be an integral closed subscheme of $\delta$-dimension $k$. Let $\xi \in Z$ be its generic point. We want to compare the coefficient $n$ of $[Z]$ in the expression $\sum n_i[Z(s_i)]_k$ with the coefficient $m$ of $[Z]$ in the expression $[Z(s)]_k$. Choose a generator $s_\xi \in \mathcal{L}_\xi$. Write $A = \mathcal{O}_{Y, \xi}$, $L = \mathcal{L}_\xi$. Then $L = As_\xi$. Write $s = f s_\xi$ for some (unique) $f \in A$. Hypothesis (3) means that $f : A \to A$ is injective. Since $\dim_\delta(Y) \leq k + 1$ and $\dim_\delta(Z) = k$ we have $\dim(A) = 0$ or $1$. We have $$ m = \text{length}_A(A/(f)) $$ which is finite in either case. \medskip\noindent If $\dim(A) = 0$, then $f : A \to A$ being injective implies that $f \in A^*$. Hence in this case $m$ is zero. Moreover, the condition $\dim(A) = 0$ means that $\xi$ does not lie on any irreducible component of $\delta$-dimension $k + 1$, i.e., $n = 0$ as well. \medskip\noindent Now, let $\dim(A) = 1$. Since $A$ is a Noetherian local ring it has finitely many minimal primes $\mathfrak q_1, \ldots, \mathfrak q_t$. These correspond 1-1 with the $Y_i$ passing through $\xi'$. Moreover $n_i = \text{length}_{A_{\mathfrak q_i}}(A_{\mathfrak q_i})$. Also, the multiplicity of $[Z]$ in $[Z(s_i)]_k$ is $\text{length}_A(A/(f, \mathfrak q_i))$. Hence the equation to prove in this case is $$ \text{length}_A(A/(f)) = \sum \text{length}_{A_{\mathfrak q_i}}(A_{\mathfrak q_i}) \text{length}_A(A/(f, \mathfrak q_i)) $$ which follows from Lemma \ref{lemma-additivity-divisors-restricted}.
Write $$ [Y]_{k + 1} = \sum n_i[Y_i] $$ where $Y_i \subset Y$ are the irreducible components of $Y$ of $\delta$-dimension $k + 1$ and $n_i > 0$. By assumption the restriction $s_i = s|_{Y_i} \in \Gamma(Y_i, \mathcal{L}|_{Y_i})$ is not zero, and hence is a regular section. By Lemma \ref{lemma-compute-c1} we see that $[Z(s_i)]_k$ represents $c_1(\mathcal{L}|_{Y_i})$. Hence by definition $$ c_1(\mathcal{L}) \cap [Y]_{k + 1} = \sum n_i[Z(s_i)]_k $$ Thus the result follows from Lemma \ref{lemma-prepare-geometric-cap}.
To prove this equality of cycles, we may work locally on $Y$. Hence we may assume $Y$ is affine and $s = p/q$ for some nonzero sections $p \in \Gamma(Y, \mathcal{L})$ and $q \in \Gamma(Y, \mathcal{O})$. If we can show both $$ f^*\text{div}_\mathcal{L}(p) = \sum n_i\text{div}_{f^*\mathcal{L}|_{X_i}}(p_i) \quad\text{and}\quad f^*\text{div}_\mathcal{O}(q) = \sum n_i\text{div}_{\mathcal{O}_{X_i}}(q_i) $$ (with obvious notations) then we win by the additivity, see Divisors, Lemma \ref{divisors-lemma-c1-additive}. Thus we may assume that $s \in \Gamma(Y, \mathcal{L})$. In this case we may apply the equality (\ref{equation-equal-as-cycles}) to see that $$ [Z(f^*(s))]_{k + r - 1} = \sum n_i\text{div}_{f^*\mathcal{L}|_{X_i}}(s_i) $$ where $f^*(s) \in f^*\mathcal{L}$ denotes the pullback of $s$ to $X$. On the other hand we have $$ f^*\text{div}_\mathcal{L}(s) = f^*[Z(s)]_{k - 1} = [f^{-1}(Z(s))]_{k + r - 1}, $$ by Lemmas \ref{lemma-compute-c1} and \ref{lemma-pullback-coherent}. Since $Z(f^*(s)) = f^{-1}(Z(s))$ we win.
Write $\alpha = \sum n_i[W_i]$. We will show that $$ f^*(c_1(\mathcal{L}) \cap [W_i]) = c_1(f^*\mathcal{L}) \cap f^*[W_i] $$ in $\CH_{k + r - 1}(X)$ by producing a rational equivalence on the closed subscheme $f^{-1}(W_i)$ of $X$. By the discussion in Remark \ref{remark-infinite-sums-rational-equivalences} this will prove the equality of the lemma is true. \medskip\noindent Let $W \subset Y$ be an integral closed subscheme of $\delta$-dimension $k$. Consider the closed subscheme $W' = f^{-1}(W) = W \times_Y X$ so that we have the fibre product diagram $$ \xymatrix{ W' \ar[r] \ar[d]_h & X \ar[d]^f \\ W \ar[r] & Y } $$ We have to show that $f^*(c_1(\mathcal{L}) \cap [W]) = c_1(f^*\mathcal{L}) \cap f^*[W]$. Choose a nonzero meromorphic section $s$ of $\mathcal{L}|_W$. Let $W'_i \subset W'$ be the irreducible components of $\delta$-dimension $k + r$. Write $[W']_{k + r} = \sum n_i[W'_i]$ with $n_i$ the multiplicity of $W'_i$ in $W'$ as per definition. So $f^*[W] = \sum n_i[W'_i]$ in $Z_{k + r}(X)$. Since each $W'_i \to W$ is dominant we see that $s_i = s|_{W'_i}$ is a nonzero meromorphic section for each $i$. By Lemma \ref{lemma-prepare-flat-pullback-cap-c1} we have the following equality of cycles $$ h^*\text{div}_{\mathcal{L}|_W}(s) = \sum n_i\text{div}_{f^*\mathcal{L}|_{W'_i}}(s_i) $$ in $Z_{k + r - 1}(W')$. This finishes the proof since the left hand side is a cycle on $W'$ which pushes to $f^*(c_1(\mathcal{L}) \cap [W])$ in $\CH_{k + r - 1}(X)$ and the right hand side is a cycle on $W'$ which pushes to $c_1(f^*\mathcal{L}) \cap f^*[W]$ in $\CH_{k + r - 1}(X)$.
The last equation follows from the first since $f_*[X] = [R(X) : R(Y)][Y]$ by definition. It turns out that we can re-use Lemma \ref{lemma-proper-pushforward-alteration} to prove this. Namely, since we are trying to prove an equality of cycles, we may work locally on $Y$. Hence we may assume that $\mathcal{L} = \mathcal{O}_Y$. In this case $s$ corresponds to a rational function $g \in R(Y)$, 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 the aforementioned 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 subscheme $W \subset X$ 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 an integral closed subscheme). 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}.
First, let us examine what happens if we replace $s_i$ by $us_i$ for some unit $u$ in $B_i$. Then $f_i$ gets replaced by $u^{-1} f_i$. Thus the first part of the first expression of the lemma is unchanged and in the second part we add $$ -\text{ord}_{B_i}(g_i)\text{div}(u|_{Z_i}) $$ (where $u|_{Z_i}$ is the image of $u$ in the residue field) by Divisors, Lemma \ref{divisors-lemma-divisor-meromorphic-well-defined} and in the second expression we add $$ \text{div}(\partial_{B_i}(u^{-1}, g_i)) $$ by bi-linearity of the tame symbol. These terms agree by property (\ref{item-normalization}) of the tame symbol. \medskip\noindent Let $Z \subset X$ be an irreducible closed with $\dim_\delta(Z) = n - 2$. To show that the coefficients of $Z$ of the two cycles of the lemma is the same, we may do a replacement $s_i \mapsto us_i$ as in the previous paragraph. In exactly the same way one shows that we may do a replacement $t_i \mapsto vt_i$ for some unit $v$ of $B_i$. \medskip\noindent Since we are proving the equality of cycles we may argue one coefficient at a time. Thus we choose an irreducible closed $Z \subset X$ with $\dim_\delta(Z) = n - 2$ and compare coefficients. Let $\xi \in Z$ be the generic point and set $A = \mathcal{O}_{X, \xi}$. This is a Noetherian local domain of dimension $2$. Choose generators $\sigma$ and $\tau$ for $\mathcal{L}_\xi$ and $\mathcal{N}_\xi$. After shrinking $X$, we may and do assume $\sigma$ and $\tau$ define trivializations of the invertible sheaves $\mathcal{L}$ and $\mathcal{N}$ over all of $X$. Because $Z_i$ is locally finite after shrinking $X$ we may assume $Z \subset Z_i$ for all $i \in I$ and that $I$ is finite. Then $\xi_i$ corresponds to a prime $\mathfrak q_i \subset A$ of height $1$. We may write $s_i = a_i \sigma$ and $t_i = b_i \tau$ for some $a_i$ and $b_i$ units in $A_{\mathfrak q_i}$. By the remarks above, it suffices to prove the lemma when $a_i = b_i = 1$ for all $i$. \medskip\noindent Assume $a_i = b_i = 1$ for all $i$. Then the first expression of the lemma is zero, because we choose $\sigma$ and $\tau$ to be trivializing sections. Write $s = f\sigma$ and $t = g \tau$ with $f$ and $g$ in the fraction field of $A$. By the previous paragraph we have reduced to the case $f_i = f$ and $g_i = g$ for all $i$. Moreover, for a height $1$ prime $\mathfrak q$ of $A$ which is not in $\{\mathfrak q_i\}$ we have that both $f$ and $g$ are units in $A_\mathfrak q$ (by our choice of the family $\{Z_i\}$ in the discussion preceding the lemma). Thus the coefficient of $Z$ in the second expression of the lemma is $$ \sum\nolimits_i \text{ord}_{A/\mathfrak q_i}(\partial_{B_i}(f, g)) $$ which is zero by the key Lemma \ref{lemma-milnor-gersten-low-degree}.
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 subschemes 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 subschemes $Z_j \subset X$ 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 subschemes with $\dim_\delta(W_j) = k$. Since $D$ is the zero scheme of $s$ we see that $D \cap W_j$ is the zero scheme 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.
Assume $Z \subset X$ as in (1). Then set $\mathcal{F} = \mathcal{O}_Z$. The assumption that $D \cap Z$ is an effective Cartier divisor is equivalent to the assumption that $s : \mathcal{F} \to \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}$ is injective. Moreover $[Z]_{k + 1} = [\mathcal{F}]_{k + 1}]$ and $[D \cap Z]_k = [\mathcal{O}_{D \cap Z}]_k = [i^*\mathcal{F}]_k$. See Lemma \ref{lemma-cycle-closed-coherent}. Hence part (1) follows from part (2). \medskip\noindent Write $[\mathcal{F}]_{k + 1} = \sum m_j[W_j]$ with $m_j > 0$ and pairwise distinct integral closed subschemes $W_j \subset X$ of $\delta$-dimension $k + 1$. The assumption that $s : \mathcal{F} \to \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}$ is injective implies that $W_j \not \subset D$ for all $j$. By definition we see that $$ i^*[\mathcal{F}]_{k + 1} = \sum [D \cap W_j]_k. $$ We claim that $$ \sum [D \cap W_j]_k = [i^*\mathcal{F}]_k $$ as cycles. Let $Z \subset D$ be an integral closed subscheme of $\delta$-dimension $k$. Let $\xi \in Z$ be its generic point. Let $A = \mathcal{O}_{X, \xi}$. Let $M = \mathcal{F}_\xi$. Let $f \in A$ be an element generating the ideal of $D$, i.e., such that $\mathcal{O}_{D, \xi} = A/fA$. By assumption $\dim(\text{Supp}(M)) = 1$, the map $f : M \to M$ is injective, and $\text{length}_A(M/fM) < \infty$. Moreover, $\text{length}_A(M/fM)$ is the coefficient of $[Z]$ in $[i^*\mathcal{F}]_k$. On the other hand, let $\mathfrak q_1, \ldots, \mathfrak q_t$ be the minimal primes in the support of $M$. Then $$ \sum \text{length}_{A_{\mathfrak q_i}}(M_{\mathfrak q_i}) \text{ord}_{A/\mathfrak q_i}(f) $$ is the coefficient of $[Z]$ in $\sum [D \cap W_j]_k$. Hence we see the equality by Lemma \ref{lemma-additivity-divisors-restricted}.
Suppose $\alpha = [W']$ for some integral closed subscheme $W' \subset X'$. Let $W = f(W') \subset X$. 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 subscheme $W \subset X$. 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 subschemes 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 $\alpha'$.
Write $\text{div}_\mathcal{N}(t) = \sum \text{ord}_{Z_i, \mathcal{N}}(t)[Z_i]$ for some integral closed subschemes $Z_i \subset X$ 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 Divisors, Lemmas \ref{divisors-lemma-components-locally-finite}, \ref{divisors-lemma-divisor-locally-finite}, and \ref{divisors-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 $B_i = \mathcal{O}_{X, \xi_i}$. For each $i$ we pick generators $s_i \in \mathcal{L}_{\xi_i}$ and $t_i \in \mathcal{N}_{\xi_i}$ 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 subschemes $W_j \subset X$ 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 $\CH_{n - 2}(D)$ by Lemma \ref{lemma-gysin-factors-general}. Of course capping with $c_1(\mathcal{O}_D)$ is the zero map (Lemma \ref{lemma-c1-cap-additive}).