hcju/stacksqa huggingface dataset:
During the course of the proof we may replace $R$ by $R_f$ for $f\in R$, $f\not\in \mathfrak p$ (finitely often). As a first step we do such a replacement such that some coefficient of $F$ is invertible in $R$. In particular the modules $S_n$ are now locally free of rank $d$ for $n \geq d$ by Lemma \ref{lemma-P1}. Pick any $G \in R[X, Y]_e$ such that the image of $G$ in $\kappa(\mathfrak p)[X, Y]$ is relatively prime to the image of $F(X, Y)$ (this is possible for some $e$). Apply Algebra, Lemma \ref{algebra-lemma-cokernel-flat} to the map induced by multiplication by $G$ from $S_d \to S_{d + e}$. By our choice of $G$ and Lemma \ref{lemma-rel-prime-pols} we see $S_d \otimes \kappa(\mathfrak p) \to S_{d + e} \otimes \kappa(\mathfrak p)$ is bijective. Thus, after replacing $R$ by $R_f$ for a suitable $f$ we may assume that $G : S_d \to S_{d + e}$ is bijective. This in turn implies that the image of $G$ in $\kappa(\mathfrak p')[X, Y]$ is relatively prime to the image of $F$ for all primes $\mathfrak p'$ of $R$. And then by Algebra, Lemma \ref{algebra-lemma-cokernel-flat} again we see that all the maps $G : S_d \to S_{d + e}$, $n \geq d$ are isomorphisms.
This lemma has been obsoleted by the stronger Algebraization of Formal Spaces, Lemma \ref{restricted-lemma-get-morphism-general-better}. In fact, we will deduce the lemma from it. \medskip\noindent Let $I \subset A \to C$ be given as in the statement above. Denote $d(\text{Gr}_I(C))$ and $q(\text{Gr}_I(C))$ the integers found in  Local Cohomology, Section \ref{local-cohomology-section-uniform}. Observe that $t$ is an upper bound for the minimal number of generators of $IC$ and hence we have $d(\text{Gr}_I(C)) + 1 \leq t$, see discussion in Local Cohomology, Section \ref{local-cohomology-section-uniform}. We may and do assume $t \geq 1$ since otherwise the lemma does not say anything. We claim that the lemma is true with $$ d = q(\text{Gr}_I(C)) $$ Namely, suppose that $c$, $B$, $n$, $\psi_n$ are as in the statement above. Then we see that $$ n > 2t\max(c, d) \Rightarrow n \geq 2tc + 1 \Rightarrow n \geq 2(d(\text{Gr}_I(C)) + 1)c + 1 $$ On the other hand, we have $$ n > 2t\max(c, d) \Rightarrow n > t(c + d) \Rightarrow n \geq q(C) + tc \geq q(\text{Gr}_I(C)) + (d(\text{Gr}_I(C)) + 1)c $$ Hence the assumptions of Algebraization of Formal Spaces, Lemma \ref{restricted-lemma-get-morphism-general-better} are satisfied and we obtain an $A$-algebra homomorphism $\varphi : B \to C$ which is congruent with $\psi_n$ module $I^{n - (d(\text{Gr}_I(C)) + 1)c}C$. Since \begin{align*} n - (d(\text{Gr}_I(C)) + 1)c & = \frac{n}{t} + \frac{(t - 1)n}{t} - (d(\text{Gr}_I(C)) + 1)c \\ & \geq \frac{n}{t} + \frac{(d(\text{Gr}_I(C))n}{t} - (d(\text{Gr}_I(C)) + 1)c \\ & > \frac{n}{t} + \frac{d(\text{Gr}_I(C))2tc}{t} - (d(\text{Gr}_I(C)) + 1)c \\ & = \frac{n}{t} + 2d(\text{Gr}_I(C))c - (d(\text{Gr}_I(C)) + 1)c \\ & = \frac{n}{t} + d(\text{Gr}_I(C))c - c \\ & \geq m - c \end{align*} we see that we have the congruence of $\varphi$ and $\psi_n$ module $I^{m - c}C$ as desired.
Set $K = R\lim K_n$. Let $U \in \mathcal{B}$. For each $n$ there is a spectral sequence $$ H^p(U, H^q(K_n)) \Rightarrow H^{p + q}(U, K_n) $$ which converges as $K_n$ is bounded below, see Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}. If we fix $m \in \mathbf{Z}$, then we see from our assumption (4) that only $H^p(U, H^q(K_n))$ contribute to $H^m(U, K_n)$ for $0 \leq p \leq d$ and $m - d \leq q \leq m$. By assumption (2) this implies that $H^m(U, K_{n + 1}) \to H^m(U, K_n)$ is an isomorphism as soon as $n \geq \max{n(m), \ldots, n(m - d)}$. The functor $R\Gamma(U, -)$ commutes with derived limits by Injectives, Lemma \ref{injectives-lemma-RF-commutes-with-Rlim}. Thus we have $$ H^m(U, K) = H^m(R\lim R\Gamma(U, K_n)) $$ On the other hand we have just seen that the complexes $R\Gamma(U, K_n)$ have eventually constant cohomology groups. Thus by More on Algebra, Remark \ref{more-algebra-remark-compare-derived-limit} we find that $H^m(U, K)$ is equal to $H^m(U, K_n)$ for all $n \gg 0$ for some bound independent of $U \in \mathcal{B}$. Pick such an $n$. Finally, recall that $H^m(K)$ is the sheafification of the presheaf $U \mapsto H^m(U, K)$ and $H^m(K_n)$ is the sheafification of the presheaf $U \mapsto H^m(U, K_n)$. On the elements of $\mathcal{B}$ these presheaves have the same values. Therefore assumption (3) guarantees that the sheafifications are the same too. The lemma follows.
The only interesting assertions are (4) and (5). Part (4) follows from the spectral sequence in Simplicial Spaces, Lemma \ref{spaces-simplicial-lemma-augmentation-spectral-sequence} and Homology, Lemma \ref{homology-lemma-biregular-ss-converges}. Then part (5) follows by considering the spectral sequence associated to the canonical filtration on an object $K$ of $D_{\mathcal{A}_{total}}^+(\mathcal{C}_{total})$ given by truncations. We omit the details.
Choose filtrations $F_\bullet$ on $M$ and $N$. Then consider the filtration on $Q = M \otimes_A N$ given by $$ F_n(Q) = \sum\nolimits_{i + j = n} F_i(M) \otimes_A F_j(N) $$ This is clearly a differential graded $B$-submodule. We see that $$ F_n(Q)/F_{n - 1}(Q) = \bigoplus\nolimits_{i + j = n} F_i(M)/F_{i - 1}(M) \otimes_A F_j(N)/F_{j - 1}(N) $$ for example because the filtration of $M$ is split in the category of graded $A$-modules. Since by assumption the quotients on the right hand side are isomorphic to direct sums of shifts of $A$ and $A \otimes_R B$ and since $A \otimes_A (A \otimes_R B) = A \otimes_R B$, we conclude that the left hand side is a direct sum of shifts of $A \otimes_R B$ as a differential graded $B$-module. (Warning: $Q$ does not have a structure of $(A, B)$-bimodule.) This proves the first statement of the lemma. The second statement is immediate from the definition of the functor in Differential Graded Algebra, Lemma \ref{dga-lemma-derived-bc}.
For any simplicial object $V$ we have \begin{eqnarray*} \Mor(V, \text{cosk}_1 \text{sk}_1 \Hom(\Delta[k], X)) & = & \Mor(\text{sk}_1 V, \text{sk}_1 \Hom(\Delta[k], X)) \\ & = & \Mor(i_{1!} \text{sk}_1 V, \Hom(\Delta[k], X)) \\ & = & \Mor(i_{1!} \text{sk}_1 V \times \Delta[k], X) \end{eqnarray*} The first equality by the adjointness of $\text{sk}$ and $\text{cosk}$, the second equality by the adjointness of $i_{1!}$ and $\text{sk}_1$, and the first equality by Simplicial, Definition \ref{simplicial-definition-hom-from-simplicial-set} where the last $X$ denotes the constant simplicial object with value $X$. By Simplicial, Lemma \ref{simplicial-lemma-augmentation-howto} an element in this set depends only on the terms of degree $0$ and $1$ of $i_{1!} \text{sk}_1 V \times \Delta[k]$. These agree with the degree $0$ and $1$ terms of $V \times \Delta[k]$, see Simplicial, Lemma \ref{simplicial-lemma-recovering-U-for-real}. Thus the set above is equal to $\Mor(V \times \Delta[k], X) = \Mor(V, \Hom(\Delta[k], X))$.
The proof is a variant on the proof of Cohomology of Schemes, Lemma \ref{coherent-lemma-property-irreducible}. In exactly the same manner as in that proof we see that any coherent sheaf whose support is strictly contained in $Z_0$ has property $\mathcal{P}$. \medskip\noindent Consider a coherent sheaf $\mathcal{G}$ as in (3). By Cohomology of Schemes, Lemma \ref{coherent-lemma-prepare-filter-irreducible} there exists a sheaf of ideals $\mathcal{I}$ on $Z_0$ and a short exact sequence $$ 0 \to \left((Z_0 \to X)_*\mathcal{I}\right)^{\oplus r} \to \mathcal{G} \to \mathcal{Q} \to 0 $$ where the support of $\mathcal{Q}$ is strictly contained in $Z_0$. In particular $r > 0$ and $\mathcal{I}$ is nonzero because the support of $\mathcal{G}$ is equal to $Z$. Since $\mathcal{Q}$ has property $\mathcal{P}$ we conclude that also $\left((Z_0 \to X)_*\mathcal{I}\right)^{\oplus r}$ has property $\mathcal{P}$. By (2) we deduce property $\mathcal{P}$ for $(Z_0 \to X)_*\mathcal{I}$. Slotting this into the proof of Cohomology of Schemes, Lemma \ref{coherent-lemma-property-irreducible} at the appropriate point gives the lemma. Some details omitted.
We may choose $V$ such that $f^{-1}(V) \to V$ is finite. Then we may shrink $V$ and assume that $f^{-1}(V) \to V$ is flat and of finite presentation by generic flatness (Morphisms, Proposition \ref{morphisms-proposition-generic-flatness}). Then the morphism is finite locally free by Morphisms, Lemma \ref{morphisms-lemma-finite-flat}. Since $V$ is irreducible the morphism has a fixed degree. The final statement follows from this and Morphisms, Lemma \ref{morphisms-lemma-finite-locally-free-universally-bounded}.
Decent Spaces, Definition \ref{decent-spaces-definition-very-reasonable} says that there exist $U_i \to X$ such that (2), (3) and (4) hold. Fix $i$, and set $R_i = U_i \times_X U_i$, and denote $s, t : R_i \to U_i$ the projections. For any affine open $W \subset U_i$ the open $W' = t(s^{-1}(W)) \subset U_i$ is a quasi-compact $R_i$-invariant open (see Groupoids, Lemma \ref{groupoids-lemma-constructing-invariant-opens}). Hence $W'$ is a quasi-compact scheme, $W' \to X$ is \'etale, and $W' \times_X W' = s^{-1}(W') = t^{-1}(W')$ so both projections $W' \times_X W' \to W'$ are quasi-compact. This means the family of $W' \to X$, where $W \subset U_i$ runs through the members of affine open coverings of the $U_i$ gives what we want.
This lemma was originally used in the proof of a criterion for openness of versality (Artin's Axioms, Lemma \ref{artin-lemma-SGE-implies-openness-versality}) but it got replaced by Artin's Axioms, Lemma \ref{artin-lemma-infinite-sequence-pre} from which it readily follows. Namely, after replacing $u_n$, $n \geq 1$ by a subsequence we may and do assume that there are no specializations among these points, see Properties, Lemma \ref{properties-lemma-thin-infinite-sequence}. Then we can apply Artin's Axioms, Lemma \ref{artin-lemma-infinite-sequence-pre} to finish the proof.
[Proof (sketch)] Let $U \to S$ be of finite type morphism of schemes, $x$ an object of $\mathcal{X}$ over $U$ and $u_0 \in U$ a finite type point such that $x$ is versal at $u_0$. After shrinking $U$ we may assume that $u_0$ is a closed point (Morphisms, Lemma \ref{morphisms-lemma-point-finite-type}) and $U = \Spec(A)$ with $U \to S$ mapping into an affine open $\Spec(\Lambda)$ of $S$. We will use Artin's Axioms, Lemma \ref{artin-lemma-dual-openness} to prove the lemma. Let $\mathcal{F}$ be the coherent module on $X_A = \Spec(A) \times_S X$ flat over $A$ corresponding to the given object $x$. \medskip\noindent Choose $E(\mathcal{F})$ and $e_\mathcal{F}$ as in Remark \ref{remark-construction-E}. The description of the cohomology sheaves of $E(\mathcal{F})$ shows that $$ \Ext^1(E(\mathcal{F}), \mathcal{F} \otimes_A M) = \Ext^1(\mathcal{F}, \mathcal{F} \otimes_A M) $$ for any $A$-module $M$. Using this and using Deformation Theory, Lemma \ref{defos-lemma-inf-ext-rel-ringed-topoi} we have an isomorphism of functors $$ T_x(M) = \Ext^1_{X_A}(E(\mathcal{F}), \mathcal{F} \otimes_A M) $$ By Lemma \ref{lemma-ob-is-obstruction} given any surjection $A' \to A$ of $\Lambda$-algebras with square zero kernel $I$ we have an obstruction class $$ \xi_{A'} \in \Ext^2_{X_A}(E(\mathcal{F}), \mathcal{F} \otimes_A I) $$ Apply Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-compute-ext} to the computation of the Ext groups $\Ext^i_{X_A}(E(\mathcal{F}), \mathcal{F} \otimes_A M)$ for $i \leq m$ with $m = 2$. We omit the verification that $E(\mathcal{F})$ is in $D^-_{\textit{Coh}}$; hint: use Cotangent, Lemma \ref{cotangent-lemma-cotangent-finite}. We find a perfect object $K \in D(A)$ and functorial isomorphisms $$ H^i(K \otimes_A^\mathbf{L} M) \longrightarrow \Ext^i_{X_A}(E(\mathcal{F}), \mathcal{F} \otimes_A M) $$ for $i \leq m$ compatible with boundary maps. This object $K$, together with the displayed identifications above gives us a datum as in Artin's Axioms, Situation \ref{artin-situation-dual}. Finally, condition (iv) of Artin's Axioms, Lemma \ref{artin-lemma-dual-obstruction} holds by a variant of Deformation Theory, Lemma \ref{defos-lemma-verify-iv-ringed-topoi} whose formulation and proof we omit. Thus Artin's Axioms, Lemma \ref{artin-lemma-dual-openness} applies and the lemma is proved.
Let $\mathcal{F} = (M \otimes_A \mathcal{O}_X)^{**}$ be the double dual and let $\mathcal{F}' \subset \mathcal{F}$ be the image of the evaluation map $M \otimes_A \mathcal{O}_X \to \mathcal{F}$. Then we have a short exact sequence $$ 0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{Q} \to 0 $$ Since $X$ is normal, the local rings $\mathcal{O}_{X, x}$ are discrete valuation rings for points of codimension $1$ (see Properties, Lemma \ref{properties-lemma-criterion-normal}). Hence $\mathcal{Q}_x = 0$ for such points by More on Algebra, Lemma \ref{more-algebra-lemma-cokernel-map-double-dual-dvr}. Thus $\mathcal{Q}$ is supported in finitely many closed points and is globally generated by Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-support-dimension-0}. We obtain the exact sequence $$ 0 \to H^0(X, \mathcal{F}') \to H^0(X, \mathcal{F}) \to H^0(X, \mathcal{Q}) \to 0 $$ because $\mathcal{F}'$ is generated by global sections (Resolution of Surfaces, Lemma \ref{resolve-lemma-globally-generated}). Since $X \to \Spec(A)$ is an isomorphism over the complement of the closed point, and since $M$ is reflexive, we see that the maps $$ M \to H^0(X, \mathcal{F}') \to H^0(X, \mathcal{F}) $$ induce isomorphisms after localization at any nonmaximal prime of $A$. Hence these maps are isomorphisms by More on Algebra, Lemma \ref{more-algebra-lemma-check-isomorphism-via-depth-and-ass} and the fact that reflexive modules over normal rings have property $(S_2)$ (More on Algebra, Lemma \ref{more-algebra-lemma-reflexive-over-normal}). Thus we conclude that $\mathcal{Q} = 0$ as desired.
The schemes $X$, $X'$, $E$, $Z$ are smooth and projective over $k$ and hence we have $K'_0(X) = K_0(X) = K_0(\textit{Vect}(X)) = K_0(D^b_{\textit{Coh}}(X)))$ and similarly for the other $3$. See Derived Categories of Schemes, Lemmas \ref{perfect-lemma-Noetherian-Kprime}, \ref{perfect-lemma-Kprime-K}, and \ref{perfect-lemma-K-is-old-K}. We will switch between these versions at will in this proof. Consider the short exact sequence $$ 0 \to \mathcal{F} \to \pi^*\mathcal{C}_{Z/X} \to \mathcal{C}_{E/X'} \to 0 $$ of finite locally free $\mathcal{O}_E$-modules defining $\mathcal{F}$. Observe that $\mathcal{C}_{E/X'} = \mathcal{O}_{X'}(-E)|_E$ is the restriction of the invertible $\mathcal{O}_X$-module $\mathcal{O}_{X'}(-E)$. Let $\alpha \in K_0(X)$ be an element such that $i^*\alpha = [\mathcal{C}_{Z/X}]$ in $K_0(Z)$. Let $\alpha' = b^*\alpha - [\mathcal{O}_{X'}(-E)]$. Then $j^*\alpha' = [\mathcal{F}]$. We deduce that $j^*\lambda^i(\alpha') = [\wedge^i(\mathcal{F})]$ by Weil Cohomology Theories, Lemma \ref{weil-lemma-lambda-operations}. This means that $[\mathcal{O}_E] \cdot \alpha' = [\wedge^i\mathcal{F}]$ in $K_0(X)$, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-projection-formula}. Let $r$ be the maximum codimension of an irreducible component of $Z$ in $X$. A computation which we omit shows that $H^{-i}(Lb^*\mathcal{O}_Z) = \wedge^i\mathcal{F}$ for $i \geq 0, 1, \ldots, r - 1$ and zero in other degrees. It follows that in $K_0(X)$ we have \begin{align*} [Lb^*\mathcal{O}_Z] & = \sum\nolimits_{i = 0, \ldots, r - 1} (-1)^i[\wedge^i\mathcal{F}] \\ & = \sum\nolimits_{i = 0, \ldots, r - 1} (-1)^i[\mathcal{O}_E] \lambda^i(\alpha') \\ & = [\mathcal{O}_E] \left(\sum\nolimits_{i = 0, \ldots, r - 1} (-1)^i \lambda^i(\alpha')\right) \end{align*} This proves the lemma with $\alpha'' = \sum_{i = 0, \ldots, r - 1} (-1)^i \lambda^i(\alpha')$.
Let $\mathcal{I} \subset \mathcal{O}_X$ be the quasi-coherent ideal sheaf of denominators of $s$, see Divisors, Definition \ref{divisors-definition-regular-meromorphic-ideal-denominators}. By Divisors, Lemma \ref{divisors-lemma-blowing-up-denominators} we get (2), (3), and (4). By Divisors, Lemma \ref{divisors-lemma-blow-up-integral-scheme} we get (1). By Divisors, Lemma \ref{divisors-lemma-blowing-up-projective} the morphism $\pi$ is projective. We still have to prove (5). By Chow Homology, Lemma \ref{chow-lemma-equal-c1-as-cycles} we have $$ \pi_*(\text{div}_{\mathcal{L}'}(s')) = \text{div}_\mathcal{L}(s). $$ Hence it suffices to show that $\text{div}_{\mathcal{L}'}(s') = [D]_{n - 1} - [E]_{n - 1}$. This follows from the equality $s' = 1_D \otimes 1_E^{-1}$ and additivity, see Divisors, Lemma \ref{divisors-lemma-c1-additive}.
Note that the quasi-coherent ideal sheaf $\mathcal{I} = \mathcal{I}_{D_1} + \mathcal{I}_{D_2}$ defines the scheme theoretic intersection $D_1 \cap D_2 \subset X$. Since $Z$ is a union of connected components of $D_1 \cap D_2$ we see that for every $z \in Z$ the kernel of $\mathcal{O}_{X, z} \to \mathcal{O}_{Z, z}$ is equal to $\mathcal{I}_z$. Let $b : X' \to X$ be the blowup of $X$ in $Z$. (So Zariski locally around $Z$ it is the blowup of $X$ in $\mathcal{I}$.) Denote $E = b^{-1}(Z)$ the corresponding effective Cartier divisor, see Divisors, Lemma \ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}. Since $Z \subset D_1$ we have $E \subset f^{-1}(D_1)$ and hence $D_1 = D_1' + E$ for some effective Cartier divisor $D'_1 \subset X'$, see Divisors, Lemma \ref{divisors-lemma-difference-effective-Cartier-divisors}. Similarly $D_2 = D_2' + E$. This takes care of assertions (1) -- (5). \medskip\noindent Note that if $W'$ is as in (7) (a) or (7) (b), then the image $W$ of $W'$ is contained in $D_1 \cap D_2$. If $W$ is not contained in $Z$, then $b$ is an isomorphism at the generic point of $W$ and we see that $\dim_\delta(W) = \dim_\delta(W') = n - 1$ which contradicts the assumption that $\dim_\delta(D_1 \cap D_2 \setminus Z) \leq n - 2$. Hence $W \subset Z$. This means that to prove (6) and (7) we may work locally around $Z$ on $X$. \medskip\noindent Thus we may assume that $X = \Spec(A)$ with $A$ a Noetherian domain, and $D_1 = \Spec(A/a)$, $D_2 = \Spec(A/b)$ and $Z = D_1 \cap D_2$. Set $I = (a, b)$. Since $A$ is a domain and $a, b \not = 0$ we can cover the blowup by two patches, namely $U = \Spec(A[s]/(as - b))$ and $V = \Spec(A[t]/(bt -a))$. These patches are glued using the isomorphism $A[s, s^{-1}]/(as - b) \cong A[t, t^{-1}]/(bt - a)$ which maps $s$ to $t^{-1}$. The effective Cartier divisor $E$ is described by $\Spec(A[s]/(as - b, a)) \subset U$ and $\Spec(A[t]/(bt - a, b)) \subset V$. The closed subscheme $D'_1$ corresponds to $\Spec(A[t]/(bt - a, t)) \subset U$. The closed subscheme $D'_2$ corresponds to $\Spec(A[s]/(as -b, s)) \subset V$. Since ``$ts = 1$'' we see that $D'_1 \cap D'_2 = \emptyset$. \medskip\noindent Suppose we have a prime $\mathfrak q \subset A[s]/(as - b)$ of height one with $s, a \in \mathfrak q$. Let $\mathfrak p \subset A$ be the corresponding prime of $A$. Observe that $a, b \in \mathfrak p$. By the dimension formula we see that $\dim(A_{\mathfrak p}) = 1$ as well. The final assertion to be shown is that $$ \text{ord}_{A_{\mathfrak p}}(a) \text{ord}_{A_{\mathfrak p}}(b) > \text{ord}_{B_{\mathfrak q}}(a) \text{ord}_{B_{\mathfrak q}}(s) $$ where $B = A[s]/(as - b)$. By Algebra, Lemma \ref{algebra-lemma-quasi-finite-extension-dim-1} we have $\text{ord}_{A_{\mathfrak p}}(x) \geq \text{ord}_{B_{\mathfrak q}}(x)$ for $x = a, b$. Since $\text{ord}_{B_{\mathfrak q}}(s) > 0$ we win by additivity of the $\text{ord}$ function and the fact that $as = b$.
Since we are proving an equality of cycles we may work locally on $X$. Hence this reduces to a finite sum, and by induction to a sum of two effective Cartier divisors $D = D_1 + D_2$. By Chow Homology, Lemma \ref{chow-lemma-compute-c1} we see that $D_1 = \text{div}_{\mathcal{O}_X(D_1)}(1_{D_1})$ where $1_{D_1}$ denotes the canonical section of $\mathcal{O}_X(D_1)$. Of course we have the same statement for $D_2$ and $D$. Since $1_D = 1_{D_1} \otimes 1_{D_2}$ via the identification $\mathcal{O}_X(D) = \mathcal{O}_X(D_1) \otimes \mathcal{O}_X(D_2)$ we win by Divisors, Lemma \ref{divisors-lemma-c1-additive}.
Part (2) is a trivial consequence of part (1). Let us write $[D]_{n - 1} = \sum n_a[Z_a]$ and $[D']_{n - 1} = \sum m_b[Z_b]$ with $Z_a$ the irreducible components of $D$ and $[Z_b]$ the irreducible components of $D'$. According to Chow Homology, Definition \ref{chow-definition-gysin-homomorphism}, we have $i^*D' = \sum m_b i^*[Z_b]$ and $(i')^*D = \sum n_a(i')^*[Z_a]$. By assumption, none of the irreducible components $Z_b$ is contained in $D$, and hence $i^*[Z_b] = [Z_b\cap D]_{n - 2}$ by definition. Similarly $(i')^*[Z_a] = [Z_a \cap D']_{n - 2}$. Hence we are trying to prove the equality of cycles $$ \sum n_a[Z_a \cap D']_{n - 2} = \sum m_b[Z_b \cap D]_{n - 2} $$ which are indeed supported on $D \cap D'$. Let $W \subset X$ be an integral closed subscheme with $\dim_\delta(W) = n - 2$. Let $\xi \in W$ be its generic point. Set $R = \mathcal{O}_{X, \xi}$. It is a Noetherian local domain. Note that $\dim(R) = 2$. Let $f \in R$, resp.\ $f' \in R$ be an element defining the ideal of $D$, resp.\ $D'$. By assumption $\dim(R/(f, f')) = 0$. Let $\mathfrak q'_1, \ldots, \mathfrak q'_t \subset R$ be the minimal primes over $(f')$, let $\mathfrak q_1, \ldots, \mathfrak q_s \subset R$ be the minimal primes over $(f)$. The equality above comes down to the equality $$ \sum_{i = 1, \ldots, s} \text{length}_{R_{\mathfrak q_i}}(R_{\mathfrak q_i}/(f)) \text{ord}_{R/\mathfrak q_i}(f') = \sum_{j = 1, \ldots, t} \text{length}_{R_{\mathfrak q'_j}}(R_{\mathfrak q'_j}/(f')) \text{ord}_{R/\mathfrak q'_j}(f). $$ By Chow Homology, Lemma \ref{chow-lemma-length-multiplication}  applied with $M = R/(f)$ the left hand side of this equation is equal to $$ \text{length}_R(R/(f, f')) - \text{length}_R(\Ker(f' : R/(f) \to R/(f))) $$ OK, and now we note that $\Ker(f' : R/(f) \to R/(f))$ is canonically isomorphic to $((f) \cap (f'))/(ff')$ via the map $x \bmod (f) \mapsto f'x \bmod (ff')$. Hence the left hand side is $$ \text{length}_R(R/(f, f')) - \text{length}_R((f) \cap (f')/(ff')) $$ Since this is symmetric in $f$ and $f'$ we win.