hcju/stacksqa huggingface dataset:
Let us explain how to prove (1) in a more elementary way and part (2) using previously developed general theory. \medskip\noindent Proof of (1). Since formation of $\text{Tor}$ commutes with localization we may assume $X$ is affine. Hence $X = \Spec(A)$ for some Noetherian ring $A$ and $\mathcal{F}$, $\mathcal{G}$ correspond to finite $A$-modules $M$ and $N$ (Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian}). By Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-tensor-product} we may compute the $\text{Tor}$'s by first computing the $\text{Tor}$'s of $M$ and $N$ over $A$, and then taking the associated $\mathcal{O}_X$-module. Since the modules $\text{Tor}_p^A(M, N)$ are finite by Algebra, Lemma \ref{algebra-lemma-tor-noetherian} we conclude. \medskip\noindent By Derived Categories of Schemes, Lemma \ref{perfect-lemma-identify-pseudo-coherent-noetherian} the assumption is equivalent to asking $L$ and $K$ to be (locally) pseudo-coherent. Then $L \otimes_{\mathcal{O}_X}^\mathbf{L} K$ is pseudo-coherent by Cohomology, Lemma \ref{cohomology-lemma-tensor-pseudo-coherent}.
The result on stalks was discussed above and it implies the support condition. The $\text{Tor}$'s are coherent by Lemma \ref{lemma-tensor-coherent}. The vanishing of negative $\text{Tor}$'s is immediate from the construction. The vanishing of $\text{Tor}_p$ for $p > \dim(X)$ can be seen as follows: the local rings $\mathcal{O}_{X, x}$ are regular (as $X$ is nonsingular) of dimension $\leq \dim(X)$ (Algebra, Lemma \ref{algebra-lemma-dimension-prime-polynomial-ring}), hence $\mathcal{O}_{X, x}$ has finite global dimension $\leq \dim(X)$ (Algebra, Lemma \ref{algebra-lemma-finite-gl-dim-finite-dim-regular}) which implies that $\text{Tor}$-groups of modules vanish beyond the dimension (More on Algebra, Lemma \ref{more-algebra-lemma-finite-gl-dim-tor-dimension}).
Let $(A, \mathfrak m, \kappa) = (\mathcal{O}_{X, \xi}, \mathfrak m_\xi, \kappa(\xi))$ where $\xi \in Z$ is the generic point. Then $\dim(A) = \dim(X) - \dim(Z)$, see Varieties, Lemma \ref{varieties-lemma-dimension-locally-algebraic}. Let $I, J \subset A$ cut out the trace of $V$ and $W$ in $\Spec(A)$. Set $\overline{I} = I + \mathfrak m^2/\mathfrak m^2$. Then $\dim_\kappa \overline{I} \leq \dim(X) - \dim(V)$ with equality if and only if $A/I$ is regular (this follows from the lemma cited above and the definition of regular rings, see Algebra, Definition \ref{algebra-definition-regular-local} and the discussion preceding it). Similarly for $\overline{J}$. If the multiplicity is $1$, then $\text{length}_A(A/I + J) = 1$, hence $I + J = \mathfrak m$, hence $\overline{I} + \overline{J} = \mathfrak m/\mathfrak m^2$. Then we get equality everywhere (because the intersection is proper). Hence we find $f_1, \ldots, f_a \in I$ and $g_1, \ldots g_b \in J$ such that $\overline{f}_1, \ldots, \overline{g}_b$ is a basis for $\mathfrak m/\mathfrak m^2$. Then $f_1, \ldots, g_b$ is a regular system of parameters and a regular sequence (Algebra, Lemma \ref{algebra-lemma-regular-ring-CM}). The same lemma shows $A/(f_1, \ldots, f_a)$ is a regular local ring of dimension $\dim(X) - \dim(V)$, hence $A/(f_1, \ldots, f_a) \to A/I$ is an isomorphism (if the kernel is nonzero, then the dimension of $A/I$ is strictly less, see Algebra, Lemmas \ref{algebra-lemma-regular-domain} and \ref{algebra-lemma-one-equation}). We conclude $I = (f_1, \ldots, f_a)$ and $J = (g_1, \ldots, g_b)$ by symmetry. Thus the Koszul complex $K_\bullet(A, f_1, \ldots, f_a)$ on $f_1, \ldots, f_a$ is a resolution of $A/I$, see More on Algebra, Lemma \ref{more-algebra-lemma-regular-koszul-regular}. Hence \begin{align*} \text{Tor}_p^A(A/I, A/J) & = H_p(K_\bullet(A, f_1, \ldots, f_a) \otimes_A A/J) \\ & = H_p(K_\bullet(A/J, f_1 \bmod J, \ldots, f_a \bmod J)) \end{align*} Since we've seen above that $f_1 \bmod J, \ldots, f_a \bmod J$ is a regular system of parameters in the regular local ring $A/J$ we conclude that there is only one cohomology group, namely $H_0 = A/(I + J) = \kappa$. This finishes the proof.
Both the left and side and the right hand side are additive in short exact sequences of modules of dimension $\leq d$, see Lemma \ref{lemma-multiplicity-ses} and Algebra, Lemma \ref{algebra-lemma-length-additive}. Hence by Algebra, Lemma \ref{algebra-lemma-filter-Noetherian-module} it suffices to prove this when $M = A/\mathfrak q$ for some prime $\mathfrak q$ of $A$ with $\dim(A/\mathfrak q) \leq d$. This case is obvious.
Set $A = \mathcal{O}_{X, \xi}$, $B = \mathcal{O}_{V, \xi}$, and $C = \mathcal{O}_{W, \xi}$. By Auslander-Buchsbaum (Algebra, Proposition \ref{algebra-proposition-Auslander-Buchsbaum}) we can find a finite free resolution $F_\bullet \to B$ of length $$ \text{depth}(A) - \text{depth}(B) = \dim(A) - \dim(B) = \dim(C) $$ First equality as $A$ and $B$ are Cohen-Macaulay and the second as $V$ and $W$ intersect properly. Then $F_\bullet \otimes_A C$ is a complex of finite free modules representing $B \otimes_A^\mathbf{L} C$ hence has cohomology modules with support in $\{\mathfrak m_A\}$. By the Acyclicity lemma (Algebra, Lemma \ref{algebra-lemma-acyclic}) which applies as $C$ is Cohen-Macaulay we conclude that $F_\bullet \otimes_A C$ has nonzero cohomology only in degree $0$. This finishes the proof.
Since $f_1, \ldots, f_r$ is a regular sequence the Koszul complex $K_\bullet(f_1, \ldots, f_r)$ is a resolution of $A/I$ over $A$, see More on Algebra, Lemma \ref{more-algebra-lemma-noetherian-finite-all-equivalent}. Thus the right hand side is equal to $$ \sum (-1)^i\text{length}_A H_i(K_\bullet(f_1, \ldots, f_r) \otimes_A M) $$ Now the result follows immediately from Theorem \ref{theorem-multiplicity-with-koszul} if $I$ is an ideal of definition. In general, we replace $A$ by $\overline{A} = A/\text{Ann}(M)$ and $f_1, \ldots, f_r$ by $\overline{f}_1, \ldots, \overline{f}_r$ which is allowed because $$ K_\bullet(f_1, \ldots, f_r) \otimes_A M = K_\bullet(\overline{f}_1, \ldots, \overline{f}_r) \otimes_{\overline{A}} M $$ Since $e_I(M, r) = e_{\overline{I}}(M, r)$ where $\overline{I} = (\overline{f}_1, \ldots, \overline{f}_r) \subset \overline{A}$ is an ideal of definition the result follows from Theorem \ref{theorem-multiplicity-with-koszul} in this case as well.
The equality $$ e(X, V\cdot W, Z) = e_{(f_1, \ldots, f_c)}(\mathcal{O}_{W, \xi}, c) $$ follows from the more general Lemma \ref{lemma-one-ideal-ci}. To see that $e_{(f_1, \ldots, f_c)}(\mathcal{O}_{W, \xi}, c)$ is $> 0$ or equivalently that $e_{(f_1, \ldots, f_c)}(\mathcal{O}_{W, \xi}, c)$ is the leading coefficient of the Hilbert polynomial it suffices to show that the dimension of $\mathcal{O}_{W, \xi}$ is $c$, because the degree of the Hilbert polynomial is equal to the dimension by Algebra, Proposition \ref{algebra-proposition-dimension}. Say $\dim(V) = r$, $\dim(W) = s$, and $\dim(X) = n$. Then $\dim(Z) = r + s - n$ as the intersection is proper. Thus the transcendence degree of $\kappa(\xi)$ over $\mathbf{C}$ is $r + s - n$, see Algebra, Lemma \ref{algebra-lemma-dimension-prime-polynomial-ring}. We have $r + c = n$ because $V$ is cut out by a regular sequence in a neighbourhood of $\xi$, see Divisors, Lemma \ref{divisors-lemma-Noetherian-scheme-regular-ideal} and then Lemma \ref{lemma-pullback-by-regular-immersion} applies (for example). Thus $$ \dim(\mathcal{O}_{W, \xi}) = s - (r + s - n) = s - ((n - c) + s - n) = c $$ the first equality by Algebra, Lemma \ref{algebra-lemma-dimension-at-a-point-finite-type-field}.
In this case the image of $f_1$ in $\mathcal{O}_{W, \xi}$ is nonzero by properness of intersection, hence a nonzerodivisor divisor. Moreover, $\mathcal{O}_{W, \xi}$ is a Noetherian local domain of dimension $1$. Thus $$ \text{length}_{\mathcal{O}_{X, \xi}} (\mathcal{O}_{W, \xi}/f_1^t\mathcal{O}_{W, \xi}) = t \text{length}_{\mathcal{O}_{X, \xi}} (\mathcal{O}_{W, \xi}/f_1\mathcal{O}_{W, \xi}) $$ for all $t \geq 1$, see Algebra, Lemma \ref{algebra-lemma-ord-additive}. This proves the lemma.
This follows immediately from Lemma \ref{lemma-intersection-multiplicity-CM}. Alternatively, we can deduce it from Lemma \ref{lemma-multiplicity-with-lci}. Namely, by Algebra, Lemma \ref{algebra-lemma-reformulate-CM} we see that $f_1, \ldots, f_c$ is a regular sequence in $\mathcal{O}_{W, \xi}$. Then Algebra, Lemma \ref{algebra-lemma-regular-quasi-regular} shows that $f_1, \ldots, f_c$ is a quasi-regular sequence. This easily implies the length of $\mathcal{O}_{W, \xi}/(f_1, \ldots, f_c)^t$ is $$ {c + t \choose c} \text{length}_{\mathcal{O}_{X, \xi}} (\mathcal{O}_{W, \xi}/ f_1\mathcal{O}_{W, \xi} + \ldots + f_c\mathcal{O}_{W, \xi}). $$ Looking at the leading coefficient we conclude.
First we observe that $X \times a$ is an effective Cartier divisor in $X \times \mathbf{P}^1$ and that $W_a$ is the scheme theoretic intersection of $W$ with $X \times a$. Hence the equality in (1)(a) is immediate from the definitions and the calculation of intersection multiplicity in case of a Cartier divisor given in Lemma \ref{lemma-multiplicity-with-effective-Cartier-divisor}. Part (1)(b) holds because $W_a \to X \times \mathbf{P}^1 \to X$ maps isomorphically onto its image which is how we viewed $W_a$ as a closed subscheme of $X$ in Section \ref{section-rational-equivalence}. Parts (2) and (3) are formal consequences of part (1) and the definitions.
Set $n = \dim(X)$, $a = n - r$, $b = n - s$. Observe that $\dim(T) = r + s - n = n - a - b$ by the assumption that the intersections are transversal. Let $(A, \mathfrak m, \kappa) = (\mathcal{O}_{X, \xi}, \mathfrak m_\xi, \kappa(\xi))$ where $\xi \in T$ is the generic point. Then $\dim(A) = a + b$, see Varieties, Lemma \ref{varieties-lemma-dimension-locally-algebraic}. Let $I_0, I, J_0, J \subset A$ cut out the trace of $Y_{i_0}$, $Y$, $Z_{j_0}$, $Z$ in $\Spec(A)$. Then $\dim(A/I) = \dim(A/I_0) = b$ and $\dim(A/J) = \dim(A/J_0) = a$ by the same reference. Set $\overline{I} = I + \mathfrak m^2/\mathfrak m^2$. Then $I \subset I_0 \subset \mathfrak m$ and $J \subset J_0 \subset \mathfrak m$ and $I + J = \mathfrak m$. By Lemma \ref{lemma-transversal} and its proof we see that $I_0 = (f_1, \ldots, f_a)$ and $J_0 = (g_1, \ldots, g_b)$ where $f_1, \ldots, g_b$ is a regular system of parameters for the regular local ring $A$. Since $I + J = \mathfrak m$, the map $$ I \oplus J \to \mathfrak m/\mathfrak m^2 = \kappa f_1 \oplus \ldots \oplus \kappa f_a \oplus \kappa g_1 \oplus \ldots \oplus \kappa g_b $$ is surjective. We conclude that we can find $f_1', \ldots, f_a' \in I$ and $g'_1, \ldots, g_b' \in J$ whose residue classes in $\mathfrak m/\mathfrak m^2$ are equal to the residue classes of $f_1, \ldots, f_a$ and $g_1, \ldots, g_b$. Then $f'_1, \ldots, g'_b$ is a regular system of parameters of $A$. By Algebra, Lemma \ref{algebra-lemma-regular-ring-CM} we find that $A/(f'_1, \ldots, f'_a)$ is a regular local ring of dimension $b$. Thus any nontrivial quotient of $A/(f'_1, \ldots, f'_a)$ has strictly smaller dimension (Algebra, Lemmas \ref{algebra-lemma-regular-domain} and \ref{algebra-lemma-one-equation}). Hence $I = (f'_1, \ldots, f'_a) = I_0$. By symmetry $J = J_0$. This proves (2), (3), and (4). Finally, the coefficient of $T$ in $[Y]_r \cdot [Z]_s$ is the coefficient of $T$ in $Y_{i_0} \cdot Z_{j_0}$ which is $1$ by Lemma \ref{lemma-transversal}.
By linearity and symmetry in $X$ and $Y$, it suffices to prove this when $\alpha = [V]$ for some subvariety $V \subset X$ of dimension $s$ and $\beta = [W_a]_s - [W_b]_s$ for some closed subvariety $W \subset Y \times \mathbf{P}^1$ of dimension $s + 1$ which intersects $Y \times a$ and $Y \times b$ properly. In this case the lemma follows if we can prove $$ [(V \times W)_a]_{r + s} = [V] \times [W_a]_s $$ and similarly with $a$ replaced by $b$. Namely, then we see that $\alpha \times \beta = [(V \times W)_a]_{r + s} - [(V \times W)_b]_{r + s}$ as desired. To see the displayed equality we note the equality $$ V \times W_a = (V \times W)_a $$ of schemes. The projection $V \times W_a \to W_a$ induces a bijection of irreducible components (see for example Varieties, Lemma \ref{varieties-lemma-bijection-irreducible-components}). Let $W' \subset W_a$ be an irreducible component with generic point $\zeta$. Then $V \times W'$ is the corresponding irreducible component of $V \times W_a$ (see Lemma \ref{lemma-dimension-product-varieties}). Let $\xi$ be the generic point of $V \times W'$. We have to show that $$ \text{length}_{\mathcal{O}_{Y, \zeta}}(\mathcal{O}_{W_a, \zeta}) = \text{length}_{\mathcal{O}_{X \times Y, \xi}}( \mathcal{O}_{V \times W_a, \xi}) $$ In this formula we may replace $\mathcal{O}_{Y, \zeta}$ by $\mathcal{O}_{W_a, \zeta}$ and we may replace $\mathcal{O}_{X \times Y, \zeta}$ by $\mathcal{O}_{V \times W_a, \zeta}$ (see Algebra, Lemma \ref{algebra-lemma-length-independent}). As $\mathcal{O}_{W_a, \zeta} \to \mathcal{O}_{V \times W_a, \xi}$ is flat, by Algebra, Lemma \ref{algebra-lemma-pullback-module} it suffices to show that $$ \text{length}_{\mathcal{O}_{V \times W_a, \xi}}( \mathcal{O}_{V \times W_a, \xi}/ \mathfrak m_\zeta\mathcal{O}_{V \times W_a, \xi}) = 1 $$ This is true because the quotient on the right is the local ring $\mathcal{O}_{V \times W', \xi}$ of a variety at a generic point hence equal to $\kappa(\xi)$.
By linearity we may assume $\alpha = [V]$ and $\beta = [W]$. Then (1) says that $\text{pr}_Y^{-1}(W) = X \times W$ and $\text{pr}_X^{-1}(V) = V \times Y$. This is clear. Part (2) holds because $X \times W \cap V \times Y = V \times W$ and $\dim(V \times W) = r + s$ by Lemma \ref{lemma-dimension-product-varieties}. \medskip\noindent Proof of (3). Let $\xi$ be the generic point of $V \times W$. Since the projections $X \times W \to W$ is smooth as a base change of $X \to \Spec(\mathbf{C})$, we see that $X \times W$ is nonsingular at every point lying over the generic point of $W$, in particular at $\xi$. Similarly for $V \times Y$. Hence $\mathcal{O}_{X \times W, \xi}$ and $\mathcal{O}_{V \times Y, \xi}$ are Cohen-Macaulay local rings and Lemma \ref{lemma-intersection-multiplicity-CM} applies. Since $V \times Y \cap X \times W = V \times W$ scheme theoretically the proof is complete.
Let us explain how to prove (1) in a more elementary way and part (2) using more general theory. As (2) implies (1) the reader can skip the proof of (1). \medskip\noindent Proof of (1). Choose an affine open $\Spec(A) \subset X$. Then $A$ is a Noetherian $\mathbf{C}$-algebra and $\mathcal{F}$, $\mathcal{G}$ correspond to finite $A$-modules $M$ and $N$ (Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-Noetherian}). By Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-tensor-product} we may compute $\text{Tor}_i$ over $\mathcal{O}_X$ by first computing the $\text{Tor}$'s of $M$ and $N$ over $A$, and then taking the associated $\mathcal{O}_X$-module. For the $\text{Tor}_i$ over $\mathcal{O}_{X \times X}$ we compute the tor of $A$ and $M \otimes_\mathbf{C} N$ over $A \otimes_\mathbf{C} A$ and then take the associated $\mathcal{O}_{X \times X}$-module. Hence on this affine patch we have to prove that $$ \text{Tor}_i^{A \otimes_\mathbf{C} A}(A, M \otimes_\mathbf{C} N) = \text{Tor}_i^A(M, N) $$ To see this choose resolutions $F_\bullet \to M$ and $G_\bullet \to M$ by finite free $A$-modules (Algebra, Lemma \ref{algebra-lemma-resolution-by-finite-free}). Note that $\text{Tot}(F_\bullet \otimes_\mathbf{C} G_\bullet)$ is a resolution of $M \otimes_\mathbf{C} N$ as it computes Tor groups over $\mathbf{C}$! Of course the terms of $F_\bullet \otimes_\mathbf{C} G_\bullet$ are finite free $A \otimes_\mathbf{C} A$-modules. Hence the left hand side of the displayed equation is the module $$ H_i(A \otimes_{A \otimes_\mathbf{C} A} \text{Tot}(F_\bullet \otimes_\mathbf{C} G_\bullet)) $$ and the right hand side is the module $$ H_i(\text{Tot}(F_\bullet \otimes_A G_\bullet)) $$ Since $A \otimes_{A \otimes_\mathbf{C} A} (F_p \otimes_\mathbf{C} G_q) = F_p \otimes_A G_q$ we see that these modules are equal. This defines an isomorphism over the affine open $\Spec(A) \times \Spec(A)$ (which is good enough for the application to equality of intersection numbers). We omit the proof that these isomorphisms glue. \medskip\noindent Proof of (2). The second statement follows from the first by the projection formula as stated in Derived Categories of Schemes, Lemma \ref{perfect-lemma-cohomology-base-change}. To see the first, represent $K$ and $M$ by K-flat complexes $\mathcal{K}^\bullet$ and $\mathcal{M}^\bullet$. Since pullback and tensor product preserve K-flat complexes (Cohomology, Lemmas \ref{cohomology-lemma-tensor-product-K-flat} and \ref{cohomology-lemma-pullback-K-flat}) we see that it suffices to show $$ \Delta^*\text{Tot}( \text{pr}_1^*\mathcal{K}^\bullet \otimes_{\mathcal{O}_{X \times X}} \text{pr}_2^*\mathcal{M}^\bullet) = \text{Tot}( \mathcal{K}^\bullet \otimes_{\mathcal{O}_X} \mathcal{M}^\bullet) $$ Thus it suffices to see that there are canonical isomorphisms $$ \Delta^*(\text{pr}_1^*\mathcal{K} \otimes_{\mathcal{O}_{X \times X}} \text{pr}_2^*\mathcal{M}) \longrightarrow \mathcal{K} \otimes_{\mathcal{O}_X} \mathcal{M} $$ whenever $\mathcal{K}$ and $\mathcal{M}$ are $\mathcal{O}_X$-modules (not necessarily quasi-coherent or flat). We omit the details.
By linearity it suffices to prove this when $\alpha = [V]$ and $\beta = [W]$ for some closed subvarieties $V \subset X$ and $W \subset Y$ which intersect properly. Recall that $V \times W$ is a closed subvariety of dimension $r + s$. Observe that scheme theoretically we have $V \cap W = \Delta^{-1}(V \times W)$ as well as $\Delta(V \cap W) = \Delta \cap V \times W$. This proves (1). \medskip\noindent Proof of (2). Let $Z \subset V \cap W$ be an irreducible component with generic point $\xi$. We have to show that the coefficient of $Z$ in $\alpha \cdot \beta$ is the same as the coefficient of $\Delta(Z)$ in $[\Delta] \cdot \alpha \times \beta$. The first is given by the integer $$ \sum (-1)^i \text{length}_{\mathcal{O}_{X, \xi}} \text{Tor}_i^{\mathcal{O}_X}(\mathcal{O}_V, \mathcal{O}_W)_\xi $$ and the second by the integer $$ \sum (-1)^i \text{length}_{\mathcal{O}_{X \times Y, \Delta(\xi)}} \text{Tor}_i^{\mathcal{O}_{X \times Y}}( \mathcal{O}_\Delta, \mathcal{O}_{V \times W})_{\Delta(\xi)} $$ However, by Lemma \ref{lemma-tor-and-diagonal} we have $$ \text{Tor}_i^{\mathcal{O}_X}(\mathcal{O}_V, \mathcal{O}_W)_\xi \cong \text{Tor}_i^{\mathcal{O}_{X \times Y}}( \mathcal{O}_\Delta, \mathcal{O}_{V \times W})_{\Delta(\xi)} $$ as $\mathcal{O}_{X \times X, \Delta(\xi)}$-modules. Thus equality of lengths (by Algebra, Lemma \ref{algebra-lemma-length-independent} to be precise). \medskip\noindent Part (2) implies (3) because $\text{pr}_{1, *} \circ \Delta_* = \text{id}$ by Lemma \ref{lemma-compose-pushforward}.
We are going to use Lemma \ref{lemma-tor-sheaf} without further mention. This implies that \begin{align*} V \cdot W & = \sum (-1)^i [\text{Tor}_i(\mathcal{O}_V, \mathcal{O}_W)]_{b + c - n} \\ U \cdot (V \cdot W) & = \sum (-1)^{i + j} [ \text{Tor}_j(\mathcal{O}_U, \text{Tor}_i(\mathcal{O}_V, \mathcal{O}_W)) ]_{a + b + c - 2n} \\ U \cdot V & = \sum (-1)^i [\text{Tor}_i(\mathcal{O}_U, \mathcal{O}_V)]_{a + b - n} \\ (U \cdot V) \cdot W & = \sum (-1)^{i + j} [ \text{Tor}_j(\text{Tor}_i(\mathcal{O}_U, \mathcal{O}_V), \mathcal{O}_W)) ]_{a + b + c - 2n} \end{align*} where $\dim(U) = a$, $\dim(V) = b$, $\dim(W) = c$, $\dim(X) = n$. The assumptions in the lemma guarantee that the coherent sheaves in the formulae above satisfy the required bounds on dimensions of supports in order to make sense of these. Now consider the object $$ K = \mathcal{O}_U \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{O}_V \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{O}_W $$ of the derived category $D_{\textit{Coh}}(\mathcal{O}_X)$. We claim that the expressions obtained above for $U \cdot (V \cdot W)$ and $(U \cdot V) \cdot W$ are equal to $$ \sum (-1)^k [H^k(K)]_{a + b + c - 2n} $$ This will prove the lemma. By symmetry it suffices to prove one of these equalities. To do this we represent $\mathcal{O}_U$ and $\mathcal{O}_V \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_W$ by K-flat complexes $M^\bullet$ and $L^\bullet$ and use the spectral sequence associated to the double complex $K^\bullet \otimes_{\mathcal{O}_X} L^\bullet$ in Homology, Section \ref{homology-section-double-complex}. This is a spectral sequence with $E_2$ page $$ E_2^{p, q} = \text{Tor}_{-p}(\mathcal{O}_U, \text{Tor}_{-q}(\mathcal{O}_V, \mathcal{O}_W)) $$ converging to $H^{p + q}(K)$ (details omitted; compare with More on Algebra, Example \ref{more-algebra-example-tor}). Since lengths are additive in short exact sequences we see that the result is true.
The statement that $[f^*\mathcal{F}]_{r + e}$ and $[f^*\mathcal{G}]_{s + e}$ intersect properly is immediate from the assumption that $f$ has relative dimension $e$. By Lemmas \ref{lemma-tor-sheaf} and \ref{lemma-pullback} it suffices to show that $$ f^*\text{Tor}_i^{\mathcal{O}_Y}(\mathcal{F}, \mathcal{G}) = \text{Tor}_i^{\mathcal{O}_X}(f^*\mathcal{F}, f^*\mathcal{G}) $$ as $\mathcal{O}_X$-modules. This follows from Cohomology, Lemma \ref{cohomology-lemma-pullback-tensor-product} and the fact that $f^*$ is exact, so $Lf^*\mathcal{F} = f^*\mathcal{F}$ and similarly for $\mathcal{G}$.
By linearity we may assume that $\alpha = [V]$ and $\beta = [W]$ for some closed subvarieties $V, W \subset Y$ of dimension $r, s$. Say $f$ has relative dimension $e$. Then the lemma is a special case of Lemma \ref{lemma-flat-pull-back-and-intersections-sheaves} because $[V] = [\mathcal{O}_V]_r$, $[W] = [\mathcal{O}_W]_r$, $f^*[V] = [f^{-1}(V)]_{r + e} = [f^*\mathcal{O}_V]_{r + e}$, and $f^*[W] = [f^{-1}(W)]_{s + e} = [f^*\mathcal{O}_W]_{s + e}$.
Let $\text{pr} : P \times \mathbf{P}^1 \to P$ be the projection. Since we are proving an equality of cycles it suffices to think of $\alpha$, resp.\ $\beta$ as a cycle on $P$, resp.\ $P \times \mathbf{P}^1$ and prove the result for pushing forward by $\text{pr}$. Because $\text{pr}^*X = X \times \mathbf{P}^1$ and $\text{pr}$ defines an isomorphism of $C'_0$ onto $C$ the projection formula (Lemma \ref{lemma-projection-formula-flat}) gives $$ \text{pr}_*([C'_0]_r \cdot X \times \mathbf{P}^1) = [C]_r \cdot X = \alpha $$ On the other hand, we have $[C'_0]_r = C' \cdot P \times 0$ as cycles on $P \times \mathbf{P}^1$ by Lemma \ref{lemma-rational-equivalence-and-intersection}. Hence $$ [C'_0]_r \cdot X \times \mathbf{P}^1 = (C' \cdot P \times 0) \cdot X \times \mathbf{P}^1 = (C' \cdot X \times \mathbf{P}^1) \cdot P \times 0 $$ by associativity (Lemma \ref{lemma-associative}) and commutativity of the intersection product. It remains to show that the intersection product of $C' \cdot X \times \mathbf{P}^1$ with $P \times 0$ on $P \times \mathbf{P}^1$ is equal (as a cycle) to the intersection product of $\beta$ with $X \times 0$ on $X \times \mathbf{P}^1$. Write $C' \cdot X \times \mathbf{P}^1 = \sum n_k[E_k]$ and hence $\beta = \sum n_k[E_k]$ for some subvarieties $E_k \subset X \times \mathbf{P}^1 \subset P \times \mathbf{P}^1$. Then both intersections are equal to $\sum m_k[E_{k, 0}]$ by Lemma \ref{lemma-rational-equivalence-and-intersection} applied twice. This finishes the proof.
We claim the lemma holds with $U = \mathbf{P}(V) \setminus X$. For a closed point $p$ of $U$ we indeed obtain a morphism $r_p|_X : X \to \mathbf{P}(W_p)$. This morphism is proper because $X$ is a proper scheme (Morphisms, Lemmas \ref{morphisms-lemma-locally-projective-proper} and \ref{morphisms-lemma-image-proper-scheme-closed}). On the other hand, the fibres of $r_p$ are affine lines as can be seen by a direct calculation. Hence the fibres of $r_p|X$ are proper and affine, whence finite (Morphisms, Lemma \ref{morphisms-lemma-finite-proper}). Finally, a proper morphism with finite fibres is finite (Cohomology of Schemes, Lemma \ref{coherent-lemma-characterize-finite}).