hcju/stacksqa huggingface dataset:
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 $(\pi : P \to X, \mathcal{O}_P(1))$ associated to $\mathcal{E}$. Consider the fibre product diagram $$ \xymatrix{ P_D = \mathbf{P}(\mathcal{E}|_D) \ar[r]_-{i'} \ar[d]_{\pi_D} & P \ar[d]^\pi \\ D \ar[r]^i & X } $$ Note that $\mathcal{O}_{P_D}(1)$ is the pullback of $\mathcal{O}_P(1)$. Apply the gysin map $(i')^*$ (Lemma \ref{lemma-gysin-factors}) to the displayed equation above. Applying Lemmas \ref{lemma-gysin-commutes-cap-c1} and \ref{lemma-gysin-flat-pullback} we obtain $$ \sum\nolimits_{i = 0}^r (-1)^i c_1(\mathcal{O}_{P_D}(1))^i \cap \pi_D^*(i^*\alpha_{r - i}) = 0 $$ in the chow group of $P_D$. By the characterization of Lemma \ref{lemma-determine-intersections} we conclude.
It is immediate that (2) implies (1). Let $\alpha \in \CH_k(X)$. 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 $\mathcal{E}$. Denote $\pi' : P' \to X'$ the base change of $\pi$ by $f$. Using Lemma \ref{lemma-c1-center} and the properties of bivariant classes we obtain \begin{align*} 0 & = c \cap \left(\sum\nolimits_{i = 0}^r (-1)^i c_1(\mathcal{O}_P(1))^i \cap \pi^*(\alpha_{r - i})\right) \\ & = \sum\nolimits_{i = 0}^r (-1)^i c_1(\mathcal{O}_{P'}(1))^i \cap (\pi')^*(c \cap \alpha_{r - i}) \end{align*} in the Chow group of $P'$ (calculation omitted). Hence we see that $c \cap \alpha_j$ is equal to $c_j(f^*\mathcal{E}) \cap (c \cap \alpha)$ by the characterization of Lemma \ref{lemma-determine-intersections}. This proves the lemma.
This should hold for any triple $(X, \mathcal{E}, \mathcal{L})$. In particular it should hold when $X$ is integral and by Lemma \ref{lemma-bivariant-zero} it is enough to prove it holds when capping with $[X]$ for such $X$. Thus assume that $X$ is integral. Let $(\pi : P \to X, \mathcal{O}_P(1))$, resp.\ $(\pi' : P' \to X, \mathcal{O}_{P'}(1))$ be the projective space bundle associated to $\mathcal{E}$, resp.\ $\mathcal{E} \otimes \mathcal{L}$. Consider the canonical morphism $$ \xymatrix{ P \ar[rd]_\pi \ar[rr]_g & & P' \ar[ld]^{\pi'} \\ & X & } $$ see Constructions, Lemma \ref{constructions-lemma-twisting-and-proj}. It has the property that $g^*\mathcal{O}_{P'}(1) = \mathcal{O}_P(1) \otimes \pi^* {\mathcal L}$. This means that we have $$ \sum\nolimits_{i = 0}^r (-1)^i (\xi + x)^i \cap \pi^*(c_{r - i}(\mathcal{E} \otimes \mathcal{L}) \cap [X]) = 0 $$ in $\CH_*(P)$, where $\xi$ represents $c_1(\mathcal{O}_P(1))$ and $x$ represents $c_1(\pi^*\mathcal{L})$. By simple algebra this is equivalent to $$ \sum\nolimits_{i = 0}^r (-1)^i \xi^i \left( \sum\nolimits_{j = i}^r (-1)^{j - i} \binom{j}{i} x^{j - i} \cap \pi^*(c_{r - j}(\mathcal{E} \otimes \mathcal{L}) \cap [X]) \right) = 0 $$ Comparing with Equation (\ref{equation-chern-classes}) it follows from this that $$ c_{r - i}(\mathcal{E}) \cap [X] = \sum\nolimits_{j = i}^r \binom{j}{i} (-c_1(\mathcal{L}))^{j - i} \cap c_{r - j}(\mathcal{E} \otimes \mathcal{L}) \cap [X] $$ Reworking this (getting rid of minus signs, and renumbering) we get the desired relation.
By Lemma \ref{lemma-bivariant-zero} it suffices to show that if $X$ is integral then $c_j(\mathcal{E}) \cap [X] = c_j(\mathcal{F}) \cap [X]$. Let $(\pi : P \to X, \mathcal{O}_P(1))$, resp.\ $(\pi' : P' \to X, \mathcal{O}_{P'}(1))$ denote the projective space bundle associated to $\mathcal{E}$, resp.\ $\mathcal{F}$. The surjection $\mathcal{E} \to \mathcal{F}$ gives rise to a closed immersion $$ i : P' \longrightarrow P $$ over $X$. Moreover, the element $1 \in \Gamma(X, \mathcal{O}_X) \subset \Gamma(X, \mathcal{E})$ gives rise to a global section $s \in \Gamma(P, \mathcal{O}_P(1))$ whose zero set is exactly $P'$. Hence $P'$ is an effective Cartier divisor on $P$ such that $\mathcal{O}_P(P') \cong \mathcal{O}_P(1)$. Hence we see that $$ c_1(\mathcal{O}_P(1)) \cap \pi^*\alpha = i_*((\pi')^*\alpha) $$ for any cycle class $\alpha$ on $X$ by Lemma \ref{lemma-relative-effective-cartier}. By Lemma \ref{lemma-determine-intersections} we see that $\alpha_j = c_j(\mathcal{F}) \cap [X]$, $j = 0, \ldots, r - 1$ satisfy $$ \sum\nolimits_{j = 0}^{r - 1} (-1)^jc_1(\mathcal{O}_{P'}(1))^j \cap (\pi')^*\alpha_j = 0 $$ Pushing this to $P$ and using the remark above as well as Lemma \ref{lemma-pushforward-cap-c1} we get $$ \sum\nolimits_{j = 0}^{r - 1} (-1)^j c_1(\mathcal{O}_P(1))^{j + 1} \cap \pi^*\alpha_j = 0 $$ By the uniqueness of Lemma \ref{lemma-determine-intersections} we conclude that $c_r(\mathcal{E}) \cap [X] = 0$ and $c_j(\mathcal{E}) \cap [X] = \alpha_j = c_j(\mathcal{F}) \cap [X]$ for $j = 0, \ldots, r - 1$. Hence the lemma holds.
This relation really just says that $c_i(\mathcal{E}) = c_i(\mathcal{F}) + c_1(\mathcal{L})c_{i - 1}(\mathcal{F})$. By Lemma \ref{lemma-get-rid-of-trivial-subbundle} we have $c_j(\mathcal{E} \otimes \mathcal{L}^{\otimes -1}) = c_j(\mathcal{F} \otimes \mathcal{L}^{\otimes -1})$ for $j = 0, \ldots, r$ were we set $c_r(\mathcal{F} \otimes \mathcal{L}^{-1}) = 0$ by convention. Applying Lemma \ref{lemma-chern-classes-E-tensor-L} we deduce $$ \sum_{j = 0}^i \binom{r - i + j}{j} (-1)^j c_{i - j}({\mathcal E}) c_1({\mathcal L})^j = \sum_{j = 0}^i \binom{r - 1 - i + j}{j} (-1)^j c_{i - j}({\mathcal F}) c_1({\mathcal L})^j $$ Setting $c_i(\mathcal{E}) = c_i(\mathcal{F}) + c_1(\mathcal{L})c_{i - 1}(\mathcal{F})$ gives a ``solution'' of this equation. The lemma follows if we show that this is the only possible solution. We omit the verification.
By Lemma \ref{lemma-bivariant-zero} we may assume that $X$ is integral and we have to show the identity when capping against $[X]$. By induction on $r_1$. The case $r_1 = 1$ is Lemma \ref{lemma-additivity-invertible-subsheaf}. Assume $r_1 > 1$. Let $(\pi : P \to X, \mathcal{O}_P(1))$ denote the projective space bundle associated to $\mathcal{E}_1$. Note that \begin{enumerate} \item $\pi^* : \CH_*(X) \to \CH_*(P)$ is injective, and \item $\pi^*\mathcal{E}_1$ sits in a short exact sequence $0 \to \mathcal{F} \to \pi^*\mathcal{E}_1 \to \mathcal{L} \to 0$ where $\mathcal{L}$ is invertible. \end{enumerate} The first assertion follows from the projective space bundle formula and the second follows from the definition of a projective space bundle. (In fact $\mathcal{L} = \mathcal{O}_P(1)$.) Let $Q = \pi^*\mathcal{E}/\mathcal{F}$, which sits in an exact sequence $0 \to \mathcal{L} \to Q \to \pi^*\mathcal{E}_2 \to 0$. By induction we have \begin{eqnarray*} c(\pi^*\mathcal{E}) \cap [P] & = & c(\mathcal{F}) \cap c(\pi^*\mathcal{E}/\mathcal{F}) \cap [P] \\ & = & c(\mathcal{F}) \cap c(\mathcal{L}) \cap c(\pi^*\mathcal{E}_2) \cap [P] \\ & = & c(\pi^*\mathcal{E}_1) \cap c(\pi^*\mathcal{E}_2) \cap [P] \end{eqnarray*} Since $[P] = \pi^*[X]$ we win by Lemma \ref{lemma-flat-pullback-cap-cj}.
Let $C_i \subset X$, $i = 1, \ldots, t$ be the irreducible components of dimension $1$ with reduced induced scheme structure and let $m_i$ be the multiplicity of $C_i$ in $X$. Then $[X]_1 = \sum m_i[C_i]$ and $c_1(\mathcal{E}) \cap [X]_1$ is the sum of the pushforwards of the cycles $m_i c_1(\mathcal{E}|_{C_i}) \cap [C_i]$. Since we have a similar decomposition of the degree of $\mathcal{E}$ by Varieties, Lemma \ref{varieties-lemma-degree-in-terms-of-components} it suffices to prove the lemma in case $X$ is a proper curve over $k$. \medskip\noindent Assume $X$ is a proper curve over $k$. By Divisors, Lemma \ref{divisors-lemma-filter-after-modification} there exists a modification $f : X' \to X$ such that $f^*\mathcal{E}$ has a filtration whose successive quotients are invertible $\mathcal{O}_{X'}$-modules. Since $f_*[X']_1 = [X]_1$ we conclude from Lemma \ref{lemma-pushforward-cap-cj} that $$ \deg(c_1(\mathcal{E}) \cap [X]_1) = \deg(c_1(f^*\mathcal{E}) \cap [X']_1) $$ Since we have a similar relationship for the degree by Varieties, Lemma \ref{varieties-lemma-degree-birational-pullback} we reduce to the case where $\mathcal{E}$ has a filtration whose successive quotients are invertible $\mathcal{O}_X$-modules. In this case, we may use additivity of the degree (Varieties, Lemma \ref{varieties-lemma-degree-additive}) and of first Chern classes (Lemma \ref{lemma-additivity-chern-classes}) to reduce to the case discussed in the next paragraph. \medskip\noindent Assume $X$ is a proper curve over $k$ and $\mathcal{E}$ is an invertible $\mathcal{O}_X$-module. By Divisors, Lemma \ref{divisors-lemma-quasi-projective-Noetherian-pic-effective-Cartier} we see that $\mathcal{E}$ is isomorphic to $\mathcal{O}_X(D) \otimes \mathcal{O}_X(D')^{\otimes -1}$ for some effective Cartier divisors $D, D'$ on $X$ (this also uses that $X$ is projective, see Varieties, Lemma \ref{varieties-lemma-dim-1-proper-projective} for example). By additivity of degree under tensor product of invertible sheaves (Varieties, Lemma \ref{varieties-lemma-degree-tensor-product}) and additivity of $c_1$ under tensor product of invertible sheaves (Lemma \ref{lemma-c1-cap-additive} or \ref{lemma-chern-classes-E-tensor-L}) we reduce to the case $\mathcal{E} = \mathcal{O}_X(D)$. In this case the left hand side gives $\deg(D)$ (Varieties, Lemma \ref{varieties-lemma-degree-effective-Cartier-divisor}) and the right hand side gives $\deg([D]_0)$ by Lemma \ref{lemma-geometric-cap}. Since $$ [D]_0 = \sum\nolimits_{x \in D} \text{length}_{\mathcal{O}_{X, x}}(\mathcal{O}_{D, x}) [x] = \sum\nolimits_{x \in D} \text{length}_{\mathcal{O}_{D, x}}(\mathcal{O}_{D, x}) [x] $$ by definition, we see $$ \deg([D]_0) = \sum\nolimits_{x \in D} \text{length}_{\mathcal{O}_{D, x}}(\mathcal{O}_{D, x}) [\kappa(x) : k] = \dim_k \Gamma(D, \mathcal{O}_D) = \deg(D) $$ The penultimate equality by Algebra, Lemma \ref{algebra-lemma-pushdown-module} using that $D$ is affine.
We will prove this by induction on $d$. If $d = 0$, then the result is true by Varieties, Lemma \ref{varieties-lemma-chi-tensor-finite}. Assume $d > 0$. \medskip\noindent Let $Z_i \subset Z$, $i = 1, \ldots, t$ be the irreducible components of dimension $d$ with reduced induced scheme structure and let $m_i$ be the multiplicity of $Z_i$ in $Z$. Then $[Z]_d = \sum m_i[Z_i]$ and $c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_d) \cap [Z]_d$ is the sum of the cycles $m_i c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_d) \cap [Z_i]$. Since we have a similar decomposition for $(\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z)$ by Varieties, Lemma \ref{varieties-lemma-numerical-polynomial-leading-term} it suffices to prove the lemma in case $Z = X$ is a proper variety of dimension $d$ over $k$. \medskip\noindent By Chow's lemma there exists a birational proper morphism $f : Y \to X$ with $Y$ H-projective over $k$. See Cohomology of Schemes, Lemma \ref{coherent-lemma-chow-Noetherian} and Remark \ref{coherent-remark-chow-Noetherian}. Then $$ (f^*\mathcal{L}_1 \cdots f^*\mathcal{L}_d \cdot Y) = (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot X) $$ by Varieties, Lemma \ref{varieties-lemma-intersection-number-and-pullback} and we have $$ f_*(c_1(f^*\mathcal{L}_1) \cap \ldots \cap c_1(f^*\mathcal{L}_d) \cap [Y]) = c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_d) \cap [X] $$ by Lemma \ref{lemma-pushforward-cap-c1}. Thus we may replace $X$ by $Y$ and assume that $X$ is projective over $k$. \medskip\noindent If $X$ is a proper $d$-dimensional projective variety, then we can write $\mathcal{L}_1 = \mathcal{O}_X(D) \otimes \mathcal{O}_X(D')^{\otimes -1}$ for some effective Cartier divisors $D, D' \subset X$ by Divisors, Lemma \ref{divisors-lemma-quasi-projective-Noetherian-pic-effective-Cartier}. By additivity for both sides of the equation (Varieties, Lemma \ref{varieties-lemma-intersection-number-additive} and Lemma \ref{lemma-c1-cap-additive}) we reduce to the case $\mathcal{L}_1 = \mathcal{O}_X(D)$ for some effective Cartier divisor $D$. By Varieties, Lemma \ref{varieties-lemma-numerical-intersection-effective-Cartier-divisor} we have $$ (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot X) = (\mathcal{L}_2 \cdots \mathcal{L}_d \cdot D) $$ and by Lemma \ref{lemma-geometric-cap} we have $$ c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_d) \cap [X] = c_1(\mathcal{L}_2) \cap \ldots \cap c_1(\mathcal{L}_d) \cap [D]_{d - 1} $$ Thus we obtain the result from our induction hypothesis.
It is clear that (1) $\Rightarrow$ (2) $\Rightarrow$ (3). Conversely, if (3) holds, then we set $X_n = \{x \in X \mid n_x = n\}$ and we get a decomposition as in (1). Namely, $X_n$ is open because given $x$ the union of the irreducible components of $X$ passing through $x$ minus the union of the irreducible components of $X$ not passing through $x$ is an open neighbourhood of $x$. If $X$ is normal, then $X$ is a disjoint union of integral schemes (Properties, Lemma \ref{properties-lemma-normal-locally-Noetherian}) and hence the properties hold. If $X$ is Cohen-Macaulay, then $\delta' : X \to \mathbf{Z}$, $x \mapsto -\dim(\mathcal{O}_{X, x})$ is a dimension function on $X$ (see Example \ref{example-CM-irreducible}). Since $\delta - \delta'$ is locally constant (Topology, Lemma \ref{topology-lemma-dimension-function-unique}) and since $\delta'(\xi) = 0$ for every generic point $\xi$ of $X$ we see that (2) holds.
We may assume $r_i \geq 1$ for all $i$. We will prove the lemma by induction on $\sum (r_i - 1)$. If this integer is $0$, then $\mathcal{E}_i$ is invertible for all $i$ and we conclude by taking $\pi = \text{id}_X$. If not, then we can pick an $i$ such that $r_i > 1$ and consider the morphism $\pi_i : P_i = \mathbf{P}(\mathcal{E}_i) \to X$. We have a short exact sequence $$ 0 \to \mathcal{F} \to \pi_i^*\mathcal{E}_i \to \mathcal{O}_{P_i}(1) \to 0 $$ of finite locally free $\mathcal{O}_{P_i}$-modules of ranks $r_i - 1$, $r_i$, and $1$. Observe that $\pi_i^*$ is injective on chow groups after any base change by the projective bundle formula (Lemma \ref{lemma-chow-ring-projective-bundle}). By the induction hypothesis applied to the finite locally free $\mathcal{O}_{P_i}$-modules $\mathcal{F}$ and $\pi_{i'}^*\mathcal{E}_{i'}$ for $i' \not = i$, we find a morphism $\pi : P \to P_i$ with properties stated as in the lemma. Then the composition $\pi_i \circ \pi : P \to X$ does the job. Some details omitted.
Choose a morphism $\pi : P \to X$ as in Lemma \ref{lemma-splitting-principle}. By the injectivity of $\pi^*$ (after any base change) it suffices to prove the relation between the Chern classes of $\mathcal{E}$ and $\mathcal{E}^\vee$ after pulling back to $P$. Thus we may assume there exist invertible $\mathcal{O}_X$-modules ${\mathcal L}_i$, $i = 1, \ldots, r$ and a filtration $$ 0 = \mathcal{E}_0 \subset \mathcal{E}_1 \subset \mathcal{E}_2 \subset \ldots \subset \mathcal{E}_r = \mathcal{E} $$ such that $\mathcal{E}_i/\mathcal{E}_{i - 1} \cong \mathcal{L}_i$. Then we obtain the dual filtration $$ 0 = \mathcal{E}_r^\perp \subset \mathcal{E}_1^\perp \subset \mathcal{E}_2^\perp \subset \ldots \subset \mathcal{E}_0^\perp = \mathcal{E}^\vee $$ such that $\mathcal{E}_{i - 1}^\perp/\mathcal{E}_i^\perp \cong \mathcal{L}_i^{\otimes -1}$. Set $x_i = c_1(\mathcal{L}_i)$. Then $c_1(\mathcal{L}_i^{\otimes -1}) = - x_i$ by Lemma \ref{lemma-c1-cap-additive}. By Lemma \ref{lemma-chern-filter-by-linebundles} we have $$ c(\mathcal{E}) = \prod\nolimits_{i = 1}^r (1 + x_i) \quad\text{and}\quad c(\mathcal{E}^\vee) = \prod\nolimits_{i = 1}^r (1 - x_i) $$ in $A^*(X)$. The result follows from a formal computation which we omit.
Arguing exactly as in the proof of Lemma \ref{lemma-chern-classes-dual} we may assume we have invertible $\mathcal{O}_X$-modules ${\mathcal L}_i$, $i = 1, \ldots, r$ ${\mathcal N}_i$, $i = 1, \ldots, s$ filtrations $$ 0 = \mathcal{E}_0 \subset \mathcal{E}_1 \subset \mathcal{E}_2 \subset \ldots \subset \mathcal{E}_r = \mathcal{E} \quad\text{and}\quad 0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \mathcal{F}_2 \subset \ldots \subset \mathcal{F}_s = \mathcal{F} $$ such that $\mathcal{E}_i/\mathcal{E}_{i - 1} \cong \mathcal{L}_i$ and such that $\mathcal{F}_j/\mathcal{F}_{j - 1} \cong \mathcal{N}_j$. Ordering pairs $(i, j)$ lexicographically we obtain a filtration $$ 0 \subset \ldots \subset \mathcal{E}_i \otimes \mathcal{F}_j + \mathcal{E}_{i - 1} \otimes \mathcal{F} \subset \ldots \subset \mathcal{E} \otimes \mathcal{F} $$ with successive quotients $$ \mathcal{L}_1 \otimes \mathcal{N}_1, \mathcal{L}_1 \otimes \mathcal{N}_2, \ldots, \mathcal{L}_1 \otimes \mathcal{N}_s, \mathcal{L}_2 \otimes \mathcal{N}_1, \ldots, \mathcal{L}_r \otimes \mathcal{N}_s $$ By Lemma \ref{lemma-chern-filter-by-linebundles} we have $$ c(\mathcal{E}) = \prod (1 + x_i), \quad c(\mathcal{F}) = \prod (1 + y_j), \quad\text{and}\quad c(\mathcal{F}) = \prod (1 + x_i + y_j), $$ in $A^*(X)$. The result follows from a formal computation which we omit.
Here $c_{top}(\mathcal{E})$ is the bivariant class defined in Remark \ref{remark-top-chern-class}. By its very definition, in order to verify the formula, we may assume that $\mathcal{E}$ has constant rank. We may similarly assume $\mathcal{N}'$ and $\mathcal{N}$ have constant ranks, say $r'$ and $r$, so $\mathcal{E}$ has rank $r - r'$ and $c_{top}(\mathcal{E}) = c_{r - r'}(\mathcal{E})$. Observe that $p^*\mathcal{E}$ has a canonical section $$ s \in \Gamma(N, p^*\mathcal{E}) = \Gamma(X, p_*p^*\mathcal{E}) = \Gamma(X, \mathcal{E} \otimes_{\mathcal{O}_X} \text{Sym}(\mathcal{N}^\vee) \supset \Gamma(X, \SheafHom(\mathcal{N}, \mathcal{E})) $$ corresponding to the surjection $\mathcal{N} \to \mathcal{E}$ given in the statement of the lemma. The vanishing scheme of this section is exactly $N' \subset N$. Let $Y \subset X$ be an integral closed subscheme of $\delta$-dimension $n$. Then we have \begin{enumerate} \item $p^*[Y] = [p^{-1}(Y)]$ since $p^{-1}(Y)$ is integral of $\delta$-dimension $n + r$, \item $(p')^*[Y] = [(p')^{-1}(Y)]$ since $(p')^{-1}(Y)$ is integral of $\delta$-dimension $n + r'$, \item the restriction of $s$ to $p^{-1}Y$ has vanishing scheme $(p')^{-1}Y$ and the closed immersion $(p')^{-1}Y \to p^{-1}Y$ is a regular immersion (locally cut out by a regular sequence). \end{enumerate} We conclude that $$ (p')^*[Y] = c_{r - r'}(p^*\mathcal{E}) \cap p^*[Y] \quad\text{in}\quad \CH_*(N) $$ by Lemma \ref{lemma-top-chern-class}. This proves the lemma.
Let $\alpha^\bullet : \mathcal{E}_1^\bullet \to \mathcal{E}_2^\bullet$ be a map of bounded complexes of finite locally free $\mathcal{O}_X$-modules representing $E_1 \to E_2$. Then the cone $C(\alpha)^\bullet$ represents $E_3$. Since $C(\alpha)^n = \mathcal{E}_2^n \oplus \mathcal{E}_1^{n + 1}$ the formulas follow from the additivity and multiplicativity of Lemmas \ref{lemma-chern-character-additive} and \ref{lemma-additivity-chern-classes}.
In the case where $E$ is locally free of rank $r$ this is Lemma \ref{lemma-chern-classes-E-tensor-L}. The reader can deduce the lemma from this special case by a formal computation. An alternative is to use the splitting principle of Remark \ref{remark-splitting-principle-perfect}. In this case one ends up having to prove the following algebra fact: if we write formally $$ \frac{\prod_{a = 1, \ldots, n} (1 + x_a)}{\prod_{n = 1, \ldots, m} (1 + y_b)} = 1 + c_1 + c_2 + c_3 + \ldots $$ with $c_i$ homogeneous of degree $i$ in $\mathbf{Z}[x_i, y_j]$ then we have $$ \frac{\prod_{a = 1, \ldots, n} (1 + x_a + t)}{\prod_{b = 1, \ldots, m} (1 + y_b + t)} = \sum\nolimits_{i \geq 0} \sum\nolimits_{j = 0}^i \binom{r - i + j}{j} c_{i - j} t^j $$ where $r = n - m$. We omit the details.
After representing $E$ and $F$ by bounded complexes of finite locally free $\mathcal{O}_X$-modules this follows by a compuation from the corresponding result for vector bundles in Lemmas \ref{lemma-chern-classes-tensor-product} and \ref{lemma-chern-character-multiplicative} by a calculation. A better proof is probably to use the splitting principle as in Remark \ref{remark-splitting-principle-perfect} and reduce the lemma to computations in polynomial rings  which we describe in the next paragraph. \medskip\noindent Let $A$ be a commutative ring (for us this will be the subring of the bivariant chow ring of $X$ generated by Chern classes). Let $S$ be a finite set together with maps $\epsilon : S \to \{\pm 1\}$ and $f : S \to A$. Define $$ P_p(S, f , \epsilon) = \sum\nolimits_{s \in S} \epsilon(s) f(s)^p $$ in $A$. Given a second triple $(S', \epsilon', f')$ the equality that has to be shown for $P_p$ is the equality $$ P_p(S \times S', f + f' , \epsilon \epsilon') =  \sum\nolimits_{p_1 + p_2 = p} {p \choose p_1} P_{p_1}(S, f, \epsilon) P_{p_2}(S', f', \epsilon') $$ To see this is true, one reduces to the polynomial ring on variables $S \amalg S'$ and one shows that each term $f(s)^if'(s')^j$ occurs on the left and right hand side with the same coefficient. To verify the formulas for $c_1(E \otimes_{\mathcal{O}_X}^\mathbf{L} F)$ and $c_2(E \otimes_{\mathcal{O}_X}^\mathbf{L} F)$ we use the splitting principle to reduce to checking these formulae in a torsion free ring. Then we use the relationship between $P_j(E)$ and $c_i(E)$ proved in Remark \ref{remark-splitting-principle-perfect}. For example $$ c_1(E \otimes F) = P_1(E \otimes F) = r(F)P_1(E) + r(E)P_1(F) = r(F)c_1(E) + r(E)c_1(F) $$ the middle equation because $r(E) = P_0(E)$ by definition. Similarly, we have \begin{align*} & 2c_2(E \otimes F) \\ & = c_1(E \otimes F)^2 - P_2(E \otimes F) \\ & = (r(F)c_1(E) + r(E)c_1(F))^2 - r(F)P_2(E) - P_1(E)P_1(F) - r(E)P_2(F) \\ & = (r(F)c_1(E) + r(E)c_1(F))^2 - r(F)(c_1(E)^2 - 2c_2(E)) - c_1(E)c_1(F) - \\ & \quad r(E)(c_1(F)^2 - 2c_2(F)) \end{align*} which the reader can verify agrees with the formula in the statement of the lemma up to a factor of $2$.
We are going to use the material of Section \ref{section-pre-derived} without further mention. \medskip\noindent Assume $E_2|_{X_1 \cap X_2}$ is zero. Consider a morphism of schemes $X' \to X$ which is locally of finite type and denote $i'_j : X'_j \to X'$ the base change of $i_j$. By Lemma \ref{lemma-exact-sequence-closed-chow} we can write any element $\alpha' \in \CH_k(X')$ as $i'_{1, *}\alpha'_1 + i'_{2, *}\alpha'_2$ where $\alpha'_2 \in \CH_k(X'_2)$ is well defined up to an element in the image of pushforward by $X'_1 \cap X'_2 \to X'_2$. Then we can set $P'_p(E_2) \cap \alpha' = P_p(E_2) \cap \alpha'_2 \in \CH_{k - p}(X'_2)$. This is well defined by our assumption that $E_2$ restricts to zero on $X_1 \cap X_2$. \medskip\noindent If $E_2|_{X_1 \cap X_2}$ is isomorphic to a finite locally free $\mathcal{O}_{X_1 \cap X_2}$-module of rank $< p$ sitting in cohomological degree $0$, then $c_p(E_2|_{X_1 \cap X_2}) = 0$ by rank considerations and we can argue in exactly the same manner.
First, assume $E|_{X_1}$ is zero. With notations as in the proof of Lemma \ref{lemma-silly} the lemma in this case follows from \begin{align*} P_p(E) \cap \alpha' & = i'_{1, *}(P_p(E) \cap \alpha'_1) + i'_{2, *}(P_p(E) \cap \alpha'_2) \\ & = i'_{1, *}(P_p(E|_{X_1}) \cap \alpha'_1) + i'_{2, *}(P'_p(E_2) \cap \alpha') \\ & = i'_{2, *}(P'_p(E_2) \cap \alpha') \end{align*} The case where $E|_{X_1}$ is isomorphic to a finite locally free $\mathcal{O}_{X_1}$-module of rank $< p$ sitting in cohomological degree $0$ is similar.
Let $\alpha \in \CH_k(X)$. We may write $$ \alpha = \alpha_1 + \alpha_2 $$ with $\alpha_i \in \CH_k(X_i)$; we are omitting the pushforwards by the closed immersions $X_i \to X$. The reader then checks that $c'_p(E_2) \cap \alpha = c_p(E_2) \cap \alpha_2$, $c \cap c'_p(E_2) \cap \alpha = c \cap c_p(E_2) \cap \alpha_2$, $c \cap \alpha = c \cap \alpha_1 + c \cap \alpha_2$, and $c'_p(Lf_2^*E_2) \cap c \cap \alpha = c_p(Lf_2^*E_2) \cap c \cap \alpha_2$. We conclude by Lemma \ref{lemma-commutative-chern-perfect}.
The statement makes sense because the zero sheaf has rank $< 1$ and hence the classes $c'_p(E_2)$ are defined for all $p \geq 1$. The equalities follow immediately from the characterization of the classes produced by Lemma \ref{lemma-silly} and the corresponding result for capping with the Chern classes of $E_2$ given in Remark \ref{remark-splitting-principle-perfect}.