hcju/stacksqa huggingface dataset:
Bilinearity follows immediately from the linearity of pushforward and pullback and the bilinearity of the intersection product. To prove associativity, say we have $X, Y, Z, W$ and $c \in \text{Corr}(X, Y)$, $c' \in \text{Corr}(Y, Z)$, and $c'' \in \text{Corr}(Z, W)$. Then we have \begin{align*} c'' \circ (c' \circ c) & = \text{pr}^{134}_{14, *}( \text{pr}^{134, *}_{13} \text{pr}^{123}_{13, *}(\text{pr}^{123, *}_{12}c \cdot \text{pr}^{123, *}_{23}c') \cdot \text{pr}^{134, *}_{34}c'') \\ & = \text{pr}^{134}_{14, *}( \text{pr}^{1234}_{134, *} \text{pr}^{1234, *}_{123}(\text{pr}^{123, *}_{12}c \cdot \text{pr}^{123, *}_{23}c') \cdot \text{pr}^{134, *}_{34}c'') \\ & = \text{pr}^{134}_{14, *}( \text{pr}^{1234}_{134, *} (\text{pr}^{1234, *}_{12}c \cdot \text{pr}^{1234, *}_{23}c') \cdot \text{pr}^{134, *}_{34}c'') \\ & = \text{pr}^{134}_{14, *} \text{pr}^{1234}_{134, *} ((\text{pr}^{1234, *}_{12}c \cdot \text{pr}^{1234, *}_{23}c') \cdot \text{pr}^{1234, *}_{34}c'') \\ & = \text{pr}^{1234}_{14, *}( (\text{pr}^{1234, *}_{12}c \cdot \text{pr}^{1234, *}_{23}c') \cdot \text{pr}^{1234, *}_{34}c'') \end{align*} Here we use the notation $$ p^{1234}_{134} : X \times Y \times Z \times W \to X \times Z \times W \quad\text{and}\quad p^{134}_{14} : X \times Z \times W \to X \times W $$ the projections and similarly for other indices. The first equality is the definition of the composition. The second equality holds because $\text{pr}^{134, *}_{13} \text{pr}^{123}_{13, *} = \text{pr}^{1234}_{134, *} \text{pr}^{1234, *}_{123}$ by Chow Homology, Lemma \ref{chow-lemma-flat-pullback-proper-pushforward}. The third equality holds because intersection product commutes with the gysin map for $p^{1234}_{123}$ (which is given by flat pullback), see Chow Homology, Lemma \ref{chow-lemma-lci-gysin-product}. The fourth equality follows from the projection formula for $p^{1234}_{134}$, see Chow Homology, Lemma \ref{chow-lemma-projection-formula}. The fourth equality is that proper pushforward is compatible with composition, see Chow Homology, Lemma \ref{chow-lemma-compose-pushforward}. Since intersection product is associative by Chow Homology, Lemma \ref{chow-lemma-associative} this concludes the proof of associativity of composition of correspondences. \medskip\noindent We omit the proofs of (2) and (3) as these are essentially proved by carefully bookkeeping where various cycles live and in what (co)dimension. \medskip\noindent The statement on pushforward and pullback of cycles means that $(c' \circ c)^*(\alpha) = c^*((c')^*(\alpha))$ and $(c' \circ c)_*(\alpha) = (c')_*(c_*(\alpha))$. This follows on combining (1), (2), and (3).
Everything is clear from the construction and Lemma \ref{lemma-composition-correspondences} except for the existence of identity morphisms. Given a smooth projective scheme $X$ consider the class $[\Delta]$ of the diagonal $\Delta \subset X \times X$ in $\text{Corr}^0(X, X)$. We note that $\Delta$ is equal to the graph of the identity $\text{id}_X : X \to X$ which is a fact we will use below. \medskip\noindent To prove that $[\Delta]$ can serve as an identity we have to show that $[\Delta] \circ c = c$ and $c' \circ [\Delta] = c'$ for any correspondences $c \in \text{Corr}^r(Y, X)$ and $c' \in \text{Corr}^s(X, Y)$. For the second case we have to show that $$ c' = \text{pr}_{13, *}(\text{pr}_{12}^*[\Delta] \cdot \text{pr}_{23}^*c') $$ where $\text{pr}_{12} : X \times X \times Y \to X \times X$ is the projection and simlarly for $\text{pr}_{13}$ and $\text{pr}_{23}$. We may write $c' = \sum a_i [Z_i]$ for some integral closed subschemes $Z_i \subset X \times Y$ and rational numers $a_i$. Thus it clearly suffices to show that $$ [Z] = \text{pr}_{13, *}(\text{pr}_{12}^*[\Delta] \cdot \text{pr}_{23}^*[Z]) $$ in the chow group of $X \times Y$ for any integral closed subscheme $Z$ of $X \times Y$. After replacing $X$ and $Y$ by the irreducible component containing the image of $Z$ under the two projections we may assume $X$ and $Y$ are integral as well. Then we have to show $$ [Z] = \text{pr}_{13, *}([\Delta \times Y] \cdot [X \times Z]) $$ Denote $Z' \subset X \times X \times Y$ the image of $Z$ by the morphism $(\Delta, 1) : X \times Y \to X \times X \times Y$. Then $Z'$ is a closed subscheme of $X \times X \times Y$ isomorphic to $Z$ and $Z' = \Delta \times Y \cap X \times Z$ scheme theoretically. By Chow Homology, Lemma \ref{chow-lemma-intersect-properly}\footnote{The reader verifies that $\dim(Z') = \dim(\Delta \times Y) + \dim(X \times Z) - \dim(X \times X \times Y)$ and that $Z'$ has a unique generic point mapping to the generic point of $Z$ (where the local ring is CM) and to some point of $X$ (where the local ring is CM). Thus all the hypothese of the lemma are indeed verified.} we conclude that $$ [Z'] = [\Delta \times Y] \cdot [X \times Z] $$ Since $Z'$ maps isomorphically to $Z$ by $\text{pr}_{13}$ also we conclude. The verification that $[\Delta] \circ c = c$ is similar and we omit it.
In the proof of Lemma \ref{lemma-category-correspondences} we have seen that this construction sends identities to identities. To finish the proof we have to show if $g : Z \to Y$ is another morphism of smooth projective schemes over $k$, then we have $[\Gamma_g] \circ [\Gamma_f] = [\Gamma_{f \circ g}]$ in $\text{Corr}^0(X, Z)$. Arguing as in the proof of Lemma \ref{lemma-category-correspondences} we see that it suffices to show $$ [\Gamma_{f \circ g}] = \text{pr}_{13, *}([\Gamma_f \times Z] \cdot [X \times \Gamma_g]) $$ in $\CH^*(X \times Z)$ when $X$, $Y$, $Z$ are integral. Denote $Z' \subset X \times Y \times Z$ the image of the closed immersion $(f \circ g, g, 1) : Z \to X \times Y \times Z$. Then $Z' = \Gamma_f \times Z \cap X \times \Gamma_g$ scheme theoretically and we conclude using Chow Homology, Lemma \ref{chow-lemma-intersect-properly} that $$ [Z'] = [\Gamma_f \times Z] \cdot [X \times \Gamma_g] $$ Since it is clear that $\text{pr}_{13, *}([Z']) = [\Gamma_{f \circ g}]$ the proof is complete.
Proof of (1). Recall that $[\Gamma_f]_*(\alpha) = \text{pr}_{2, *}([\Gamma_f] \cdot \text{pr}_1^*\alpha)$. We have $$ [\Gamma_f] \cdot \text{pr}_1^*\alpha = (f, 1)_*((f, 1)^! \text{pr}_1^*\alpha) = (f, 1)_*((f, 1)^! \text{pr}_1^!\alpha) = (f, 1)_*(f^!\alpha) $$ The first equality by Chow Homology, Lemma \ref{chow-lemma-intersect-regularly-embedded}. The second by  Chow Homology, Lemma \ref{chow-lemma-lci-gysin-flat}. The third because $\text{pr}_1 \circ (f, 1) = f$ and Chow Homology, Lemma \ref{chow-lemma-lci-gysin-composition}. Then we coclude because $\text{pr}_{2, *} \circ (f, 1)_* = 1_*$ by Chow Homology, Lemma \ref{chow-lemma-compose-pushforward}. \medskip\noindent Proof of (2). Recall that $[\Gamma_f]_*(\beta) = \text{pr}_{1, *}([\Gamma_f] \cdot \text{pr}_2^*\beta)$. Arguing exactly as above we have $$ [\Gamma_f] \cdot \text{pr}_2^*\beta = (f, 1)_*\beta $$ Thus the result follows as before. \medskip\noindent Proof of (3). Proved in exactly the same manner as above.
Recall that $\text{Corr}^r(W, \Spec(k)) = \CH_{-r}(W)$ for any smooth projective scheme $W$ over $k$ and given $c \in \text{Corr}^s(W', W)$ the composition with $c$ agrees with pullback by $c$ as a map $\CH_{-r}(W) \to \CH_{-r - s}(W')$ (Lemma \ref{lemma-composition-correspondences}). Finally, we have Lemma \ref{lemma-functor-and-cycles} which tells us how to convert this into usual pushforward and pullback of cycles. We have $$ (a \otimes \text{id})^* \eta_Y^* [Y] = (a \otimes \text{id})^* [\Delta_Y] = (f \times \text{id})_*\Delta_Y = [\Gamma_f] $$ and the other way around we get $$ (\text{id} \otimes a^t)^* \eta_X^* [X] = (\text{id} \otimes a^t)^* [\Delta_X] = (\text{id} \times f)^![\Delta_X] = [\Gamma_f] $$ The last equality follows from Chow Homology, Lemma \ref{chow-lemma-lci-gysin-easy}. In other words, going either way around the diagram we obtain the element of $\text{Corr}^d(X \times Y, \Spec(k))$ corresponding to the cycle $\Gamma_f \subset X \times Y$.
Let $M = (X, p, m)$ be a motive and let $a \in \Mor(M, M)$ be a projector. Then $a = a \circ a$ both in $\Mor(M, M)$ as well as in $\text{Corr}^0(X, X)$. Set $N = (X, a, m)$. Since we have $a = p \circ a \circ a$ in $\text{Corr}^0(X, X)$ we see that $a : N \to M$ is a morphism of $M_k$. Next, suppose that $b : (Y, q, n) \to M$ is a morphism such that $(1 - a) \circ b = 0$. Then $b = a \circ b$ as well as $b = b \circ q$. Hence $b$ is a morphism $b : (Y, q, n) \to N$. Thus we see that the projector $1 - a$ has a kernel, namely $N$ and we find that $M_k$ is Karoubian, see Homology, Definition \ref{homology-definition-karoubian}.
We will use Lemma \ref{lemma-contravariant-functor} without further mention. The structure morphism $X \to \Spec(k)$ gives a correspondence $a \in \text{Corr}^0(\Spec(k), X)$. On the other hand, the rational point $x$ is a morphism $\Spec(k) \to X$ which gives a correspondence $b \in \text{Corr}^0(X, \Spec(k))$. We have $b \circ a = 1$ as a correspondence on $\Spec(k)$. The composition $a \circ b$ corresponds to the graph of the composition $X \to x \to X$ which is $c_0 = [x \times X]$. Thus $a = a \circ b \circ a = c_0 \circ a$ and $b = a \circ b \circ a = b \circ c_0$. Hence, unwinding the definitions, we see that $a$ and $b$ are mutually inverse morphisms $a : (\Spec(k), 1, 0) \to (X, c_0, 0)$ and $b : (X, c_0, 0) \to (\Spec(k), 1, 0)$. \medskip\noindent We will proceed exactly as above to prove the second statement. Denote $$ a' \in \text{Corr}^1(\Spec(k), X) = \CH^1(X) $$ the class of the point $x$. Denote $$ b' \in \text{Corr}^{-1}(X, \Spec(k)) = \CH_1(X) $$ the class of $[X]$. We have $b' \circ a' = 1$ as a correspondence on $\Spec(k)$ because $[x] \cdot [X] = [x]$ on $X = \Spec(k) \times X \times \Spec(k)$. Computing the intersection product $\text{pr}_{12}^*b' \cdot \text{pr}_{23}^*a'$ on $X \times \Spec(k) \times X$ gives the cycle $X \times \Spec(k) \times x$. Hence the composition $a' \circ b'$ is equal to $c_2$ as a correspondence on $X$. Thus $a' = a' \circ b \circ a' = c_2 \circ a'$ and $b' = b' \circ a' \circ b' = b' \circ c_2$. Recall that $$ \Mor((\Spec(k), 1, -1), (X, c_2, 0)) = c_2 \circ \text{Corr}^1(\Spec(k), X) \subset \text{Corr}^1(\Spec(k), X) $$ and $$ \Mor((X, c_2, 0), (\Spec(k), 1, -1)) = \text{Corr}^{-1}(X, \Spec(k)) \circ c_2 \subset \text{Corr}^{-1}(X, \Spec(k)) $$ Hence, we see that $a'$ and $b'$ are mutually inverse morphisms $a' : (\Spec(k), 1, -1) \to (X, c_0, 0)$ and $b' : (X, c_0, 0) \to (\Spec(k), 1, -1)$.
Let $(Y, p, m)$ and $(Z, q, n)$ be motives. If $n = m$, then a direct sum is given by $(Y \amalg Z, p + q, m)$, with obvious notation. Details omitted. \medskip\noindent Suppose that $n < m$. Let $X$, $c_2$ be as in Example \ref{example-decompose-P1}. Then we consider \begin{align*} (Z, q, n) & = (Z, q, m) \otimes (\Spec(k), 1, -1) \otimes \ldots \otimes (\Spec(k), 1, -1) \\ & \cong (Z, q, m) \otimes (X, c_2, 0) \otimes \ldots \otimes (X, c_2, 0) \\ & \cong (Z \times X^{m - n}, q \otimes c_2 \otimes \ldots \otimes c_2, m) \end{align*} where we have used Lemma \ref{lemma-inverse-h2}. This reduces us to the case discussed in the first paragraph.
Denote $U$ in $\mathcal{C}$ the invertible object which is assumed to exist in the statement of the lemma. We extend $F$ to motives by setting $$ F(X, p, m) = \left(\text{the image of the projector }F(p)\text{ in }F(X)\right) \otimes U^{\otimes -m} $$ which makes sense because $U$ is invertible and because $\mathcal{C}$ is Karoubian. An important feature of this choice is that $F(X, c_2, 0) = U$. Observe that \begin{align*} F((X, p, m) \otimes (Y, q, n)) & = F(X \times Y, p \otimes q, m + n) \\ & = \left(\text{the image of }F(p \otimes q)\text{ in }F(X \times Y)\right) \otimes U^{\otimes -m - n} \\ & = F(X, p, m) \otimes F(Y, q, n) \end{align*} Thus we see that our rule is compatible with tensor products on the level of objects (details omitted). \medskip\noindent Next, we extend $F$ to morphisms of motives. Suppose that $$ a \in \Hom((Y, p, m), (Z, q, n)) = q \circ \text{Corr}^{n - m}(Y, Z) \circ p \subset \text{Corr}^{n - m}(Y, Z) $$ is a morphism. If $n = m$, then $a$ is a correspondence of degree $0$ and we can use $F(a) : F(Y) \to F(Z)$ to get the desired map $F(Y, p, m) \to F(Z, q, n)$. If $n < m$ we get canonical identifications \begin{align*} s : F((Z, q, n)) & \to F(Z, q, m) \otimes U^{m - n} \\ & \to F(Z, q, m) \otimes F(X, c_2, 0) \otimes \ldots \otimes F(X, c_2, 0) \\ & \to F((Z, q, m) \otimes (X, c_2, 0) \otimes \ldots \otimes (X, c_2, 0)) \\ & \to F((Z \times X^{m - n}, q \otimes c_2 \otimes \ldots \otimes c_2, m)) \end{align*} Namely, for the first isomorphism we use the definition of $F$ on motives above. For the second, we use the choice of $U$. For the third we use the compatibility of $F$ on tensor products of motives. The fourth is the definition of tensor products on motives. On the other hand, since we similarly have an isomorphism $$ \sigma : (Z, q, n) \to (Z \times X^{m - n}, q \otimes c_2 \otimes \ldots \otimes c_2, m) $$ (see proof of Lemma \ref{lemma-additive}). Composing $a$ with this isomorphism gives $$ \sigma \circ a \in \Hom((Y, p, m), (Z \times X^{m - n}, q \otimes c_2 \otimes \ldots \otimes c_2, m)) $$ Putting everything together we obtain $$ s^{-1} \circ F(\sigma \circ a) : F(Y, p, m) \to F(Z, q, n) $$ If $n > m$ we similarly define isomorphisms $$ t : F((Y, p, m)) \to F((Y \times X^{n - m}, p \otimes c_2 \otimes \ldots \otimes c_2, n)) $$ and $$ \tau : (Y, p, m)) \to (Y \times X^{n - m}, p \otimes c_2 \otimes \ldots \otimes c_2, n) $$ and we set $F(a) = F(a \circ \tau^{-1}) \circ t$. We omit the verification that this construction defines a functor of symmetric monoidal categories.
We will use Lemma \ref{lemma-composition-correspondences} without further mention. We compute $$ \Hom(\mathbf{1}, h(X) \otimes h(X)(d)) = \text{Corr}^d(\Spec(k), X \times X) = \CH^d(X \times X) $$ Here we have $\eta = [\Delta]$. On the other hand, we have $$ \Hom(h(X)(d) \otimes h(X), \mathbf{1}) = \text{Corr}^{-d}(X \times X, \Spec(k)) = \CH_d(X \times X) $$ and here we have the class $\epsilon = [\Delta]$ of the diagonal as well. The composition of the correspondence $[\Delta] \otimes 1$ with $1 \otimes [\Delta]$ either way is the correspondence $[\Delta] = 1$ in $\text{Corr}^0(X, X)$ which proves the required diagrams of Categories, Definition \ref{categories-definition-dual} commute. Namely, observe that $$ [\Delta] \otimes 1 \in \text{Corr}^d(X, X \times X \times X) = \CH^{2d}(X \times X \times X \times X) $$ is given by the class of the cycle $\text{pr}^{1234, -1}_{23}(\Delta) \cap \text{pr}^{1234, -1}_{14}(\Delta)$ with obvious notation. Similarly, the class $$ 1 \otimes [\Delta] \in \text{Corr}^{-d}(X \times X \times X, X) = \CH^{2d}(X \times X \times X \times X) $$ is given by the class of the cycle $\text{pr}^{1234, -1}_{23}(\Delta) \cap \text{pr}^{1234, -1}_{14}(\Delta)$. The composition $(1 \otimes [\Delta]) \circ ([\Delta] \otimes 1)$ is by definition the pushforward $\text{pr}^{12345}_{15, *}$ of the intersection product $$ [\text{pr}^{12345, -1}_{23}(\Delta) \cap \text{pr}^{12345, -1}_{14}(\Delta)] \cdot [\text{pr}^{12345, -1}_{34}(\Delta) \cap \text{pr}^{12345, -1}_{15}(\Delta)] = [\text{small diagonal in } X^5] $$ which is equal to $\Delta$ as desired. We omit the proof of the formula for the composition in the other order.
Let $M = (X, p, m)$ be an object of $M_k$. Then $M$ is a summand of $(X, 0, m) = h(X)(m)$. By Homology, Lemma \ref{homology-lemma-Karoubian-dual} it suffices to show that $h(X)(m) = h(X) \otimes \mathbf{1}(m)$ has a dual. By construction $\mathbf{1}(-m)$ is a left dual of $\mathbf{1}(m)$. Hence it suffices to show that $h(X)$ has a left dual, see Categories, Lemma \ref{categories-lemma-tensor-dual}. Let $X = \coprod X_i$ be the decomposition of $X$ into irreducible components. Then $h(X) = \bigoplus h(X_i)$ and it suffices to show that $h(X_i)$ has a left dual, see Homology, Lemma \ref{homology-lemma-additive-dual}. This follows from Lemma \ref{lemma-dual}.
Any object $L$ of $M_k$ is a summand of $h(X)(m)$ for some smooth projective scheme $X$ over $k$ and some $m \in \mathbf{Z}$. Observe that the Chow groups of $M \otimes h(X)(m)$ are the same as the Chow groups of of $M \otimes h(X)$ up to a shift in degrees. Hence our assumption implies that $c \otimes 1 : M \otimes L \to N \otimes L$ induces an isomorphism on Chow groups for every object $L$ of $M_k$. By Lemma \ref{lemma-chow-groups-representable} we see that $$ \Hom_{M_k}(\mathbf{1}, M \otimes L) \to \Hom_{M_k}(\mathbf{1}, N \otimes L) $$ is an isomorphism for every $L$. Since every object of $M_k$ has a left dual (Lemma \ref{lemma-dual-general}) we conclude that $$ \Hom_{M_k}(K, M) \to \Hom_{M_k}(K, N) $$ is an isomorphism for every object $K$ of $M_k$, see Categories, Lemma \ref{categories-lemma-left-dual}. We conclude by the Yoneda lemma (Categories, Lemma \ref{categories-lemma-yoneda}).
By Lemma \ref{lemma-manin} it suffices to show that our map defines an isomorphism on Chow groups of motives after taking the product with any smooth projective scheme $Z$. Observe that $P \times Z \to X \times Z$ is the projective bundle associated to the pullback of $\mathcal{E}$ to $X \times Z$. Hence the statement on Chow groups is true by the projective space bundle formula given in Chow Homology, Lemma \ref{chow-lemma-chow-ring-projective-bundle}. Namely, pushforward of cycles along $[\Gamma_p]$ is given by pullback of cycles by $p$ according to Lemma \ref{lemma-functor-and-cycles} and Chow Homology, Lemma \ref{chow-lemma-lci-gysin-flat}. Hence pushforward along $c_i$ sends $\alpha$ to $c_1(\mathcal{O}_P(1))^i \cap p^*\alpha$. Some details omitted.
Observe that $(p \times p)^*[\Delta_X] = [P \times_X P]$. Since $\Delta_P \subset P \times_X P \subset P \times P$ and since capping with Chern classes commutes with proper pushforward (Chow Homology, Lemma \ref{chow-lemma-pushforward-cap-cj}) it suffices to show that the class of $\Delta_P \subset P \times_X P$ in $\CH^*(P \times_X P)$ is equal to $$ \left(\sum\nolimits_{i = 0, \ldots, r - 1} {r - 1 \choose i} c_{r - 1 - i}(q_1^*\mathcal{S}^\vee) \cap c_1(q_2^*\mathcal{O}_P(1))^i\right) \cap [P \times_X P] $$ where $q_i : P \times_X P \to P$, $i = 1, 2$ are the projections. Set $q = p \circ q_1 = p \circ q_2 : P \times_X P \to X$. Consider the maps $$ q_1^*\mathcal{S} \otimes q_2^*\mathcal{O}_P(-1) \to q^*\mathcal{E} \otimes q^*\mathcal{E}^\vee \to \mathcal{O}_{P \times_X P} $$ where the final arrow is the pullback by $q$ of the evaluation map $\mathcal{E} \otimes_{\mathcal{O}_X} \mathcal{E}^\vee \to \mathcal{O}_X$. The source of the composition is a module locally free of rank $r - 1$ and a local calculation shows that this map vanishes exactly along $\Delta_P$. By Chow Homology, Lemma \ref{chow-lemma-top-chern-class} the class $[\Delta_P]$ is the top Chern class of the dual $$ q_1^*\mathcal{S}^\vee \otimes q_2^*\mathcal{O}_P(1) $$ The desired result follows from Chow Homology, Lemma \ref{chow-lemma-chern-classes-E-tensor-L}.
Say $\dim(X) = d$ and $\dim(Y) = e$. By axiom (B) we have $H^{2d + 2e}(X \times Y) = H^{2d}(X) \otimes H^{2e}(Y)$ and by axiom (A)(d) this is $1$-dimensional. By Lemma \ref{lemma-degrees-cycles-classical} this $1$-dimensional vector space generated by the class $\gamma([x \times y])$ of a closed point $(x, y)$ and $\int_{X \times Y} \gamma([x \times y]) = 1$. Since $\gamma([x \times y]) = \gamma([x]) \otimes \gamma([y])$ by axioms (C)(a) and (C)(c) and since $\int_X \gamma([x]) = 1$ and $\int_Y \gamma([y]) = 1$ we conclude.
Recall that $\Delta^* : H^*(X \times X) \to H^*(X)$ is equal to the cup product map $H^*(X) \otimes_F H^*(X) \to H^*(X)$, see Remark \ref{remark-replace-cup-product-classical}. On the other hand we have $\gamma([\Delta]) = \Delta_*\gamma([X]) = \Delta_*1$ by axiom (C)(b) and the fact that $\gamma([X]) = 1$. Namely, $[X] \cdot [X] = [X]$ hence by axiom (C)(c) the cohomology class $\gamma([X])$ is $0$ or $1$ in the $1$-dimensional $F$-algebra $H^0(X)$; here we have also used axioms (A)(d) and (A)(b). But $\gamma([X])$ cannot be zero as $[X] \cdot [x] = [x]$ for a closed point $x$ of $X$ and we have the nonvanishing of $\gamma([x])$ by Lemma \ref{lemma-degrees-cycles-classical}. Hence $$ \int_{X \times X} \gamma([\Delta]) \cup a \otimes b = \int_{X \times X} \Delta_*1 \cup a \otimes b = \int_X a \cup b $$ by the definition of $\Delta_*$. On the other hand, we have $$ \int_{X \times X} (\sum e_{i, j} \otimes e'_{2d -i , j}) \cup a \otimes b = \sum (\int_X a \cup e_{i, j})(\int_X e'_{2d - i, j} \cup b) $$ by Lemma \ref{lemma-trace-product-classical}; note that we made two switches of order so that the sign is $1$. Thus if we choose $a$ such that $\int_X a \cup e_{i, j} = 1$ and all other pairings equal to zero, then we conclude that $\int_X e'_{2d - i, j} \cup b = \int_X a \cup b$ for all $b$, i.e., $e'_{2d - i, j} = a$. This proves the lemma.
Equality on the right. We have $[\Delta] \cdot [\Delta] = \Delta_*(\Delta^![\Delta])$ (Chow Homology, Lemma \ref{chow-lemma-intersect-regularly-embedded}). Since $\Delta_*$ preserves degrees of $0$-cycles it suffices to compute the degree of $\Delta^![\Delta]$. The class $\Delta^![\Delta]$ is given by capping $[\Delta]$ with the top Chern class of the normal sheaf of $\Delta \subset X \times X$ (Chow Homology, Lemma \ref{chow-lemma-gysin-fundamental}). Since the conormal sheaf of $\Delta$ is $\Omega_{X/k}$ (Morphisms, Lemma \ref{morphisms-lemma-differentials-diagonal}) we see that the normal sheaf is equal to the tangent sheaf $\mathcal{T}_X = \SheafHom_{\mathcal{O}_X}(\Omega_{X/k}, \mathcal{O}_X)$ as desired. \medskip\noindent Equality on the left. By Lemma \ref{lemma-degrees-cycles-classical} we have \begin{align*} \deg([\Delta] \cdot [\Delta]) & = \int_{X \times X} \gamma([\Delta]) \cup \gamma([\Delta]) \\ & = \int_{X \times X} \Delta_*1 \cup \gamma([\Delta]) \\ & = \int_{X \times X} \Delta_*(\Delta^*\gamma([\Delta])) \\ & = \int_X \Delta^*\gamma([\Delta]) \end{align*} Write $\gamma([\Delta]) = \sum  e_{i, j} \otimes e'_{2d - i , j}$ as in Lemma \ref{lemma-class-diagonal-classical}. Recalling that $\Delta^*$ is given by cup product we obtain $$ \int_X \sum\nolimits_{i, j} e_{i, j} \cup e'_{2d - i, j} = \sum\nolimits_{i, j} \int_X e_{i, j} \cup e'_{2d - i, j} = \sum\nolimits_{i, j} (-1)^i = \sum (-1)^i\beta_i $$ as desired.