Proof. See [Reference Fulton and Pragacz19, §5.1, p. 54–55], [Reference Gelfand, Kapranov and Zelevinsky20, Lemma 4.12], or [Reference Golubitsky and Guillemin21, Ex. V (4), p. 145].

Proof. First we prove the lemma for the total Hom space $H = \lvert \operatorname {Hom}_X(\mathscr {F},\mathscr {E})\rvert $ . Denote $\pi \colon H = \lvert \operatorname {Hom}_X(\mathscr {F},\mathscr {E})\rvert \to X$ the projection, and let $\mathbb {D}_\ell : = D_\ell (\tau _H)\subset H$ be the degeneracy locus for the tautological map $\tau _H \colon \pi ^* \mathscr {F} \to \pi ^* \mathscr {E}$ . As the statement is local, we may assume $X = \operatorname {Spec} A$ , $\mathscr {F} = W \otimes _\Bbbk A$ , and $\mathscr {E} = V \otimes _\Bbbk A$ , where A is a $\Bbbk $ -algebra and $W, V$ are $\Bbbk $ -vector spaces. Then $H = \operatorname {Hom}(W, V) \times _\Bbbk X$ is the flat base change of $\operatorname {Hom}_\Bbbk (W,V)$ along $X \to \operatorname {Spec} \Bbbk $ , and $\mathbb {D} := \mathbb {D}_\ell \backslash \mathbb {D}_{\ell -1} = D \times _\Bbbk X$ . The desired result holds for H and $\mathbb {D}$ by Lemma 2.2.

In general, the map $\sigma \colon \mathscr {F} \to \mathscr {E}$ induces a section map $s_{\sigma } \colon X \to H$ , such that $\sigma = s_{\sigma }^* \tau _H$ and $D = \mathbb {D} \times _{X} H$ . Since s is the section of a smooth separated morphism, it is a regular closed immersion. Since H and X are Cohen–Macaulay, $\mathbb {D} \hookrightarrow H$ is a regular immersion, and the intersection $D = \mathbb {D} \times _X H \hookrightarrow X$ has the expected codimension; therefore the inclusion $D \hookrightarrow X$ is also a regular immersion, with normal bundle $N_{D/X} = s_{\sigma }^* N_{\mathbb {D}/H}$ . Finally, $s_{\sigma }^* N_{\mathbb {D}/H} = K^\vee \otimes C$ holds, since $K^\vee = s_{\sigma }^* \operatorname {Coker}\left (\tau _H^\vee \right )$ and $C = s_{\sigma }^* \operatorname {Coker}(\tau _H)$ .

Proof. Part (1) follows from [Reference Fulton18, Proposition 3.1(a)], and part (2) follows from [Reference Fulton18, Theorem 3.3], which could also be viewed as a special case of [Reference Fulton18, Proposition 14.6.5]. For part (3), to agree with Fulton’s notation, let $E = \mathscr {E}^\vee $ be the dual vector bundle, so $\mathbb {P}_{\text {sub}}(E) = \mathbb {P}(\mathscr {E})$ . From part (2), for any $\beta \in \operatorname {CH}_k(\mathbb {P}(\mathscr {E}))$ there exist unique $\alpha _i \in \operatorname {CH}_{k-(r-1)+i}(X)$ , $i \in [0,r-1]$ , such that

$$ \begin{align*} \beta = \sum_{i=0}^{r-1} \zeta^i \cdot \pi^* \alpha_i. \end{align*} $$

It follows from the definition of Segre classes $s_i(E) \cap \alpha : = \pi _*\left (\zeta ^{i+r-1} \cdot \pi ^* \alpha \right )$ that

$$ \begin{align*} \pi_* \left(\zeta^{j} \cdot \beta\right) = \sum_{i=0}^{j} s_{i}(E) \cap \alpha_{r-1-j+i}, \qquad \text{for } j=0,1,\dotsc, r-1. \end{align*} $$

Then the desired results follow from solving $\alpha _{i}$ s using $1 = c(E)s(E) = (1+c_1(E)+ c_2(E) + \dotsb )(1+ s_1(E)+s_2(E) + \dotsb )$ .

Proof. (a) In fact, from [Reference Jiang, Leung and Xie25, Remark. 2.5], the Euler sequence for $\mathbb {P}(\mathscr {N}_i)$ is equivalent to

(3.5) $$ \begin{align} 0 \to \mathscr{N}_j \to p^* \mathscr{N}_i \to \mathscr{O}_{\mathbb{P}\left(\mathscr{N}_i\right)}(1) \to 0, \end{align} $$

where $\mathscr {N}_i = \mathscr {E}\rvert _Z$ and $\mathscr {N}_j \simeq \Omega _{\mathbb {P}(\mathscr {E})/X}(1)\rvert _{\mathbb {P}\left (\mathscr {N}_i\right )}$ . Therefore the excess bundle for diagram 3.1 is given by $ \mathscr {O}_{\mathbb {P}\left (\mathscr {N}_i\right )}(1)$ . Now from [Reference Fulton18, Theorem 6.3, Propositions 6.2(1) and 6.6], one has:

$$ \begin{align*} \pi^* i_* (\underline{\hphantom{A}}) = j_* \pi^!_{\mathbb{P}\left(\mathscr{N}_i\right)} (\underline{\hphantom{A}}) = j_* \left(c_1\left(\mathscr{O}_{\mathbb{P}\left(\mathscr{N}_i\right)}(1)\right) \cap p^*(\underline{\hphantom{A}})\right). \end{align*} $$

(b) Since $\iota \colon \mathcal {H}_s \hookrightarrow \mathbb {P}(\mathscr {E})$ is a divisor of $\mathscr {O}_{\mathbb {P}(\mathscr {E})}(1)$ , we have $\iota ^* \iota _* (\underline {\hphantom {A}}) = \zeta \cdot (\underline {\hphantom {A}})$ , and

$$ \begin{align*} \pi_{i*} \pi_j^* \alpha = q_{i+1*} \iota_* \left( \iota^* \left(q_j^* \alpha\right) \right) = q_{i+1*} \left(\zeta \cdot \zeta^j \cdot q^* \alpha\right) = q_{i+1*} q_{j+1}^* \alpha =\delta_{i,j} \alpha. \end{align*} $$

(c) Since $j^*j_*\varepsilon = c_{r-1}\left (\mathscr {N}_j\right ) \cap \varepsilon $ , from the Euler sequence (3.5) and $\mathscr {N}_i = \mathscr {E}\rvert _Z$ ,

$$ \begin{align*} c_{r-1}(\mathscr{N}_j) = \sum_{i=0}^{r-1} (-1)^i \zeta^i p^* c_{r-1-i}(\mathscr{E}) = (-1)^{r-1} \zeta^{r-1} + \text{(lower order terms of}\ \zeta^{i}). \end{align*} $$

Therefore $j^*j_*\varepsilon = c_{r-1}\left (\mathscr {N}_j\right ) \cap \varepsilon = 0$ and $p_{1*} \varepsilon = \dotsb = p_{r-1*} \varepsilon =0$ imply $p_{0 *} \varepsilon = p_*\left (\zeta ^{r-1} \cdot \varepsilon \right ) + p_* \left (\left (\text {lower-order terms of }\zeta ^{i}\right ) \cap \varepsilon \right ) = \pm p_*\left ( c_{r-1}\left (\mathscr {N}_j\right ) \cap \varepsilon \right ) + p_* \left (\left (\text {lower-order terms of }\zeta ^{i}\right )\right. \left.\cap \varepsilon \right ) = 0$ . Hence $\varepsilon = \sum _{i=0}^{r-1} p_i^* p_{i*} \varepsilon = 0$ .

(d)(i) Over the open subscheme $U = X \backslash Z$ , $\mathcal {H}_s\rvert _{\pi ^{-1}(U)} = \mathbb {P}(\mathscr {G}\rvert _U)$ is a projective bundle with fiber $\mathbb {P}^{r-2}$ . In fact, over U there is an exact sequence of vector bundles $0 \to \mathscr {O}_U \to \mathscr {E}\rvert _U \to \mathscr {G}\rvert _U \to 0$ . Then the linear subbundle $\mathbb {P}( \mathscr {G}\rvert _U) \subset \mathbb {P}(\mathscr {E}\rvert _U)$ is a divisor representing the class $\zeta = c_1 (\mathscr {O}(1))$ . For any $\beta \in \operatorname {CH}(\mathbb {P}( \mathscr {G}\rvert _U))$ , by Theorem 2.4 applied to $\mathbb {P}(\mathscr {G}_U)$ there exists a unique $\alpha _i \in \operatorname {CH}(U)$ such that $\beta = \sum _{i=0}^{r-2} (\iota ^*\zeta )^i \cdot \pi ^* \alpha _i$ . Therefore the following holds:

$$ \begin{align*} \iota_* \beta = \sum_{i=0}^{r-2} \iota_* \left(\left(\iota^*(\zeta)^i \cdot \iota^*(q^* \alpha)\right)\right. = \sum_{i=0}^{r-2} \zeta^{i+1} \cdot q^* \alpha_i, \end{align*} $$

where the last equality follows from the projection formula and $\iota _* \iota ^* (\underline {\hphantom {A}}) = \zeta \cdot (\underline {\hphantom {A}})$ . From the uniqueness statement of Theorem 2.4 applied to $\mathbb {P}(\mathscr {E})$ , we know that

$$ \begin{align*} \alpha_i = q_{i+1*} \iota_* \beta = \pi_{i *} \beta. \end{align*} $$

Therefore, over U, we have $\beta = \sum _{i=0}^{r-2} \pi _i^* \pi _{i*} \beta .$ Now for any $\beta \in \operatorname {CH}_k(\mathcal {H}_s)$ , $\left (\beta - \sum _{i=0}^{r-2} \pi _i^* \pi _{i*} \beta \right )\big \rvert _U = 0$ . From the exact sequence $\operatorname {CH}(\mathbb {P}(\mathscr {N}_i)) \to \operatorname {CH}(\mathcal {H}_s) \to \operatorname {CH}(\mathcal {H}_{s}\rvert _U) \to 0$ , there exists an $\varepsilon \in \operatorname {CH}(\mathbb {P}(\mathscr {N}_i))$ such that $\beta - \sum _{i=0}^{r-2} \pi _i^* \pi _{i*} \beta = j_* \varepsilon $ .

(d)(ii) From part (d)(i) we know that $\beta = j_* \varepsilon $ , for $\varepsilon \in \operatorname {CH}_k(\mathbb {P}(\mathscr {N}_i))$ . Since the ambient square of diagram (3.1) is flat, by the flat base-change formula we have

$$ \begin{align*} i_* p_{i+1 *} = q_{i+1 *} (\iota \circ j)_* = \pi_{i *} j_*, \qquad \text{for } i = 0,1,\dotsc, r-2. \end{align*} $$

Therefore $i_* (p_{i+1 *} \varepsilon ) = \pi _{i } * \beta = 0$ for $i \in [0,r-2]$ . Notice that since $\varepsilon = \sum _{i=0}^{r-1} p_i^* p_{i *} \varepsilon $ , one has

$$ \begin{align*} j_* \left(p_0^* p_{0*} \varepsilon\right) = j_*( \varepsilon) - j_* \left(\zeta \cdot \sum_{i=0}^{r-2} p_{i}^* p_{i+1 *} \varepsilon \right) = j_*( \varepsilon) - \sum_{i=0}^{r-2} \pi_{i}^* i_* (p_{i+1 *} \varepsilon) = j_*( \varepsilon). \end{align*} $$

Here the second equality follows from the key formula of part (a). Now $j^* j_* \left (p_0^* p_{0*} \varepsilon \right ) = j^* j_* \varepsilon =0$ . By part (c), $p_0^* p_{0*} \varepsilon =0$ , hence $\beta = j_* \varepsilon =0$ .

Proof of Theorem 3.1. The fact $gf =0$ follows from part (a). The surjectivity of g is part (d)(i). By part (b), a left inverse of f is given by $h \colon \left (\oplus _{i=0}^{r-2} \alpha _i, \varepsilon \right ) \mapsto \oplus _{i=0}^{r-2} p_{i+1*} \varepsilon .$ In fact, $hf$ is

$$ \begin{align*} \oplus_{i=0}^{r-2} \gamma_i \mapsto \oplus_{i=0}^{r-2} p_{i+1*} \left(\sum_{j=0}^{r-2} \zeta^{j+1} \cdot p^* \gamma_j\right) = \oplus_{i=0}^{r-2} \left(p_{i+1*} \sum_{j=0}^{r-2} p_{j+1}^* \gamma_i\right) = \oplus_{i=0}^{r-2} \gamma_i. \end{align*} $$

To show the exactness of formula (3.2), suppose that for $\alpha _i \in \operatorname {CH}(X)$ and $\varepsilon \in \operatorname {CH}(\mathbb {P}(\mathscr {N}_i))$ , we have $\sum _{i=0}^{r-2} \pi _i^* \alpha _i + j_* \varepsilon = 0.$ Then similar to part (d)(ii), from part (b), for all $i \in [0,r-2]$ ,

$$ \begin{align*} \alpha_i = - \pi_{i*} ( j_* \varepsilon) = - i_{*} p_{i+1*} \varepsilon \in \operatorname{CH}(X). \end{align*} $$

Now consider $ \varepsilon ^{\prime } = \varepsilon - \sum _{i=0}^{r-2} p_{i+1}^* p_{i+1*} \varepsilon. $ Then similar to the proof of part (d)(ii), we have

$$ \begin{align*} j_* \varepsilon^{\prime} = j_*( \varepsilon) - j_* \left(\zeta \cdot \sum_{i=0}^{r-2} p_{i}^* p_{i+1 *} \varepsilon \right) = j_*( \varepsilon) - \sum_{i=0}^{r-2} \pi_{i}^* i_* (p_{i+1 *} \varepsilon) = j_*( \varepsilon) + \sum_{i=0}^{r-2} \pi_{i}^* \alpha_i =0, \end{align*} $$

and $p_{1*} \varepsilon ^{\prime } = \dotsb = p_{r-1*} \varepsilon ^{\prime } =0$ (since $\varepsilon ^{\prime } =p_0^* p_{0*} \varepsilon $ ). Therefore by part (c), $ \varepsilon ^{\prime } =0$ . Hence $(\oplus _i \alpha _i, \varepsilon ) = \left ( -\oplus _i i_{*} \gamma _i, \sum _{i=0}^{r-2} p_{i+1}^*\gamma _i\right )$ for $\gamma _i= p_{i+1*} \varepsilon $ . Hence the sequence (3.2) is exact.

To prove the last statement, we show that for any $\beta \in \operatorname {CH}(\mathcal {H}_s)$ , there exists a unique $\varepsilon \in \operatorname {CH}(\mathbb {P}(\mathscr {N}_i))$ such that $p_{1*} \varepsilon = \dotsb = p_{r-1*} \varepsilon =0$ , and

$$ \begin{align*} \beta = \sum_{i=0}^{r-2} \pi_i^* \pi_{i*} \beta + j_* \varepsilon. \end{align*} $$

In fact, for any expression $\beta = \sum _{i=0}^{r-2} \pi _i^* \alpha _i + j_* \varepsilon ,$ by replacing $\varepsilon $ with $\varepsilon - \sum _{i=0}^{r-2} p_{i+1}^* p_{i+1*} \varepsilon $ and $\alpha _i$ with $\alpha _i+ i_* (p_{i+1 *} \varepsilon )$ , we may assume $p_{1*} \varepsilon = \dotsb = p_{r-1*} \varepsilon =0$ . Hence by the projective bundle formula, $\varepsilon = p^* \gamma $ for a unique $\gamma \in \operatorname {CH}(Z)$ . Now by the flat base-change formula,

$$ \begin{align*} \pi_{i*} (j_* p^* \gamma) = q_{i+1*} (\iota \circ j)_* p^* \gamma = q_{i+1*} q^* (i_* \gamma) = 0, \qquad i \in [0,r-2]. \end{align*} $$

Therefore $\pi _{i*} \beta = \pi _{i*} \left ( \sum _{i=0}^{r-2} \pi _i^* \alpha _i + j_* p^* \gamma \right ) = \alpha _i$ for all $i=0,1,\dotsc , r-2$ . Hence we have established the identification

$$ \begin{align*} \operatorname{CH}_{k}(\mathcal{H}_s) & =\bigoplus_{i=0}^{r-2} \pi_i^* \operatorname{CH}_{k-(r-2)+i}(X) \oplus j_* \left(\operatorname{CH}_{k}(\mathbb{P}(\mathscr{N}_i))_{p_{1*} = \dotsb = p_{r-1*} =0}\right) \\ &= \bigoplus_{i=0}^{r-2} \pi_i^* \operatorname{CH}_{k-(r-2)+i}(X) \oplus j_* p^*\operatorname{CH}_{k-(r-1)}(Z), \end{align*} $$

where $\operatorname {CH}_{k}(\mathbb {P}(\mathscr {N}_i))_{p_{1*} = \dotsb =p_{r-1*} =0}$ denotes the subgroup $\{\gamma \in \operatorname {CH}_{k}(\mathbb {P}(\mathscr {N}_i)) \mid p_{1*} \gamma = \dotsb =p_{r-1*} \gamma =0 \}$ of $\operatorname {CH}_{k}(\mathbb {P}(\mathscr {N}_i) )$ . Moreover, the projection maps to the first $(r-1)$ -summands are respectively given by $\beta \mapsto \alpha _i =\pi _{i*} \beta $ , for $i = 0,1,\dotsc , r-2$ . For the formula of the projection to the last summand, it suffices to notice that $p_* j^* \pi _i^* (\underline {\hphantom {A}})= p_* \left (p_i^*(i^*(\underline {\hphantom {A}}))\right ) = 0$ for $i \in [0, r-2]$ and that $p_* j^*j_*p^*(\underline {\hphantom {A}}) = p_* \left ( c_{r-1} \left (\Omega _{\mathbb {P}(\mathscr {E})}(1)\right ) \cap p^*(\underline {\hphantom {A}})\right ) = (-1)^{r-1} \operatorname {Id}$ .□

Proof. By Manin’s identity principle, it suffices to notice that for any smooth T, the schemes $Z \times T \subset X \times T$ and $\mathcal {H}_s \times T$ are also in the same situation as the Cayley trick (Theorem 3.4). Hence the identities of Remark 3.3 hold for the Chow motives.

Proof of the first part of Theorem 3.6(1). The equality $\Gamma _* \Gamma ^* = \operatorname {Id}$ follows exactly the same line of proof as [Reference Lee, Lin and Wang35, Theorem 2.1], as already mentioned in [Reference Lee, Lin and Wang35, §2.3]. We sketch the proof here for completeness. For any class $[W^{\prime }] \in \operatorname {CH}_k(X^{\prime })$ , by Chow’s moving lemma (if allowing negative coefficients), we may assume it is represented by a cycle $W^{\prime }$ which intersects $P^{\prime }$ transversely. Therefore $\pi ^{\prime *}[W^{\prime }] = \left [\widetilde {W}\right ]$ by [Reference Fulton18, Corollary 6.7.2], where $\widetilde {W}$ is the blowup of $W^{\prime }$ along $W^{\prime } \cap P^{\prime }$ . Hence $\Gamma ^*[W^{\prime }] = \pi _* \left [\widetilde {W}\right ] = [W]$ , where W is the image of $\widetilde {W}$ and is also the proper transform of $W^{\prime }$ along the birational rational map $f^{-1}$ . Now we have

$$ \begin{align*} \pi^* [W] = \left[\widetilde{W}\right] + j_* \sum_{B} [E_{B}], \end{align*} $$

where we let $B^{\prime } \subset W^{\prime } \cap P^{\prime }$ be a component; then $E_{B} \subset E$ are k-cycles supported over components $\overline {B} = \psi ^{\prime }(B^{\prime }) \subset \psi ^{\prime }(W^{\prime } \cap P^{\prime }) \subset S$ . A direct computation of dimensions shows that for a general point s, the fiber $E_{B,s}$ over s has dimension

$$ \begin{align*} \dim E_{B,s} \ge \dim E_{B} - \dim \left(\overline{B}\right) \ge \dim E_{B} - \dim ({B^{\prime}}) = k - (k - (n+1)) = n+1. \end{align*} $$

Now $E_{B,s}$ must contain positive fibers of of $p^{\prime }_s \colon \mathbb {P}_s^n \times \mathbb {P}_s^m \to \mathbb {P}_s^m$ , as $n+1> n \ge m$ . Hence $\pi _* j_*[E_B] = p_*[E_B] = 0$ , and $\Gamma _* \Gamma ^* [W^{\prime }] = \pi ^{\prime }_* \pi ^* [W] =\pi ^{\prime }_* \left [\widetilde {W}\right ] = [W^{\prime }]$ .□

Proof of the second part of Theorem 3.6(1). It follows from Lemma 2.7 that

$$ \begin{align*} c_{m}(\mathscr{V}) &= (-1)^m c_m\left(\mathscr{O}_{P}(1) \boxtimes \Omega_{P^{\prime}/S}(1)\right) = (-1)^m \sum_{t=0}^{m} \zeta^t \cdot c_{m-t}\left(\Omega_{P^{\prime}/S}(1)\right)\\[-3pt] &= (-1)^m \sum_{t=0}^{m} (-1)^{m-t} \sum_{s=0}^{m-t} c_{s}(F^{\prime}) \cdot (\zeta')^{m-s-t} \cdot \zeta^t \end{align*} $$

and

$$ \begin{align*} c_{n}(\mathscr{V}^{\prime}) &= (-1)^n c_n\left(\Omega_{P/S}(1)\boxtimes \mathscr{O}_{P^{\prime}}(1) \right ) = (-1)^n \sum_{j=0}^{n} c_{n-j}\left(\Omega_{P/S}(1)\right)\cdot (\zeta^{\prime})^j \\[-3pt] &= (-1)^n \sum_{j=0}^{n} (-1)^{n-j} \sum_{i=0}^{n-j} c_{i}(F) \cdot \zeta^{n-i-j} \cdot (\zeta^{\prime})^{j}. \end{align*} $$

The map $\Phi _* \circ \Phi ^*$ is given by the convolution of correspondences

$$ \begin{align*} c_{m}(\mathscr{V}) *c_{n}(\mathscr{V}^{\prime}) := p_{13*}\left(p_{12}^*(c_{n}(\mathscr{V}^{\prime})) \cdot p_{23}^* (c_{m}(\mathscr{V})) \right) \in \operatorname{CH}^{m}(P^{\prime} \times_S P^{\prime}), \end{align*} $$

where $p_{ij}$ are the obvious projections from $P^{\prime } \times _S P \times _S P^{\prime }$ to the corresponding factors and the cohomological degree m is computed via $m+n -\dim (P/S) = m$ . To avoid confusion, we denote the product $P^{\prime } \times _S P \times _S P^{\prime }$ by $P_1' \times _S P \times _S P_2'$ , and denote the relative $\mathscr {O}(1)$ -classes of $P_1'$ and $P_2'$ by $\zeta ^{\prime }_1$ and $\zeta ^{\prime }_2$ , respectively. Therefore,

$$ \begin{align*} c_{m}(\mathscr{V}) *c_{n}(\mathscr{V}^{\prime}) = p_{13*}\!\left(\!\sum_{j=0}^{n} \sum_{t=0}^{m}\! (-1)^{j+t} \!\sum_{s=0}^{m-t} \sum_{i=0}^{n-j}\! c_{s}\!\left(F_2'\right) \cdot c_{i}(F) \cdot \zeta^{n+t-i-j} \cdot\!\left(\zeta_1'\right)^{j} \cdot \!\left(\zeta_2'\right)^{m-s-t} \!\right)\!. \end{align*} $$

Since $p_{13*}(\zeta ^k) = 0$ for all $0 \le k \le n-1$ , the only terms inside the parentheses that could survive $p_{13*}$ are the ones whose indices satisfy $t-i-j \ge 0$ . Thus we may assume that the indices of the summation satisfy $j \le t \le m$ and $0 \le i \le t-j$ . From the definition of the Segre class of F, we have $p_{13*}\left (\zeta ^{n+k}\right ) = s_k(F)$ , hence

$$ \begin{align*} c_{m}(\mathscr{V}) *c_{n}(\mathscr{V}^{\prime}) = \sum_{j=0}^{m} \sum_{t=j}^{m} (-1)^{j+t} \sum_{s=0}^{m-t} \sum_{i=0}^{t-j} c_{i}(F) \cdot s_{t-i-j}(F) \cdot c_{s}\left(F_2'\right) \cdot\left(\zeta_1'\right)^{j} \cdot \left(\zeta_2'\right)^{m-s-t}. \end{align*} $$

From $c(F) s(F) = 1$ , we know that $\sum _{i=0}^{t-j} c_{i}(F) \cdot s_{t-i-j}(F) = 0$ unless $t=j$ , in which case $c_0(F) s_0(F) =1$ . Hence this expression reduces to

$$ \begin{align*} c_{m}(\mathscr{V}) *c_{n}(\mathscr{V}^{\prime}) &= \sum_{j=0}^{m} \sum_{s=0}^{m-j} c_{s}\left(F_2'\right) \cdot\left(\zeta_1'\right)^{j} \cdot \left(\zeta_2'\right)^{m-j-s} = \sum_{j=0}^{m} c_{m-j}\left(\mathcal{T}_{P_2'/S}(-1)\right) \cdot\left(\zeta_1'\right)^{j} \\ & = c_m \left(\mathscr{O}_{P_1'}(1) \boxtimes \mathcal{T}_{P_2'/S}(-1)\right). \end{align*} $$

(For the second equality, we used Lemma 2.7.) On the other hand, the diagonal $\Delta _{P^{\prime }} \subset P^{\prime } \times _S P^{\prime }$ is the zero locus of a regular section s of the rank m vector bundle $\mathscr {O}_{P^{\prime }}(1) \boxtimes \mathcal {T}_{P^{\prime }/S}(-1)$ ; the section s under the canonical identification

$$ \begin{align*} \Gamma\left(P^{\prime} \times_S P^{\prime}, \mathscr{O}_{P^{\prime}}(1) \boxtimes \mathcal{T}_{P^{\prime}/S}(-1)\right) = \Gamma\left(S, F^{\prime\vee} \otimes F^{\prime}\right) = \operatorname{Hom}_S(F^{\prime},F^{\prime}) \end{align*} $$

corresponds to $1_{F^{\prime }} \colon F^{\prime } \to F^{\prime }$ . Hence $ [\Delta _{P^{\prime }}] = c_m\left (\mathscr {O}_{P^{\prime }}(1) \boxtimes \mathcal {T}_{P^{\prime }/S}(-1)\right )$ , and therefore

$$ \begin{align*} c_{m}(\mathscr{V}) *c_{n}(\mathscr{V}^{\prime}) = [\Delta_{P^{\prime}}], \qquad \text{hence } \Phi_* \, \Phi^* = \operatorname{Id}_{\operatorname{CH}(P^{\prime})}. \\[-36pt]\end{align*} $$

□

Proof. Statement (1) follows directly from the projection formula [Reference Fulton18, Theorem 3.2(c)] and Theorem 2.4(1). For statement (2), notice that for any $0 \le i \le m$ ,

$$ \begin{align*} \Phi_* \left(\zeta^{n-m+i}\right)& = p^{\prime}_* \left(c_{m}(\mathscr{V}) \cup p^*\zeta^{n-m+i}\right) \\ &= p^{\prime}_*\left((-1)^m \sum_{t=0}^{m} (-1)^{m-t} \sum_{s=0}^{m-t} c_{s}(F^{\prime}) \cdot (\zeta^{\prime})^{m-s-t} \cdot \zeta^{n-m+i+t}\right) \\ & = \sum_{t=m-i}^{m} (-1)^{t} \sum_{s=0}^{m-t} c_{s}(F^{\prime}) \cdot s_{t-(m-i)}(F) \cdot (\zeta^{\prime})^{m-s-t} \\ & = (-1)^{m-i} (\zeta^{\prime})^{i} + (\text{lower-order terms}). \end{align*} $$

(For example, $\Phi _* (\zeta ^m) = (\zeta ^{\prime })^{n} + (\text {lower-order terms})$ and $\Phi _* (\zeta ^n) = (-1)^n (\zeta ^{\prime })^{0}$ .) This computation together with statement (1) shows that $\Phi _* \left (\zeta ^{n-m+i} \cap \psi ^* \theta \right ) = \left (\pm \zeta ^{\prime i} + (\text {lower-order terms})\right ) \cap \psi ^{\prime *} \theta $ for all $\theta \in \operatorname {CH}(S)$ , which implies the injectivity of $\Phi _*$ on $\operatorname {CH}(P)_{m}$ . The same computation in the case $i<0$ shows that $\Phi _*$ is zero on $\operatorname {CH}(P)_{\Phi _*=0}$ . A similar computation shows

$$ \begin{align*} \Phi^*\left(\zeta^{\prime i}\right) = \pm \zeta^{n-m+i} + (\text{lower-order terms}) \end{align*} $$

for $0 \le i \le m$ , which implies statement (3).

Proof. Let $\gamma = \sum _{i=0}^{n} \zeta ^i \cdot \psi ^* \theta _i$ for $\theta _i \in \operatorname {CH}(S)$ . Then from the lemma, $\Phi _* \gamma = 0$ implies $\theta _{n-m} = \theta _{n-m+1} = \dotsb = \theta _{n} = 0$ . On the other hand, $i^* i_* \gamma = c_{m+1}\left (\mathscr {N}_{P/X}\right ) \cap \gamma = c_{m+1}(F^{\prime }\otimes \mathscr {O}_P(-1)) \cap \gamma = 0$ , and

$$ \begin{align*} c_{m+1}(F^{\prime}\otimes \mathscr{O}_P(-1)) & = \sum_{i=0}^{m+1} c_{m+1-i}(F^{\prime}) (-\zeta)^i \\ &= (-1)^{m+1} \zeta^{m+1} + (\text{lower-order terms}). \end{align*} $$

Hence by the uniqueness of an expression of the form $\sum _{i=0}^{n} \zeta ^i \cdot \psi ^*(\underline {\hphantom {A}})$ , one can inductively show $\theta _{n-m-1} = 0, \theta _{n-m-2} = 0, \dotsc , \theta _0 = 0$ . Therefore $\gamma =0$ .

Proof of Theorem 3.6(2). To show that the sequence is a complex, simply observe that for any $\gamma ^{\prime } \in \operatorname {CH}(P)$ ,

$$ \begin{align*} i_*\Phi^*\gamma^{\prime} + \Gamma^*\left(-i^{\prime}_* \gamma^{\prime}\right) &= i_*p_*(c_n(\mathscr{V}^{\prime})\cap p^{\prime*}\gamma^{\prime}) - \Gamma^*\,i^{\prime}_*\, \gamma^{\prime} \\ &= \pi_*j_*(c_n(\mathscr{V}^{\prime})\cap p^{\prime*}\gamma^{\prime}) - \Gamma^*i^{\prime}_* \gamma^{\prime} \\ & \overset{\text{(k.f.)}}{=} \pi_*\pi^{\prime*}i^{\prime}_* \gamma^{\prime} - \pi_*\pi^{\prime*}i^{\prime}_* \gamma^{\prime} = 0. \end{align*} $$

(Here and later, ‘(k.f.)’ means the key formula (2.2) for blowup.)

For any $\alpha \in \operatorname {CH}_k(X)$ , $\alpha - \Gamma ^* \Gamma _* \alpha = 0$ on $\operatorname {CH}_k(X \backslash P)$ . Then from the exact sequence $\operatorname {CH}_k(P) \to \operatorname {CH}_k(X) \to \operatorname {CH}_k(X\backslash P) \to 0$ , there exists an element $\gamma \in \operatorname {CH}_k(P)$ such that $\alpha = \Gamma ^* \Gamma _* \alpha + i_* \gamma $ . This establishes the surjectivity of the last map of the sequence of Theorem 3.6(2). The injectivity of the first map and the left inverse statement follow directly from $\Phi _* \Phi _* = \operatorname {Id}$ . To show that the sequence is exactness in the middle, assume $(\gamma , \alpha ^{\prime }) \in \operatorname {CH}(P)\oplus \operatorname {CH}(X^{\prime })$ such that $i_*\gamma + \Gamma ^* \alpha ^{\prime }=0$ ; we want to find $\gamma ^{\prime }$ such that $(\gamma , \alpha ^{\prime }) = \left (\Phi ^* \gamma ^{\prime }, -i^{\prime }_* \gamma \right )$ , since

$$ \begin{align*} \alpha^{\prime} & = \Gamma_* \Gamma^* \alpha^{\prime} = - \Gamma_* i_* \gamma = - \pi^{\prime}_* \pi^* i_* \gamma \overset{\text{(k.f.)}}{=} -\pi^{\prime}_* j_* (c_{m}(\mathscr{V}) \cap p^*\gamma) \\ & = -i^{\prime}_* p^{\prime}_*(c_{m}(\mathscr{V}) \cap p^* \gamma) = -i^{\prime}_* \Phi_* (\gamma). \end{align*} $$

Define $\gamma _0 = \gamma - \Phi ^* \Phi _* \gamma .$ The goal is to show $\gamma _0 =0$ . Notice that

$$ \begin{align*} \Phi_* \gamma_0 &= \Phi_* \gamma - \Phi_* \Phi^* \Phi_* \gamma = 0 \end{align*} $$

and

$$ \begin{align*} i_*\gamma_0 & = i_* \gamma - i_* \Phi^* \Phi_* \gamma = i_* \gamma - i_* p_* (c_n{\mathscr{V}^{\prime}} \cap p^{\prime*} (\Phi^* \gamma)) \\ & = i_* \gamma - \pi_* j_*(c_n{\mathscr{V}^{\prime}} \cap p^{\prime*} ( \Phi_* \gamma)) \overset{\text{(k.f.)}}{=} i_* \gamma - \pi_* \pi^{\prime*} i^{\prime}_* \Phi_* \gamma \\ & = i_* \gamma - \Gamma^* \left(i^{\prime}_* \Phi_* \gamma\right) = i_* \gamma + \Gamma^* \alpha^{\prime} = 0. \end{align*} $$

From Lemma 3.9, $\gamma _0 = 0$ , hence $(\gamma , \alpha ^{\prime }) = \left (\Phi ^* \gamma ^{\prime }, -i^{\prime }_* \gamma \right )$ for $\gamma ^{\prime } = \Phi _* \, \gamma $ .□

Proof of Theorem 3.6(3). As before, from Lemma 3.8(3) and the exact sequence of part (2), we obtain that for any $\alpha \in \operatorname {CH}_{k}(X)$ , there exist an $\alpha ^{\prime } \in \operatorname {CH}_{k}(X)$ and a unique $\gamma \in \operatorname {CH}_k(P)$ such that $\Phi _* \gamma = 0$ and $\alpha = \Gamma ^* \alpha ^{\prime } + i_* \gamma $ . Further notice that

$$ \begin{align*} \Gamma_* i_* \gamma = \pi^{\prime}_*\pi^*i_*\gamma \overset{\text{(k.f.)}}{=} \pi^{\prime}_*j_*(c_n(\mathscr{V}) \cap p^*\gamma) = i^{\prime}_*p^{\prime}_* j_*(c_n(\mathscr{V}) \cap p^*\gamma) = i^{\prime}_* \Phi_* \gamma = 0. \\[-15pt]\end{align*} $$

Therefore $\alpha ^{\prime } = \Gamma _* \alpha $ . Hence we have established

$$ \begin{align*} \operatorname{CH}_{k}(X) &= \Gamma^* \operatorname{CH}_{k}(X^{\prime}) \oplus i_* \left(\operatorname{CH}(P)_{\Phi_* = 0}\right) \\ & = \Gamma^* \operatorname{CH}_{k}(X^{\prime}) \oplus i_*\left(\bigoplus_{i=0}^{n-m-1} \zeta^i \cdot \psi^* \operatorname{CH}_{k-n+i}(S)\right), \\[-16pt]\end{align*} $$

and the projection to the first summand is given by $\alpha \mapsto \alpha ^{\prime } = \Gamma _* \alpha $ .□

Proof. It suffices to notice that the argument of Theorem 3.1 for these statements depends only on the properties of the normal bundles, and thus still works here. More precisely, we may assume $\mathscr {G} = \operatorname {Coker}\left (\mathscr {F} \xrightarrow {\sigma } \mathscr {E}\right )$ for simplicity; then over Z there exists a line bundle L such that there is an exact sequence of vector bundles

$$ \begin{align*} 0 \to L \to \mathscr{F}\rvert_Z \to \mathscr{E}\rvert_Z \to i^* \mathscr{G} \to 0. \end{align*} $$

Also we have a similar picture as with the Cayley trick (3.1):

Denote $G_Z: = i^* \mathscr {G}$ , which is a vector bundle on Z. Then it is easy to compute that the normal bundles are $\mathscr {N}_i = L^\vee \otimes G_Z$ , $\mathscr {N}_j = L^\vee \otimes \Omega _{\mathbb {P}\left (G_Z\right )/Z}^1(1)$ , and the excess bundle for the left square is $\mathscr {V} = p^* N_i/N_j \simeq L^\vee \otimes \mathscr {O}_{\mathbb {P}\left (G_Z\right )}(1)$ (see Lemma 4.10 for the more general situation). Therefore $\Gamma _* \Gamma ^* = p_* \left (c_{r}\left (L^\vee \otimes \Omega _{\mathbb {P}\left (G_Z\right )/Z}^1(1)\right ) \cap p^*(\underline {\hphantom {A}})\right ) = p_*(((-1)^r\zeta ^{r} + \text {lower-order terms}) \cap p^*(\underline {\hphantom {A}})) = (-1)^r \operatorname {Id}$ , and the rest of the orthogonal relations follow from Lemma 4.5. Finally, for the last identity it suffices to show the surjectivity of $\Gamma ^* + \sum \pi _i^*$ . For any $\beta \in \operatorname {CH}(\mathbb {P}(\mathscr {G})$ , $\beta ^{\prime } = \beta - \sum \pi _i^* \pi _{i*} \beta $ is supported on $\mathbb {P}_Z(G_Z)$ , and hence can always be expressed in the form $\beta ^{\prime } = j_*\left (p^* \beta _0 + (\zeta -c_1(L)) \sum _{i=0}^{r-1} \zeta ^{i} \cap p^* \beta _{i+1}\right )$ . Therefore $\beta ^{\prime } = \Gamma ^* \beta _0 + \sum _{i=0}^{r-1} \pi _i^* i_* \beta _{i+1}^{\prime }$ , and hence we are done.

Proof. The following arguments follow the strategy of Fu and Wang [Reference Fu and Wang16] for stratified Mukai flops, which is itself a generalization of [Reference Lee, Lin and Wang35]’s treatment for standard flops and flips (see also §3.2). For any class $[W] \in \operatorname {CH}_k(\mathbb {P}(\mathscr {K}))$ , by Chow’s moving lemma we may assume that W intersects transversely with $\sum _{i \ge 1} \mathbb {P}(\mathscr {K})_i$ .

First, notice that over the open subset $\mathring {X}:=X\backslash X_{1}$ , $\mathring {\mathbb {P}}(\mathscr {K}) : = \mathbb {P}(\mathscr {K})_{0} \backslash \mathbb {P}(\mathscr {K})_{1} \simeq \mathring {Z}: = X_{0} \backslash X_{1} \xrightarrow {i} \mathring {X}$ is an inclusion of codimension $r+1$ ; $\mathscr {G}$ has constant rank r over $\mathring {X} \backslash \mathring {Z}$ and constant rank $r+1$ over $\mathring {Z}$ ; and $\mathring {\Gamma } \simeq \mathbb {P}(i^*\mathscr {G}) \subset \mathring {\mathbb {P}}(\mathscr {G}) : = \mathbb {P}(\mathscr {G})_{0} \backslash \mathbb {P}(\mathscr {G})_{1}$ . Therefore we are in the situation of the variant of the Cayley trick (Lemma 4.6). Therefore if we set $\mathring {W} = W \cap \mathring {X}$ , then the cycle $r_+^* r_{+\,*} r_{-}^* \left [\mathring {W}\right ]$ is represented by a k-cycle $\mathring {\widetilde {W}}$ which maps generically one-to-one to a k-cycle which is rationally equivalent to $\mathring {W}$ , and $r_*\left [\mathring {\widetilde {W}}\right ] = \Gamma _* \Gamma ^* \left [\mathring {W}\right ] =(-1)^r \left [\mathring {W}\right ]$ .

Now back to the whole space. If we let $\widetilde {W}$ be the closure of $\mathring {\widetilde {W}}$ in $\Gamma $ , then

$$ \begin{align*} r_+^* r_{+\,*} r_{-}^* [W] = \left[\widetilde{W}\right] + \sum_{C} a_C [F_{C}], \end{align*} $$

where $\widetilde {W}$ is the k-dimensional cycle as before, mapping generically one-to-one to a k-cycle that is rationally equivalent to $(-1)^rW$ , $a_C \in \mathbb {Z}$ , and $F_{C}$ are k-dimensional irreducible schemes supported over $\pi ^{\prime }\left (W \cap \sum _{i \ge 1} \mathbb {P}(\mathscr {K})_i\right )$ . More precisely, let $C^{\prime }$ be irreducible components of $\pi ^{-1} {\pi ^{\prime }} \left (W \cap \sum _{i \ge 1} \mathbb {P}(\mathscr {K})_i\right )$ ; then the fiber $F_C$ runs through the components $\left \{C= \pi (C^{\prime }) \subset \pi ^{\prime }\left (W \cap \sum _{i \ge 1} \mathbb {P}(\mathscr {K})_i\right )\right \}$ ; here different $C^{\prime }$ s may have the same image C. For any $F_C$ , take the largest i such that there is a component $D \subset \mathbb {P}(\mathscr {K})_i$ with $B_C :=\pi r_+(F_C)= \pi ^{\prime } r_-(F_C) \subset \pi ^{\prime } (W \cap D)$ . For a general $s \in B_C$ , the fiber $F_{C,s} \subset \Gamma _{s} \simeq \mathbb {P}^i_{\kappa (s)} \times _{\kappa (s)} \mathbb {P}^{r+i}_{\kappa (s)}$ over s has dimension

$$ \begin{align*} \dim F_{C,s} &\ge \dim F_{C} - \dim (B_C) \ge \dim F_{C} - \dim r_-(F_C) \\ &\ge \dim F_{C} - \dim (W \cap D) = k -(k - \operatorname{codim}(\mathbb{P}(\mathscr{K})_i \subset \mathbb{P}(\mathscr{K})) ) \\ &= \operatorname{codim}(\mathbb{P}(\mathscr{K})_i \subset \mathbb{P}(\mathscr{K})) \ge i+1. \end{align*} $$

But since the general fiber of $\pi ^{\prime }$ over s has dimension i, then $F_{C,s}$ contains positive-dimension fibers of $r_-$ . Therefore $r_{-*} [F_C] = 0$ , and hence

$$ \begin{align*} \Gamma_* \Gamma^* [W] = r_{-*} \left(\left[\widetilde{W}\right] + \sum_{C} a_C [F_{C}]\right) = r_{-*} \left[\widetilde{W}\right] = (-1)^r[W]. \\[-40pt]\end{align*} $$

Proof. Set $[V] \in \operatorname {CH}_k(\mathbb {P}(\mathscr {G}))_{\text {tor.}}$ – that is, $[V] \in \operatorname {CH}_k(\mathbb {P}(\mathscr {G}))$ such that $\pi _{i*} [V] = 0$ for all $i \in [0,r-1]$ . By the moving lemma we may assume that V intersects transversely with $\sum _{i \ge 1} \mathbb {P}(\mathscr {G})_i$ . Similar to the proof of Lemma 4.8, by the variant of the Cayley trick (Lemma 4.6), $\Gamma ^* \Gamma _* \left [\mathring {V}\right ] = (-1)^r \left [\mathring {V}\right ]$ over $\mathring {X}:=X\backslash X_{1}$ , where $\mathring {V} = V \cap \mathring {X}$ and $[V_0] \in \mathbb {P}(\mathscr {G})_{\text {tor.}}$ . Therefore there exists $\mathring {W}$ representing $\Gamma _* \left [\mathring {V}\right ] \in \operatorname {CH}_{k-r}(\mathbb {P}(\mathscr {K}))$ such that $r_{-}^{-1}\left (\mathring {W}\right )$ is a k-dimensional cycle and $r_{+*} \left (r_{-}^{-1}\left (\mathring {W}\right )\right )$ , though supported on $\mathbb {P}(\mathscr {G})_1$ , is rationally equivalent to $(-1)^r\mathring {V}$ in $\mathbb {P}(\mathscr {G})$ .

Therefore over the whole space, we have

$$ \begin{align*} r_-^* r_{-*} r_{+}^* [V] = \left[\widetilde{V}\right] + \sum_{C} a_C [F_{C}], \end{align*} $$

where $\widetilde {V}$ is the closure of $r_{-}^{-1}\left (\mathring {W}\right )$ in $\Gamma $ , and hence $r_{+*} \widetilde {V}$ is rationally equivalent to $(-1)^rV$ , $a_C \in \mathbb {Z}$ , and $F_{C}$ are irreducible k-dimensional cycles supported over $\pi \left (V \cap \sum _{i \ge 1} \mathbb {P}(\mathscr {G})_i\right )$ . Similarly as before, for any $F_C$ , take the largest $i \ge 1$ such that there is a component $D \subset \mathbb {P}(\mathscr {G})_i$ with $B_C :=\pi r_+(F_C)= \pi ^{\prime } r_-(F_C) \subset \pi (V \cap D)$ . For a general $s \in B_C$ , the fiber $F_{C,s}$ has dimension

$$ \begin{align*} \dim F_{C,s} &\ge \dim F_{C} - \dim (B_C) \ge \dim F_{C} - \dim r_-(F_C) \\ &\ge \dim F_{C} - \dim (V \cap D) = \operatorname{codim}(\mathbb{P}(\mathscr{G})_i \subset \mathbb{P}(\mathscr{G})) \ge r+i+1. \end{align*} $$

Now since the general fiber of $\pi $ over s has dimension $r+i$ , then $F_{C,s}$ contains positive-dimension fibers of $r_+$ . Therefore $r_{+*} [F_C] = 0$ , and

$$ \begin{align*} \Gamma^* \Gamma_* [V] = r_{+*} \left(\left[\widetilde{V}\right] + \sum_{C} a_C [F_{C}]\right) = r_{+*} \left[\widetilde{V}\right] =(-1)^r [V]. \\[-43pt]\end{align*} $$

Proof of Theorem 4.1 under condition (A). The injectivity of map (4.2) follows directly from Lemmas 4.4 and 4.8; the surjectivity follows from Lemmas 4.4 and 4.9. This completes the proof.

Proof. As the statements are local, we may assume $\mathscr {G} = \operatorname {Coker} \left (F \xrightarrow {\sigma } E\right )$ , where $E,F$ are vector bundles of rank n and m. Then by our assumption on Z and Lemma 2.3, $Z \subset X$ is a closed locally complete intersection subscheme, and $\mathscr {N}_i = G_Z \otimes K_Z$ . Moreover, the image $\operatorname {im} (\sigma \rvert _Z) \subset E\rvert _Z$ is a vector sub-bundle; let us denote it by $B_Z$ . Therefore the map $\sigma \rvert _Z$ induces two short exact sequences of vector bundles over Z:

$$ \begin{align*} 0 \to K_Z^\vee \to F\rvert_Z \to B_Z \to 0, \qquad 0 \to B_Z \to E\rvert_Z \to G_Z \to 0. \end{align*} $$

Next, over $\mathbb {P}(\mathscr {G}) \subset \mathbb {P}(E)$ , the composition $\pi ^*F \xrightarrow {\pi ^*\sigma } \pi ^*E \to \mathscr {O}_{\mathbb {P}(E)}(1)$ is zero, hence $\pi ^*\sigma $ factors through a map between vector bundles $\widetilde {\sigma } \colon \pi ^* F \to \Omega _{\mathbb {P}(E)/X}^1(1)$ . The rank of $\widetilde {\sigma }$ at a point $p \in \mathbb {P}(\mathscr {G})$ agrees with the rank of $\sigma $ at $\pi (p)$ , so $\widetilde {Z}: =\pi ^{-1}(Z) = \mathbb {P}(G_Z)$ is the bottom degeneracy locus of $\widetilde {\sigma }$ . We claim that there is an exact sequence of vector bundles

$$ \begin{align*} 0 \to \pi_Z^* K_Z^\vee \to \pi_Z^* F_Z \xrightarrow{\widetilde{\sigma} }\Omega_{\mathbb{P}(E)/X}^1(1)\rvert_{\widetilde{Z}} \to \Omega_{\mathbb{P}\left(G_Z\right)/Z}^1(1) \to 0. \end{align*} $$

Then by Lemma 2.3, $\mathscr {N}_j =\Omega ^1_{\mathbb {P}\left (G_Z\right )/Z}(1) \boxtimes K_Z$ and $\mathscr {V} = \pi _Z^* \mathscr {N}_i/ \mathscr {N}_j \simeq \mathscr {O}_{\mathbb {P}\left (G_Z\right )}(1) \boxtimes K_Z.$

To prove the claim, it suffices to notice that over $\widetilde {Z}$ there is a commutative diagram

The three columns and the last two rows are exact, hence the first row is a short exact sequence. Combining with the short exact sequence of vector bundles $0 \to \pi _Z^*\left (K_Z^\vee \right ) \to \pi _Z^*(F\rvert _Z) \to \pi _Z^*(B_Z) \to 0$ , the claim follows. Notice that we do not use the condition $n \ge m$ , so the same argument works for all the other cases.

Proof. For the first two statements, notice that if we write $F = G_Z^\vee $ , $F^{\prime }=K_Z^\vee $ , with rank $n=r+i$ and $m=i$ , and $S= Z$ , then $P=\mathbb {P}(G_Z)$ , $P^{\prime }=\mathbb {P}(K_Z)$ , and $E = \Gamma _Z$ , and we are in a very similar situation as the case of standard flips (§3.2). In fact, for part (1), using the notation of the proof of Theorem 3.6, $\Psi _*$ and $\Psi ^*$ correspond to the correspondences given by $(-1)^{n}c_{n}(\mathscr {V}^{\prime })$ and $(-1)^m c_{m}(\mathscr {V})$ , respectively (instead of $c_{m}(\mathscr {V})$ for $\Phi _*$ and $c_{n}(\mathscr {V}^{\prime })$ for $\Phi ^*$ ). However, by the commutativity of the intersection product, the composition $c_{n}(\mathscr {V}^{\prime }) * c_{m}(\mathscr {V})$ is still computed by the same formula as $c_{m}(\mathscr {V}) * c_{n}(\mathscr {V}^{\prime })$ , with the role of the first and third factors of the product $P^{\prime }\times _S P \times _S P^{\prime }$ switched. Hence $c_{n}(\mathscr {V}^{\prime }) * c_{m}(\mathscr {V}) = [\Delta _{P^{\prime }}]$ , and $\Psi _* \Psi ^* = (-1)^{m+n} \operatorname {Id} = (-1)^r \operatorname {Id}$ .

For part (2), the same argument as in Lemma 3.8 works. In fact, the image of $\Psi ^*$ is the ‘sub- $\operatorname {CH}(Z)$ -module’ generated by $1,\zeta ,\dotsc ,\zeta ^i$ . Hence up to elements of $\operatorname {Im} \Psi ^* = \operatorname {Span} \left \{ 1, \zeta ^1, \dotsc , \zeta ^i \right \}$ , the map

$$ \begin{align*} \pi_{Z,a}^*(\underline{\hphantom{A}}) = \zeta^a \cdot c_{\text{top}}(\mathscr{V}) \cdot \cap \pi_Z^*(\underline{\hphantom{A}}) = \zeta^a \cdot \left(\zeta^{i+1} + \text{lower-order terms}\right) \cdot \pi_Z^*(\underline{\hphantom{A}}) \end{align*} $$

hits each element of the basis $\left \{\zeta ^{i+a+1} \bmod \operatorname {Im} \Psi ^*\right \}_{a \in \left [0,r-1\right ]}$ of the quotient $\operatorname {CH}(\mathbb {P}(G_Z))/ \operatorname {Im} \Psi ^*$ . Therefore the result follows.

For part (3), it follows directly from Lemma 4.10 that for any $\gamma \in \operatorname {CH}(\mathbb {P}(K_Z))$ ,

$$ \begin{align*} \Gamma^*k_* \gamma = r_{+*} r_-^*k_* \gamma = r_{+*} \ell_{*} \left(c_{\text{top}}(\mathscr{V}^{\prime}) \cap r_{Z-}^* \gamma\right) = j_* r_{Z+*} \left(c_{\text{top}}(\mathscr{V}^{\prime}) \cap r_{Z-}^* \gamma\right) = j_* \Psi^* \gamma, \end{align*} $$

and similarly, $\Gamma _* j_* = k_* \Psi _*$ . Also, for any $a \in [0,r-1]$ and $\alpha \in \operatorname {CH}(Z)$ ,

$$ \begin{align*} \pi_a^*(i_* \alpha) = \pi^*(\zeta^a \cdot i_* \alpha) = j_* \left(\zeta^a \cdot c_{\text{top}}(\mathscr{V}) \cap \pi_Z^*(\alpha)\right) = j_* \pi_{Z,a}^*(\alpha). \\[-38pt] \end{align*} $$

Proof of Theorem 4.1 under condition (B). Stratify the space X the same way as in the first approach, namely $X_i:= X^{\ge r+i+1}(\mathscr {G})$ for $i \ge -1$ , and similarly for $\mathbb {P}(\mathscr {G})_i$ , $\mathbb {P}(\mathscr {K})_i$ , and $\Gamma _i$ . For each $i \ge -1$ , we will denote the natural inclusions by $i_{i} \colon X_i \hookrightarrow X$ , $j_{i} \colon \mathbb {P}(\mathscr {G})_i \hookrightarrow \mathbb {P}(\mathscr {G})$ , $k_{i} \colon \mathbb {P}(\mathscr {K})_{i} \hookrightarrow \mathbb {P}(\mathscr {K})$ , and $\ell _{i} \colon \Gamma _i \hookrightarrow \Gamma $ . For $i \ge 0$ , we also denote $i_{i,i-1} \colon X_{i} \hookrightarrow X_{i-1}$ the natural inclusion, and $j_{i,i-1}$ , $k_{i,i-1}$ , and $\ell _{i,i-1}$ are defined similarly. Finally, for each pair $(i,j)$ with $j> i \ge -1$ , denote $X_{i\backslash j} : = X_{i} \backslash X_{j}$ ; $\mathbb {P}(\mathscr {G})_{i\backslash j}$ , $\mathbb {P}(\mathscr {K})_{i\backslash j}$ , and $\Gamma _{i\backslash j}$ are defined in the same manner. By abuse of notation, the inclusion $i_i \colon X_{i\backslash j} \hookrightarrow X \backslash X_j = X_{-1 \backslash j}$ is also denoted by $i_i$ , and similarly for other inclusions.

For any fixed integer $i \ge 0$ , if we assume that condition (B) of Theorem 4.1 is satisfied, then $Z: = X_{i\backslash i+1} \subset X \backslash X_{i+1} = X_{-1 \backslash i+1}$ is a locally complete intersection subscheme of codimension $(i+1)(r+i+1)$ , and $\mathscr {G}$ has constant rank $r+i+1$ over Z. Therefore the conditions of Lemma 4.10 are satisfied by $Z \subset X \backslash X_{i+1}$ and $\mathscr {G}$ , with $\mathbb {P}(G_Z) = \mathbb {P}(\mathscr {G})_{i \backslash i+1}$ , $\mathbb {P}(K_Z) = \mathbb {P}(\mathscr {K})_{i \backslash i+1}$ , and $\Gamma _Z = \Gamma _{i \backslash i+1}$ , as well as $i = i_i$ , $j=j_i$ , $k=k_i$ , and $\ell = \ell _i$ . Hence the results of Lemma 4.11 can be applied.

Now our goal is to show that the isomorphism of Lemma 4.11(2) over each stratum can indeed be integrated into an isomorphism of map (4.2) of Theorem 4.1.

Surjectivity of map (4.2). For each $i \ge -1$ , there is an exact sequence

for which if $i = i_{\max }+1$ , then the middle term is the whole space, where $i_{\max }$ is the largest number such that $X_{i_{\max }} \ne \emptyset $ . (Since X is locally Noetherian of pure dimension, there exist only finitely many strata, and such an $i_{\max }$ always exists.) Therefore inductively we see that $\operatorname {CH}(\mathbb {P}(\mathscr {G}))$ is generated by the images of $j_{i *} \colon \operatorname {CH}\left (\mathbb {P}(\mathscr {G})_{i \backslash i+1}\right ) \to \operatorname {CH}(\mathbb {P}(\mathscr {G}))$ for all strata $\mathbb {P}(\mathscr {G})_{i \backslash i+1}$ , $i \ge -1$ , where $i=-1$ corresponds to the open stratum.

Hence we need only show that the image of map (4.2) contains the image of the strata $\operatorname {CH}\left (\mathbb {P}(\mathscr {G})_{i \backslash i+1}\right )$ in $\operatorname {CH}(\mathbb {P}(\mathscr {G}))$ for each $i \ge -1$ . The open-stratum case $i=-1$ follows from the projective bundle formula. For other cases – that is, $i \ge 0$ – set $Z: = X_{i\backslash i+1} \subset X \backslash X_{i+1}$ as before, and for simplicity denote $j_{*} :=j_{i *} \colon \operatorname {CH}\left (\mathbb {P}(\mathscr {G})_{i \backslash i+1}\right ) \to \operatorname {CH}(\mathbb {P}(\mathscr {G}))$ , which agrees with the notation of Lemmas 4.10 and 4.11, and similarly for the maps $i,k,\ell $ . Then by Lemma 4.11(2), any $\alpha \in \operatorname {CH}\left (\mathbb {P}(\mathscr {G})_{i \backslash i+1}\right ) = \operatorname {CH}(\mathbb {P}(G_Z))$ can be written as $\alpha = \sum _{a=0}^{r-1} \pi _{Z,a}^* \alpha _a + \Psi ^* \gamma $ , for certain $\alpha _a \in \operatorname {CH}(Z)$ and $\gamma \in \mathbb {P}(K_Z) = \mathbb {P}(\mathscr {K})_{i\backslash i+1}$ . Therefore by Lemma 4.11(3),

$$ \begin{align*} j_* (\alpha) = j_*\left(\sum_{a=0}^{r-1} \pi_{Z,a}^* \alpha_a + \Psi^* \gamma\right) = \sum_{a=0}^{r-1} \pi_{a}^* (i_* \alpha_a) + \Gamma^* (k_* \gamma). \\[-15pt]\end{align*} $$

That is, the image of $j_*$ is contained in the image of map (4.2), and hence we are done.

Injectivity of map (4.2). This part is a little tricky; the key observation is that the excision exact sequence becomes a short exact sequence if we take the image of first map. The injectivity of $\pi _a^*$ follows from Lemma 4.4; it remains to show the injectivity of $\Gamma ^*$ . For each $i \ge -1$ , there is a commutative diagram of short exact sequences

where we recall that the maps $k_{i*} $ and $ j_{i*}$ are the inclusions to (an open subset of) the whole space:

$$ \begin{align*} k_{i*} \colon \operatorname{CH}\left(\mathbb{P}(\mathscr{K})_{i \backslash i+1}\right) \to \operatorname{CH}\left(\mathbb{P}(\mathscr{K})_{-1 \backslash i}\right), \qquad j_{i*} \colon \operatorname{CH}\left(\mathbb{P}(\mathscr{G})_{i \backslash i+1}\right) \to \operatorname{CH}\left(\mathbb{P}(\mathscr{G})_{-1 \backslash i+1}\right). \end{align*} $$

We want to show that for each $i \ge 0$ , the map $\Gamma ^*\rvert _{\operatorname {Im} k_{i*}}$ is injective. Set $Z: = X_{i\backslash i+1} \subset X \backslash X_{i+1}$ as before; then the question reduces to showing that in the commutative diagram

(which is commutative by Lemma 4.11(3)), the injection $\Psi ^*$ induces an injection $\Gamma ^*$ on the image. In fact, for any $\gamma \in \operatorname {CH}\left (\mathbb {P}(\mathscr {K})_{i\backslash i+1}\right )$ , if $\Gamma ^* k_{i *} \gamma = j_{i*} \Psi ^* \gamma = 0$ , then by Lemma 4.11(1), we have $\gamma = (-1)^r \Psi _* \Psi ^* \gamma $ . Therefore by Lemma 4.11(3),

$$ \begin{align*} k_{i*} \gamma = (-1)^r k_{i*} \Psi_* \Psi^* \gamma = (-1)^r \Gamma_* j_{i*} \Psi^* \gamma = 0. \end{align*} $$

Hence $\Gamma ^*\rvert _{\operatorname {Im} k_{i*}}$ is injective. Now by induction, starting with the case $i=0$ , when the injectivity of $\Gamma ^*\rvert _{-1 \backslash 1}$ follows from the commutative diagram

we can inductively show that $\Gamma ^*\rvert _{-1 \backslash i}$ is injective for all $i=0,1, 2, \dotsc , i_{\max }, i_{\max } +1$ , where $i_{\max }$ is the largest number such that $X_{i_{\max }} \ne \emptyset $ . Therefore $\Gamma ^* = \Gamma ^*\rvert _{-1 \backslash i_{\max }+1}$ is injective on the whole space. Notice that from the preceding argument, we also obtain $\Gamma _* \Gamma ^* = (-1)^r \operatorname {Id}$ , since it is true on the image of each stratum. Together with Lemmas 4.4 and 4.5, this completes the proof of Theorem 4.1.□