The Stacks project

Lemma 42.52.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $Z \to X$ be a closed immersion. Let

\[ E_1 \to E_2 \to E_3 \to E_1[1] \]

be a distinguished triangle of perfect objects in $D(\mathcal{O}_ X)$. Assume

  1. the restrictions $E_1|_{X \setminus Z}$ and $E_3|_{X \setminus Z}$ are isomorphic to finite locally free $\mathcal{O}_{X \setminus Z}$-modules of rank $< p_1$ and $< p_3$ placed in degree $0$, and

  2. at least one of the following is true: (a) $X$ is quasi-compact, (b) $X$ has quasi-compact irreducible components, (c) $E_3 \to E_1[1]$ can be represented by a map of locally bounded complexes of finite locally free $\mathcal{O}_ X$-modules, or (d) there exists an envelope $f : Y \to X$ such that $Lf^*E_3 \to Lf^*E_1[1]$ can be represented by a map of locally bounded complexes of finite locally free $\mathcal{O}_ Y$-modules.

With notation as in Remark 42.52.3 we have

\[ c^{(p_1 + p_3)}(Z \to X, E_2) = c^{(p_1)}(Z \to X, E_1)c^{(p_3)}(Z \to X, E_3) \]

in $A^{(p_1 + p_3)}(Z \to X)$.

Proof. Observe that the assumptions imply that $E_2|_{X \setminus Z}$ is zero, resp. isomorphic to a finite locally free $\mathcal{O}_{X \setminus Z}$-module of rank $< p_1 + p_3$. Thus the statement makes sense.

Let $f : Y \to X$ be an envelope. Expaing the left and right hand sides of the formula in the statement of the lemma we see that we have to prove some equalities of classes in $A^*(X)$ and in $A^*(Z \to X)$. By the uniqueness in Lemma 42.35.6 it suffices to prove the corresponding relations in $A^*(Y)$ and $A^*(Z \to Y)$. Since moreover the construction of the classes involved is compatible with base change (Lemma 42.50.4) we may replace $X$ by $Y$ and the distinguished triangle by its pullback.

In the proof of Lemma 42.46.7 we have seen that conditions (2)(a), (2)(b), and (2)(c) imply condition (2)(d). Combined with the discussion in the previous paragraph we reduce to the case discussed in the next paragraph.

Let $\varphi ^\bullet : \mathcal{E}_3^\bullet [-1] \to \mathcal{E}_1^\bullet $ be a map of locally bounded complexes of finite locally free $\mathcal{O}_ X$-modules representing the map $E_3[-1] \to E_1$ in the derived category. Consider the scheme $X' = \mathbf{A}^1 \times X$ with projection $g : X' \to X$. Let $Z' = g^{-1}(Z) = \mathbf{A}^1 \times Z$. Denote $t$ the coordinate on $\mathbf{A}^1$. Consider the cone $\mathcal{C}^\bullet $ of the map of complexes

\[ t g^*\varphi ^\bullet : g^*\mathcal{E}_3^\bullet [-1] \longrightarrow g^*\mathcal{E}_1^\bullet \]

over $X'$. We obtain a distinguished triangle

\[ g^*\mathcal{E}_1^\bullet \to \mathcal{C}^\bullet \to g^*\mathcal{E}_3^\bullet \to g^*\mathcal{E}_1^\bullet [1] \]

where the first three terms form a termwise split short exact sequence of complexes. Clearly $\mathcal{C}^\bullet $ is a bounded complex of finite locally free $\mathcal{O}_{X'}$-modules whose restriction to $X' \setminus Z'$ is isomorphic to a finite locally free $\mathcal{O}_{X' \setminus Z'}$-module of rank $< p_1 + p_3$ placed in degree $0$. Thus we have the localized Chern classes

\[ c_ p(Z' \to X', \mathcal{C}^\bullet ) \in A^ p(Z' \to X') \]

for $p \geq p_1 + p_3$. For any $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ consider

\[ c_ p(Z' \to X', \mathcal{C}^\bullet ) \cap g^*\alpha \in \mathop{\mathrm{CH}}\nolimits _{k + 1 - p}(\mathbf{A}^1 \times X) \]

If we restrict to $t = 0$, then the map $t g^*\varphi ^\bullet $ restricts to zero and $\mathcal{C}^\bullet |_{t = 0}$ is the direct sum of $\mathcal{E}_1^\bullet $ and $\mathcal{E}_3^\bullet $. By compatibility of localized Chern classes with base change (Lemma 42.50.4) we conclude that

\[ i_0^* \circ c^{(p_1 + p_3)}(Z' \to X', \mathcal{C}^\bullet ) \circ g^* = c^{(p_1 + p_2)}(Z \to X, E_1 \oplus E_3) \]

in $A^{(p_1 + p_3)}(Z \to X)$. On the other hand, if we restrict to $t = 1$, then the map $t g^*\varphi ^\bullet $ restricts to $\varphi $ and $\mathcal{C}^\bullet |_{t = 1}$ is a bounded complex of finite locally free modules representing $E_2$. We conclude that

\[ i_1^* \circ c^{(p_1 + p_3)}(Z' \to X', \mathcal{C}^\bullet ) \circ g^* = c^{(p_1 + p_2)}(Z \to X, E_2) \]

in $A^{(p_1 + p_3)}(Z \to X)$. Since $i_0^* = i_1^*$ by definition of rational equivalence (more precisely this follows from the formulae in Lemma 42.32.4) we conclude that

\[ c^{(p_1 + p_2)}(Z \to X, E_2) = c^{(p_1 + p_2)}(Z \to X, E_1 \oplus E_3) \]

This reduces us to the case discussed in the next paragraph.

Assume $E_2 = E_1 \oplus E_3$ and the triples $(X, Z, E_ i)$ are as in Situation 42.50.1. For $i = 1, 3$ let

\[ b_ i : W_ i \to \mathbf{P}^1_ X \quad \text{and}\quad Q_ i \quad \text{and}\quad T'_ i \subset T_ i \subset W_{i, \infty } \]

be as in the proof of Lemma 42.50.2. By definition

\[ c_ p(Z \to X, E_ i) = c'_ p(Q_ i) \]

where the right hand side is the bivariant class constructed in Lemma 42.49.1 using $W_ i, b_ i, Q_ i, T'_ i$. Set $W = W_1 \times _{b_1, \mathbf{P}^1_ X, b_2} W_2$ and consider the cartesian diagram

\[ \xymatrix{ W \ar[d]_{g_1} \ar[rd]^ b \ar[r]_{g_3} & W_3 \ar[d]^{b_3} \\ W_1 \ar[r]^{b_1} & \mathbf{P}^1_ X } \]

Of course $b^{-1}(\mathbf{A}^1)$ maps isomorphically to $\mathbf{A}^1_ X$. Observe that $T' = g_1^{-1}(T'_1) \cap g_2^{-1}(T'_2)$ still contains all the points of $W_\infty $ lying over $X \setminus Z$. By Lemma 42.49.3 we may use $W$, $b$, $g_ i^*\mathcal{Q}_ i$, and $T'$ to construct $c_ p(Z \to X, E_ i)$ for $i = 1, 3$. Also, by the stronger independence given in Lemma 42.51.2 we may use $W$, $b$, $g_1^*Q_1 \oplus g_3^*Q_3$, and $T'$ to compute the classes $c_ p(Z \to X, E_2)$. Thus the desired equality follows from Lemma 42.49.7. $\square$


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