The Stacks project

Lemma 42.52.5. 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 zero, 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.

Then we have

\[ P_ p(Z \to X, E_2) = P_ p(Z \to X, E_1) + P_ p(Z \to X, E_3) \]

for all $p \in \mathbf{Z}$ and consequently $ch(Z \to X, E_2) = ch(Z \to X, E_1) + ch(Z \to X, E_3)$.

Proof. The proof is exactly the same as the proof of Lemma 42.52.4 except it uses Lemma 42.49.8 at the very end. For $p > 0$ we can deduce this lemma from Lemma 42.52.4 with $p_1 = p_3 = 1$ and the relationship between $P_ p(Z \to X, E)$ and $c_ p(Z \to X, E)$ given in Lemma 42.52.1. The case $p = 0$ can be shown directly (it is only interesting if $X$ has a connected component entirely contained in $Z$). $\square$

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