The Stacks project

Lemma 42.47.11. In Situation 42.7.1 let $X$ be locally of finite type over $S$. Let

\[ X = X_1 \cup X_2 = X'_1 \cup X'_2 \]

be two ways of writing $X$ as a set theoretic union of closed subschemes. Let $E$, $E'$ be perfect objects of $D(\mathcal{O}_ X)$ whose Chern classes are defined. Assume that $E|_{X_1}$ and $E'|_{X'_1}$ are zero1 for $i = 1, 2$. Denote

  1. $r = P'_0(E) \in A^0(X_2 \to X)$ and $r' = P'_0(E') \in A^0(X'_2 \to X)$,

  2. $\gamma _ p = c'_ p(E|_{X_2}) \in A^ p(X_2 \to X)$ and $\gamma '_ p = c'_ p(E'|_{X'_2}) \in A^ p(X'_2 \to X)$,

  3. $\chi _ p = P'_ p(E|_{X_2}) \in A^ p(X_2 \to X)$ and $\chi '_ p = P'_ p(E'|_{X'_2}) \in A^ p(X'_2 \to X)$

the classes constructed in Lemma 42.47.1. Then we have

\[ c'_1((E \otimes _{\mathcal{O}_ X}^\mathbf {L} E')|_{X_2 \cap X'_2}) = r \gamma '_1 + r' \gamma _1 \]

in $A^1(X_2 \cap X'_2 \to X)$ and

\[ c'_2((E \otimes _{\mathcal{O}_ X}^\mathbf {L} E')|_{X_2 \cap X'_2}) = r \gamma '_2 + r' \gamma _2 + {r \choose 2} (\gamma '_1)^2 + (rr' - 1) \gamma '_1\gamma _1 + {r' \choose 2} \gamma _1^2 \]

in $A^2(X_2 \cap X'_2 \to X)$ and so on for higher Chern classes. Similarly, we have

\[ P'_ p((E \otimes _{\mathcal{O}_ X}^\mathbf {L} E')|_{X_2 \cap X'_2}) = \sum \nolimits _{p_1 + p_2 = p} {p \choose p_1} \chi _{p_1} \chi '_{p_2} \]

in $A^ p(X_2 \cap X'_2 \to X)$.

Proof. First we observe that the statement makes sense. Namely, we have $X = (X_2 \cap X'_2) \cup Y$ where $Y = (X_1 \cap X'_1) \cup (X_1 \cap X'_2) \cup (X_2 \cap X'_1)$ and the object $E \otimes _{\mathcal{O}_ X}^\mathbf {L} E'$ restricts to zero on $Y$. The actual equalities follow from the characterization of our classes in Lemma 42.47.1 and the equalities of Lemma 42.46.11. We omit the details. $\square$

[1] Presumably there is a variant of this lemma where we only assume these restrictions are isomorphic to a finite locally free modules of rank $< p$ and $< p'$.

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