Lemma 42.56.6. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. If $\psi ^2$ is as in Lemma 42.56.1 and $c$ and $ch$ are as in Remarks 42.56.4 and 42.56.5 then we have $c_ i(\psi ^2(\alpha )) = 2^ i c_ i(\alpha )$ and $ch_ i(\psi ^2(\alpha )) = 2^ i ch_ i(\alpha )$ for all $\alpha \in K_0(\textit{Vect}(X))$.
Proof. Observe that the map $\prod _{i \geq 0} A^ i(X) \to \prod _{i \geq 0} A^ i(X)$ multiplying by $2^ i$ on $A^ i(X)$ is a ring map. Hence, since $\psi ^2$ is also a ring map, it suffices to prove the formulas for additive generators of $K_0(\textit{Vect}(X))$. Thus we may assume $\alpha = [\mathcal{E}]$ for some finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$. By construction of the Chern classes of $\mathcal{E}$ we immediately reduce to the case where $\mathcal{E}$ has constant rank $r$, see Remark 42.38.10. In this case, we can choose a projective smooth morphism $p : P \to X$ such that restriction $A^*(X) \to A^*(P)$ is injective and such that $p^*\mathcal{E}$ has a finite filtration whose graded parts are invertible $\mathcal{O}_ P$-modules $\mathcal{L}_ j$, see Lemma 42.43.1. Then $[p^*\mathcal{E}] = \sum [\mathcal{L}_ j]$ and hence $\psi ^2([p^\mathcal {E}]) = \sum [\mathcal{L}_ j^{\otimes 2}]$ by definition of $\psi ^2$. Setting $x_ j = c_1(\mathcal{L}_ j)$ we have
\[ c(\alpha ) = \prod (1 + x_ j) \quad \text{and}\quad c(\psi ^2(\alpha )) = \prod (1 + 2 x_ j) \]
in $\prod A^ i(P)$ and we have
\[ ch(\alpha ) = \sum \exp (x_ j) \quad \text{and}\quad ch(\psi ^2(\alpha )) = \sum \exp (2 x_ j) \]
in $\prod A^ i(P)$. From these formulas the desired result follows. $\square$
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