The Stacks project

Lemma 45.12.5. In the situation above let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. If $\psi ^2$ is as in Chow Homology, Lemma 42.56.1 and $c^ A$ and $ch^ A$ are as in Propositions 45.12.1 and 45.12.4 then we have $c^ A_ i(\psi ^2(\alpha )) = 2^ i c^ A_ i(\alpha )$ and $ch^ A_ i(\psi ^2(\alpha )) = 2^ i ch^ A_ i(\alpha )$ 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 $r$. 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 45.12.2. 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 $x_ j = c^ A_1(\mathcal{L}_ j)$ we have

\[ c^ A(\alpha ) = \prod (1 + x_ j) \quad \text{and}\quad c^ A(\psi ^2(\alpha )) = \prod (1 + 2 x_ j) \]

in $\prod A^ i(P)$ and we have

\[ ch^ A(\alpha ) = \sum \exp (x_ j) \quad \text{and}\quad ch^ A(\psi ^2(\alpha )) = \sum \exp (2 x_ j) \]

in $\prod A^ i(P)$. From these formulas the desired result follows. $\square$

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