The Stacks project

Lemma 42.46.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $E \in D(\mathcal{O}_ X)$ be a perfect object. Assume there exists an envelope $f : Y \to X$ (Definition 42.22.1) such that $Lf^*E$ is isomorphic in $D(\mathcal{O}_ Y)$ to a locally bounded complex $\mathcal{E}^\bullet $ of finite locally free $\mathcal{O}_ Y$-modules. Then there exists unique bivariant classes $c(E) \in \prod _{p \geq 0} A^ p(X)$, $ch(E) \in \prod _{p \geq 0} A^ p(X) \otimes \mathbf{Q}$, and $P_ p(E) \in A^ p(X)$, independent of the choice of $f : Y \to X$ and $\mathcal{E}^\bullet $, such that the restriction of these classes to $Y$ are equal to $c(\mathcal{E}^\bullet )$, $ch(\mathcal{E}^\bullet )$, and $P_ p(\mathcal{E}^\bullet )$.

Proof. Fix $p \in \mathbf{Z}$. We will prove the lemma for the chern class $c_ p(E) \in A^ p(X)$ and omit the arguments for the other cases.

Let $g : T \to X$ be a morphism locally of finite type such that there exists a locally bounded complex $\mathcal{E}^\bullet $ of finite locally free $\mathcal{O}_ T$-modules representing $Lg^*E$ in $D(\mathcal{O}_ T)$. The bivariant class $c_ p(\mathcal{E}^\bullet ) \in A^ p(T)$ is independent of the choice of $\mathcal{E}^\bullet $ by Lemma 42.46.1. Let us write $c_ p(Lg^*E) \in A^ p(T)$ for this class. For any further morphism $h : T' \to T$ which is locally of finite type, setting $g' = g \circ h$ we see that $L(g')^*E = L(g \circ h)^*E = Lh^*Lg^*E$ is represented by $h^*\mathcal{E}^\bullet $ in $D(\mathcal{O}_{T'})$. We conclude that $c_ p(L(g')^*E)$ makes sense and is equal to the restriction (Remark 42.33.5) of $c_ p(Lg^*E)$ to $T'$ (strictly speaking this requires an application of Lemma 42.38.7).

Let $f : Y \to X$ and $\mathcal{E}^\bullet $ be as in the statement of the lemma. We obtain a bivariant class $c_ p(E) \in A^ p(X)$ from an application of Lemma 42.35.6 to $f : Y \to X$ and the class $c' = c_ p(Lf^*E)$ we constructed in the previous paragraph. The assumption in the lemma is satisfied because by the discussion in the previous paragraph we have $res_1(c') = c_ p(Lg^*E) = res_2(c')$ where $g = f \circ p = f \circ q : Y \times _ X Y \to X$.

Finally, suppose that $f' : Y' \to X$ is a second envelope such that $L(f')^*E$ is represented by a bounded complex of finite locally free $\mathcal{O}_{Y'}$-modules. Then it follows that the restrictions of $c_ p(Lf^*E)$ and $c_ p(L(f')^*E)$ to $Y \times _ X Y'$ are equal. Since $Y \times _ X Y' \to X$ is an envelope (Lemmas 42.22.3 and 42.22.2), we see that our two candidates for $c_ p(E)$ agree by the unicity in Lemma 42.35.6. $\square$

Comments (2)

Comment #7944 by 11k on

I think and be replaced by and . The same changes apply to that on .

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