The Stacks project

Lemma 42.46.5. 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 the Chern classes of $E$ are defined. For $g : W \to X$ locally of finite type with $W$ integral, there exists a commutative diagram

\[ \xymatrix{ W' \ar[rd]_{g'} \ar[rr]_ b & & W \ar[ld]^ g \\ & X } \]

with $W'$ integral and $b : W' \to W$ proper birational such that $L(g')^*E$ is represented by a bounded complex $\mathcal{E}^\bullet $ of locally free $\mathcal{O}_{W'}$-modules of constant rank and we have $res(c_ p(E)) = c_ p(\mathcal{E}^\bullet )$ in $A^ p(W')$.

Proof. Choose an envelope $f : Y \to X$ 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. The base change $Y \times _ X W \to W$ of $f$ is an envelope by Lemma 42.22.3. Choose a point $\xi \in Y \times _ X W$ mapping to the generic point of $W$ with the same residue field. Consider the integral closed subscheme $W' \subset Y \times _ X W$ with generic point $\xi $. The restriction of the projection $Y \times _ X W \to W$ to $W'$ is a proper birational morphism $b : W' \to W$. Set $g' = g \circ b$. Finally, consider the pullback $(W' \to Y)^*\mathcal{E}^\bullet $. This is a locally bounded complex of finite locally free modules on $W'$. Since $W'$ is integral it follows that it is bounded and that the terms have constant rank. Finally, by construction $(W' \to Y)^*\mathcal{E}^\bullet $ represents $L(g')^*E$ and by construction its $p$th chern class gives the restriction of $c_ p(E)$ by $W' \to X$. This finishes the proof. $\square$

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