The Stacks project

Remark 82.28.5. In Situation 82.2.1 let $X/B$ be good. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. If the rank of $\mathcal{E}$ is not constant then we can still define the Chern classes of $\mathcal{E}$. Namely, in this case we can write

\[ X = X_0 \amalg X_1 \amalg X_2 \amalg \ldots \]

where $X_ r \subset X$ is the open and closed subspace where the rank of $\mathcal{E}$ is $r$. If $X' \to X$ is a morphism of good algebraic spaces over $B$, then we obtain by pullback a corresponding decomposition of $X'$ and we find that

\[ \mathop{\mathrm{CH}}\nolimits _*(X') = \prod \nolimits _{r \geq 0} \mathop{\mathrm{CH}}\nolimits _*(X'_ r) \]

by our definitions. Then we simply define $c_ i(\mathcal{E})$ to be the bivariant class which preserves these direct product decompositions and acts by the already defined operations $c_ i(\mathcal{E}|_{X_ r}) \cap -$ on the factors. Observe that in this setting it may happen that $c_ i(\mathcal{E})$ is nonzero for infinitely many $i$.

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