The Stacks project

Remark 42.38.10. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. 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 $r$. By Lemma 42.35.4 we have $A^ p(X) = \prod A^ p(X_ r)$. Hence we can define $c_ p(\mathcal{E})$ to be the product of the classes $c_ p(\mathcal{E}|_{X_ r})$ in $A^ p(X_ r)$. Explicitly, if $X' \to X$ is a morphism locally of finite type, 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 $c_ p(\mathcal{E}) \in A^ p(X)$ is 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 $c_ p(\mathcal{E})$ is nonzero for infinitely many $p$. It follows that the total chern class is an element

\[ c(\mathcal{E}) = c_0(\mathcal{E}) + c_1(\mathcal{E}) + c_2(\mathcal{E}) + \ldots \in A^*(X)^\wedge \]

of the completed bivariant cohomology ring, see Remark 42.35.5. In this setting we define the “rank” of $\mathcal{E}$ to be the element $r(\mathcal{E}) \in A^0(X)$ as the bivariant operation which sends $(\alpha _ r) \in \prod \mathop{\mathrm{CH}}\nolimits _*(X'_ r)$ to $(r\alpha _ r) \in \prod \mathop{\mathrm{CH}}\nolimits _*(X'_ r)$. Note that it is still true that $c_ p(\mathcal{E})$ and $r(\mathcal{E})$ are in the center of $A^*(X)$.

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