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
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
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
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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