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I am looking for a reference on splitting principle for etale cohomology of simplicial schemes (over arbitrary field k). I found a paper by Schechtman, "On the delooping of chern character and Adams operations", where chern classes are introduced in Grothendieck style, for a given cohomology theory, and the projective bundle formula is stated as a required axiom for the cohomology theory. It will be appreciated if someone can direct me to a reference where the same is explained, possibly with more details.

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    $\begingroup$ Have you looked through arxiv.org/pdf/1007.1385.pdf? In particular, see pdf 37/144 beginning with Section 2.2 (Chern classes of algebraic bundles). $\endgroup$
    – Benjamin Dickman
    Commented May 23, 2014 at 3:39

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