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I know the famous principle (or theorem depending on how you define) of Chern classes for locally free sheaves and a morphism $f: X'\rightarrow X$,

$ch(f^* \epsilon)=f^* ch(\epsilon)$.

But if $f$ is not flat, in principle we must have higher derived pullbacks $L f^*$. So I'm wondering how to compute Chern classes/characters of the complex $Lf^*\epsilon$,

$ch(L f^*\epsilon)=?$

I can imagine if we use Grothendieck-Riemann-Roch, we may be able to say something about $f_*ch(L f^*\epsilon)$, but this is not enough.

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    $\begingroup$ Precisely what are you asking? For a locally free $\mathcal{O}_X$-module of finite rank, the derived pullback by $f$ of the $\mathcal{O}_X$-module is quasi-isomorphic to the complex concentrated in degree $0$ that equals the usual pullback of the $\mathcal{O}_X$-module. $\endgroup$
    – Jason Starr
    Commented May 21, 2018 at 17:53
  • $\begingroup$ I didn't know what you said :) Is it possible to find the concentrated complex, at least it's Chern classes? $\endgroup$
    – Mohsen Karkheiran
    Commented May 21, 2018 at 17:57

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