  We construct Chern classes of vector bundles in the universal precobordism
theory of Annala--Yokura over an arbitrary Noetherian base ring of finite Krull
dimension. As an immediate corollary of this, we show that the Grothendieck
ring of vector bundles can be recovered from the universal precobordism ring,
and that we can construct candidates for Chow rings satisfying an analogue of
the classical Grothendieck--Riemann--Roch theorem. We also strengthen the weak
projective bundle formula of Annala--Yokura to work for arbitrary projective
bundles.
