Virtual Classes in Algebraic Cobordism

Virtual fundamental classes admit a natural description as orientation or Gysin maps for quasi-smooth morphisms of derived schemes. We show that Algebraic Bordism as developed by Levine and Morel admits such orientations, and thus there exist fundamental classes in Algebraic Bordism. We then present a theory of derived algebraic bordism, which uses quasi-smooth derived schemes as
generators instead of only smooth schemes. This is the universal homology theory having orientations for quasi-smooth morphisms. Our main result is that in characteristic zero, derived algebraic bordism and algebraic bordism are equivalent.