Grothendieck-Riemann-Roch for derived schemes
  We define bivariant algebraic K-theory and bivariant derived Chow on the homotopy category of derived schemes over a smooth base. The orientation on the latter corresponds to virtual Gysin homomorphisms. We then provide a morphism between these two bivariant theories and compare the two orientations. This comparison then yields a homological and cohomological Grothendieck-Riemann-Roch formula for virtual classes. 
Parker Lowrey, Timo Sch\"urg
math.AG math.AT math.KT