grothendieck took the perspective that the riemann–roch theorem is a statement about morphisms of varieties, not the varieties themselves. he proved that there is a homomorphism from formula_1 to the chow groups of "x" coming from the chern character and todd class of "x". additionally, he proved that a proper morphism formula_7 to a smooth variety "y" determines a homomorphism formula_8 called the "pushforward". this gives two ways of determining an element in the chow group of "y" from a vector bundle on "x": starting from "x", one can first compute the pushforward in "k"-theory and then apply the chern character and todd class of "y", or one can first apply the chern character and todd class of "x" and then compute the pushforward for chow groups. the grothendieck–riemann–roch theorem says that these are equal. when "y" is a point, a vector bundle is a vector space, the class of a vector space is its dimension, and the grothendieck–riemann–roch theorem specializes to hirzebruch's theorem.