Riemann–Roch type theorems relate Euler characteristics of the cohomology of a vector bundle with their topological degrees, or more generally their characteristic classes in (co)homology or algebraic analogues thereof. The classical Riemann–Roch theorem does this for curves and line bundles, whereas the Hirzebruch–Riemann–Roch theorem generalises this to vector bundles over manifolds. The Grothendieck–Hirzebruch–Riemann–Roch theorem sets both theorems in a relative situation of a morphism between two manifolds (or more general schemes) and changes the theorem from a statement about a single bundle, to one applying to chain complexes of sheaves.
The theorem has been very influential, not least for the development of the Atiyah–Singer index theorem. Conversely, complex analytic analogues of the Grothendieck–Hirzebruch–Riemann–Roch theorem can be proved using the families index theorem. Alexander Grothendieck, its author, was rumored to have finished the proof around 1956 but did not publish his theorem because he was not satisfied with it. Instead Armand Borel and Jean-Pierre Serre, wrote up and published Grothendieck's preliminary (as he saw it) proof.
is the Chow group of cycles on X of dimension d modulo rational equivalence, tensored with the rational numbers. In case X is defined over the complex numbers, the latter group maps to the topological cohomology group
Now consider a proper morphism
between smooth quasi-projective schemes and a bounded complex of sheaves
The Grothendieck–Riemann–Roch theorem relates the push forward maps
and the pushforward
by the formula
Generalisations of the theorem can be made to the non-smooth case by considering a proper generalisation of the combination ch(—)td(X) and to the non-proper case by considering cohomology with compact support.
The arithmetic Riemann–Roch theorem extends the Grothendieck–Riemann–Roch theorem to arithmetic schemes.
The Hirzebruch–Riemann–Roch theorem is (essentially) the special case where Y is a point and the field is the field of complex numbers.
Grothendieck's version of the Riemann–Roch theorem was originally conveyed in a letter to Serre around 1956–7. It was made public at the initial Bonn Arbeitstagung, in 1957. Serre and Armand Borel subsequently organized a seminar at Princeton to understand it. The final published paper was in effect the Borel–Serre exposition.
The significance of Grothendieck's approach rests on several points. First, Grothendieck changed the statement itself: the theorem was, at the time, understood to be a theorem about a variety, whereas after Grothendieck, it was known to essentially be understood as a theorem about a morphism between varieties. In short, he applied a strong categorical approach to a hard piece of analysis. Moreover, Grothendieck introduced K-groups, as discussed above, which paved the way for algebraic K theory.