In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces. 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–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–Riemann–Roch theorem can be proved using the index theorem for families. Alexander Grothendieck gave a first proof in a 1957 manuscript, later published. Armand Borel and Jean-Pierre Serre wrote up and published Grothendieck's proof in 1958. Later, Grothendieck and his collaborators simplified and generalized the proof.
Formulation Let X be a smooth quasi-projective scheme over a field. Under these assumptions, the Grothendieck group K 0 ( X ) {\displaystyle K_{0}(X)} of bounded complexes of coherent sheaves is canonically isomorphic to the Grothendieck group of bounded complexes of finite-rank vector bundles. Using this isomorphism, consider the Chern character (a rational combination of Chern classes) as a functorial transformation:
c h : K 0 ( X ) → A ( X , Q ) , {\displaystyle \mathrm {ch} \colon K_{0}(X)\to A(X,\mathbb {Q} ),}
where A d ( X , Q ) {\displaystyle A_{d}(X,\mathbb {Q} )} 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:
H 2 dim ( X ) − 2 d ( X , Q ) . {\displaystyle H^{2\dim(X)-2d}(X,\mathbb {Q} ).}
Now consider a proper morphism f : X → Y {\displaystyle f\colon X\to Y} between smooth quasi-projective schemes and a bounded complex of sheaves F ∙ {\displaystyle {{\mathcal {F}}^{\bullet }}} on X . {\displaystyle X.}
The Grothendieck–Riemann–Roch theorem relates the pushforward map
f ! = ∑ ( − 1 ) i R i f ∗ : K 0 ( X ) → K 0 ( Y ) {\displaystyle f_{!}=\sum (-1)^{i}R^{i}f_{*}\colon K_{0}(X)\to K_{0}(Y)}
(alternating sum of higher direct images) and the pushforward
f ∗ : A ( X ) → A ( Y ) , {\displaystyle f_{*}\colon A(X)\to A(Y),}
by the formula
c h ( f ! F ∙ ) t d ( Y ) = f ∗ ( c h ( F ∙ ) t d ( X ) ) . {\displaystyle \mathrm {ch} (f_{!}{\mathcal {F}}^{\bullet })\mathrm {td} (Y)=f_{*}(\mathrm {ch} ({\mathcal {F}}^{\bullet })\mathrm {td} (X)).}
Here t d ( X ) {\displaystyle \mathrm {td} (X)} is the Todd genus of (the tangent bundle of) X. Thus the theorem gives a precise measure for the lack of commutativity of taking the push forwards in the above senses and the Chern character and shows that the needed correction factors depend on X and Y only. In fact, since the Todd genus is functorial and multiplicative in exact sequences, we can rewrite the Grothendieck–Riemann–Roch formula as
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