Grothendieck–Riemann–Roch theorem

The theorem states that for a proper morphism f: X → Y between smooth quasi-projective varieties over a field, the diagram involving the Grothendieck group K(X) and the Chow ring A(X) commutes after multiplying by the Todd class: ch(f_!(E))·Td(Y) = f_*(ch(E)·Td(X)). Here f_! is the pushforward in K-theory (derived direct image), and f_* is the pushforward in Chow groups. The theorem was proved by Alexander Grothendieck in 1958 and first published in 1960 in a Bourbaki seminar by Armand Borel and Jean-Pierre Serre¹. It extends the Hirzebruch–Riemann–Roch theorem, which is the case where Y is a point, and it also recovers the Riemann–Roch theorem for curves and surfaces.

GRR has numerous applications in algebraic geometry and number theory. It is used to compute Chern classes of direct images of sheaves, which is essential in moduli theory and the study of vector bundles on curves and surfaces. For example, it yields formulas for the degree of the determinant line bundle in families of curves, a key ingredient in the construction of the moduli space of curves and in the proof of the Bogomolov–Miyaoka–Yau inequality². In arithmetic geometry, GRR has been extended to the setting of Arakelov theory, leading to arithmetic Riemann–Roch theorems that relate heights and intersection numbers on arithmetic surfaces. The theorem also plays a role in the theory of motives and in the construction of Chern classes in higher algebraic K-theory.

Several generalizations of GRR have been developed. The most notable is the equivariant version for group actions, which is used in the study of quotient stacks and in the proof of the Atiyah–Segal theorem in K-theory³. There is also a derived version for schemes with singularities, formulated in terms of derived categories and the Grothendieck group of perfect complexes. In the context of motivic homotopy theory, a motivic Riemann–Roch theorem has been established by Marc Levine and others, relating algebraic cobordism and K-theory⁴. Another variant is the arithmetic Riemann–Roch theorem of Gillet and Soulé, which incorporates Green currents and provides a refined statement in Arakelov geometry.

One lesser-known aspect is that Grothendieck's original proof used the technique of deformation to the normal cone, which later became a central tool in intersection theory. Another is that the theorem holds for proper morphisms of schemes over a base that is not necessarily a field, provided one works with the right K-theory and Chow groups. The theorem also has a version for Deligne–Mumford stacks, which is used in the study of moduli of stable curves. A curious historical note: the theorem was first announced in a letter from Grothendieck to Serre in 1957, and the published version in Borel–Serre's Bourbaki report contains a proof that is essentially due to Grothendieck but was written up by Borel and Serre¹. The theorem is also a special case of the more general Riemann–Roch theorem for proper morphisms of schemes, which was later proved by Fulton and MacPherson using intersection theory⁵.

Chern character

A ring homomorphism from K-theory to Chow groups (or cohomology) that encodes the ranks and Chern classes of vector bundles.

Todd class

A characteristic class that appears in the Riemann–Roch formula; for a vector bundle, it is a polynomial in the Chern classes.

Proper morphism

A morphism of schemes that is separated, of finite type, and universally closed; it ensures that pushforwards of coherent sheaves are coherent.

K-theory

The Grothendieck group of vector bundles (or coherent sheaves) modulo exact sequences, used to formalize the notion of virtual dimension.