Grothendieck–Riemann–Roch theorem

Grothendieck–Riemann–Roch theorem
Name	Grothendieck–Riemann–Roch theorem
Field	Algebraic geometry
Introduced	1950s–1960s
Contributors	Alexander Grothendieck, Friedrich Hirzebruch, Jean-Pierre Serre, David Mumford

Grothendieck–Riemann–Roch theorem The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry connecting K-theory and intersection theory through characteristic classes. It generalizes the Riemann–Roch theorem of Bernhard Riemann and Gustav Roch and the Hirzebruch–Riemann–Roch theorem of Friedrich Hirzebruch, providing a functorial formula for pushforwards in algebraic K-theory and Chow groups. The theorem has deep ties to work by Alexander Grothendieck, Jean-Pierre Serre, David Mumford, and Michael Atiyah, and has influenced developments related to Grothendieck's school including the theories of scheme by Grothendieck and sheaf cohomology by Serre.

Introduction

Grothendieck formulated the theorem as part of his research program in the Séminaire de Géométrie Algébrique under the influence of Friedrich Hirzebruch, Jean-Pierre Serre, and David Mumford. The statement situates in the context of algebraic K-theory introduced by Grothendieck and topological K-theory developed by Michael Atiyah, relating algebraic cycles studied by André Weil and Oscar Zariski to characteristic classes familiar from Isaac Newton and Henri Poincaré. The theorem provides a bridge between the categorical viewpoint advanced by Grothendieck and classical calculations associated with Bernhard Riemann, Gustav Roch, and Kunihiko Kodaira.

Statement

Let f: X → Y be a proper morphism of smooth quasi-projective schemes over a field, with X and Y in the style of schemes introduced by Grothendieck and schemes studied by Alexander Grothendieck and David Mumford. The theorem asserts that the pushforward f_* on the Grothendieck group K_0(X) relates to the pushforward on Chow groups A_*(X) via the Todd class introduced by Friedrich Hirzebruch and developed in intersection theory by William Fulton. More precisely, for an element α in K_0(X) one has ch(f_*(α))·td(T_Y) = f_*(ch(α)·td(T_X)) in A_*(Y) ⊗ Q, where ch denotes the Chern character introduced by Chern and developed by Shiing-Shen Chern and the Todd class td appears in Hirzebruch's work. The formulation uses concepts from Serre's coherent sheaf theory and Grothendieck's derived functors.

Historical Context and Motivation

The theorem arose from attempts to generalize classical results such as Riemann–Roch for algebraic curves by Bernhard Riemann and Gustav Roch, and Hirzebruch's extension to complex manifolds. Grothendieck, influenced by Serre's work on coherent cohomology and Atiyah's work in topological K-theory, sought a categorical and functorial formulation. The mathematical environment included the École Normale Supérieure and institutions like Institut des Hautes Études Scientifiques and Collège de France where Grothendieck and Serre worked. The development connected to contemporaneous advances by Jean-Louis Verdier on derived categories and by Alexander Beilinson on motivic ideas and later inspired work by Pierre Deligne on the Weil conjectures and by Spencer Bloch on algebraic cycles.

Proof Outline and Methods

Grothendieck's proof strategy used deformation to the normal cone, resolution of singularities in the spirit of Heisuke Hironaka, and functorial properties of K-theory he developed. The argument employs the formalism of derived categories as later axiomatized by Jean-Louis Verdier, spectral sequences familiar from Henri Cartan and Jean Leray, and localization sequences in K-theory analogous to Quillen's higher K-theory. Key tools include Chern classes from Shiing-Shen Chern, the Todd class from Friedrich Hirzebruch, and pushforward and pullback operations formalized by Serre and Grothendieck. Later alternative proofs and expositions invoked the Riemann–Roch for singular varieties by William Fulton and the Riemann–Roch without denominators by Paolo Aluffi and Luc Illusie.

Examples and Applications

Classical instances include recovering the Riemann–Roch theorem for algebraic curves treated by Bernhard Riemann and Gustav Roch and the Hirzebruch–Riemann–Roch formula for complex manifolds used by Hirzebruch in his work on genera. The theorem informs computations in moduli problems studied by David Mumford and Michael Atiyah, including applications to the moduli of vector bundles on curves considered by Narasimhan and Seshadri and to index theorems in the style of Atiyah–Singer. It underpins computations in intersection theory as developed by William Fulton and in motivic cohomology influenced by Vladimir Voevodsky and Alexander Beilinson. Further applications appear in string theory contexts investigated by Edward Witten and in enumerative geometry studied by Maxim Kontsevich.

Generalizations and Related Results

Generalizations include the singular Riemann–Roch theorems by William Fulton and R. MacPherson, the equivariant Riemann–Roch theorems connected to works of Victor Kac and George Lusztig, and Lefschetz fixed-point formula adaptations by Atiyah and Bott. Relations extend to Quillen's higher K-theory, Beilinson's conjectures on special values inspired by Pierre Deligne, and the multiplicative formalisms in the work of Spencer Bloch and Andrei Suslin. Connections also exist with the index theorem of Michael Atiyah and Isadore Singer and with the theory of motives advanced by Alexander Grothendieck and Yuri Manin.

Technical Background and Preliminaries

Understanding the theorem requires familiarity with schemes from Alexander Grothendieck, coherent sheaves and Serre duality from Jean-Pierre Serre, algebraic K-theory initiated by Grothendieck and extended by Daniel Quillen, and intersection theory as systematized by William Fulton. One needs the theory of Chern classes derived from Shiing-Shen Chern, the Todd class arising in Friedrich Hirzebruch's work, and tools from derived categories developed by Jean-Louis Verdier. Familiarity with resolution of singularities due to Heisuke Hironaka, spectral sequence techniques of Jean Leray, and later advances by Pierre Deligne and Vladimir Voevodsky will aid deeper study.

Category:Algebraic geometry