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Grothendieck's proof of Hirzebruch-Riemann-Roch theorem

Giacomo Graziani

TL;DR

The paper presents a self-contained development of Hirzebruch-Riemann-Roch for smooth projective varieties by building the Grothendieck group framework $K_0(X)$ and the Chow ring $A_*(X)$, introducing the Chern character $\mathrm{ch}$ and Todd class $\mathrm{td}$, and proving the HRR formula $\chi(X,\mathcal{E})=\int_X \mathrm{ch}(\mathcal{E})\cdot \mathrm{td}(X)$. It starts from the projective space case and then extends to general smooth projective varieties via functorial pushforwards and closed embeddings, employing the Dévissage theorem, the Whitney sum formula, and the projective bundle Calculus. The work recovers classical curve and surface Riemann-Roch formulas as explicit low-dimensional consequences and provides a robust toolkit (K-theory, Chow groups, Chern/Todd classes) for computing Euler characteristics through intersection theory. Overall, the text ties cohomological data to intersection-theoretic invariants in a comprehensive, modern Grothendieckian framework, culminating in a full HRR proof and its classical corollaries.

Abstract

The Riemann-Roch Theorem is one of the cornerstones of algebraic geometry, connecting algebraic data (sheaf cohomology) with geometric ones (intersection theory). This survey paper provides a self-contained introduction and a complete proof of the Hirzebruch-Riemann-Roch (HRR) Theorem for smooth projective varieties over an algebraically closed field. Starting from the classical formulations for curves and surfaces, we introduce the two modern tools necessary for the generalization: the Grothendieck group $K_{0}(X)$ as the natural setting for the Euler characteristic, and the Chow ring $A_{\bullet}(X)$ as the setting for cycles and intersection theory. We then construct the fundamental bridge between these two worlds\textemdash the Chern character ($\mathrm{ch}$) and the Todd class ($\mathrm{td}$) \textemdash culminating in a full proof of the HRR formula: \[ χ(X,\mathcal{E})=\int_{X}\mathrm{ch}(\mathcal{E})\cdot\mathrm{td}(X) \] We conclude by showing how this general formula recovers the classical theorems for curves and surfaces.

Grothendieck's proof of Hirzebruch-Riemann-Roch theorem

TL;DR

The paper presents a self-contained development of Hirzebruch-Riemann-Roch for smooth projective varieties by building the Grothendieck group framework and the Chow ring , introducing the Chern character and Todd class , and proving the HRR formula . It starts from the projective space case and then extends to general smooth projective varieties via functorial pushforwards and closed embeddings, employing the Dévissage theorem, the Whitney sum formula, and the projective bundle Calculus. The work recovers classical curve and surface Riemann-Roch formulas as explicit low-dimensional consequences and provides a robust toolkit (K-theory, Chow groups, Chern/Todd classes) for computing Euler characteristics through intersection theory. Overall, the text ties cohomological data to intersection-theoretic invariants in a comprehensive, modern Grothendieckian framework, culminating in a full HRR proof and its classical corollaries.

Abstract

The Riemann-Roch Theorem is one of the cornerstones of algebraic geometry, connecting algebraic data (sheaf cohomology) with geometric ones (intersection theory). This survey paper provides a self-contained introduction and a complete proof of the Hirzebruch-Riemann-Roch (HRR) Theorem for smooth projective varieties over an algebraically closed field. Starting from the classical formulations for curves and surfaces, we introduce the two modern tools necessary for the generalization: the Grothendieck group as the natural setting for the Euler characteristic, and the Chow ring as the setting for cycles and intersection theory. We then construct the fundamental bridge between these two worlds\textemdash the Chern character () and the Todd class () \textemdash culminating in a full proof of the HRR formula: We conclude by showing how this general formula recovers the classical theorems for curves and surfaces.

Paper Structure

This paper contains 18 sections, 27 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. The $K$-theory
  3. The Grothendieck group
  4. Dévissage theorem
  5. Multiplicative structure
  6. Functoriality
  7. An important example: the Grothendieck group of projective spaces
  8. Intersection theory
  9. The Chow groups
  10. Intersection with divisors
  11. Projective bundles
  12. Chern and Todd classes
  13. Hirzebruch-Riemann-Roch for smooth projective varieties
  14. The case of $\mathbb{P}^{n}$
  15. The case of smooth projective varieties
  16. ...and 3 more sections

Key Result

Lemma 2.6

Let $\mathcal{P}$ be a category with exact sequences seen as a full subcategory of an abelian category $\mathcal{C}$ and suppose that, for every epimorphism $\pi$ in $\mathcal{P}$, $\ker\left(\pi\right)$ is still an object of $\mathcal{P}$. Let be an exact sequence in $\mathcal{P}$. Then the equality holds in $K_{0}\left(\mathcal{P}\right)$.

Theorems & Definitions (90)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • Theorem 2.8
  • ...and 80 more