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Grothendieck–Riemann–Roch theorem

For a proper map of smooth algebraic schemes, Grothendieck–Riemann–Roch equates a K-theoretic direct image with a rational Chow pushforward after Chern-character and Todd-class correction.

Version
v1 · 2026-10-07 · History
Domain-specific #
13902
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Intersection Theory, Algebraic K Theory → Mathematics
Aliases
Grothendieck–Riemann–Roch formula

Core Idea

The Grothendieck–Riemann–Roch theorem compares two pushforwards along a proper map f:X→Y of smooth quasi-projective schemes over a field. One route takes the alternating higher-direct-image K-class and its Chern character. The other Todd-corrects the Chern character on X and pushes it forward in rational Chow groups. The equality is Td(T_Y) ch_Y(f_![F]) = f_*^Chow(Td(T_X) ch_X(F)); the left Todd factor is not inverted.[^ref-2f61e6c0c1d1]

Scope of Application

Use this statement under its stated hypotheses. Keep f_![F]=Σ_i(−1)^i[R^if_*F] distinct from Chow cycle pushforward. A singular scheme, stack, or nonproper map requires a separately specified extension. The curve example uses an algebraically closed field; the cited immersion formula uses characteristic zero.[ref-2f61e6c0c1d1][ref-f4424d116eb4]

Clarity

The two f_* symbols act on different groups. The K route moves the sheaf class and records its higher direct images with alternating signs. The Chow route moves a corrected characteristic class. Equivalently, with the virtual relative tangent class T_f=T_X−f^*T_Y, one can write ch_Y(f_![F])=f_*^Chow(ch_X(F)Td(T_f)). The pullback f^* is essential.[^ref-2f61e6c0c1d1]

Manages Complexity

The theorem turns an alternating collection of higher direct images into a Chow calculation. A curve mapped to a point yields the Euler characteristic of a line bundle from degree and genus. A regular embedding uses the normal bundle's inverse Todd class. The equality does not recover individual cohomology groups or ensure easy computation.[ref-2f61e6c0c1d1][ref-f4424d116eb4]

Abstract Reasoning

Check the hypotheses and type each route. Form the alternating K pushforward, take Chern character, compute source and target Todd classes, and compare with the Chow pushforward in A(Y)⊗Q. For an immersion, the relative tangent class is the negative normal-bundle class; for a curve-to-point map, Chow pushforward takes degree. Omitting Todd factors generally destroys the equality.[ref-2f61e6c0c1d1][ref-f4424d116eb4]

Knowledge Transfer

A curve projection and regular embedding share the map, sheaf K-class, alternating direct image, Chern character, Todd correction, and Chow output, while yielding different results. It is a specific proved statement under live Formal Theorem and independently presupposes live K-theory. The broader corrected-two-route skeleton remains a future-Prime question; no direct Prime parent is proposed.[ref-2f61e6c0c1d1][ref-f4424d116eb4]

Example

Curve to a point. Over an algebraically closed field, let C be a smooth projective genus-g curve and L≅O(D) a line bundle. The K pushforward has rank χ(C,L)=h⁰−h¹. Since ch(L)=1+D and Td(T_C)=1−K_C/2, taking degree gives χ(C,L)=deg D+1−g. This is an alternating Euler characteristic, not h⁰ alone.[^ref-f4424d116eb4]

Hyperplane immersion. Over the complex numbers, include H≅P¹ in P² and use F=O_H. The K route gives ch(i_!O_H)=1−e^(−h)=h−h²/2. The normal-bundle correction gives i_*(Td(O_H(1))^(−1))=h−h²/2 in rational Chow groups. This curator-worked case specializes SGA 6's characteristic-zero immersion formula.[^ref-2f61e6c0c1d1]

Relationships to Other Abstractions

Local relationship map for Grothendieck–Riemann–Roch theoremParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Grothendieck–Riemann…DOMAINDomain-specific abstraction: K-theory — presupposesK-theoryDOMAINDomain-specific abstraction: Formal theorem — is a kind ofFormal theoremDOMAIN

Current abstraction Grothendieck–Riemann–Roch theorem Domain-specific

Parents (2) — more general patterns this builds on

  • Grothendieck–Riemann–Roch theorem is a kind of Formal theorem Domain-specific

    Grothendieck–Riemann–Roch is a proved mathematical statement with explicit hypotheses and a characteristic-class conclusion.

  • Grothendieck–Riemann–Roch theorem presupposes K-theory Domain-specific

    The theorem requires coherent-sheaf K/G-classes and their alternating higher-direct-image pushforward.

Hierarchy paths (3) — routes to 3 parentless roots

Neighborhood in Abstraction Space

Grothendieck–Riemann–Roch theorem sits in a sparse region of the domain-specific corpus (84th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

Classical curve Riemann–Roch is the point-target specialization, not the full relative theorem. Ordinary direct image f_*F omits higher terms unless they vanish. Chow pushforward is not K-theoretic direct image. Grothendieck Group is a distinct live group-completion identity, not an automatic parent merely because of the name. An uncorrected commuting square and an unrestricted singular or stack version are different claims.[ref-2f61e6c0c1d1][ref-f4424d116eb4]

References

[^ref-2f61e6c0c1d1]: Pierre Berthelot, Alexander Grothendieck, and Luc Illusie, Théorie des intersections et théorème de Riemann-Roch (SGA 6, Lecture Notes in Mathematics 225, Springer, 1971), Exposé 0 by Grothendieck, §1 equations (1.1)–(1.3) and Appendix II §5 Corollary equation (2.38 bis). The online English text edition was consulted; the cited Appendix II immersion corollary carries a characteristic-zero hypothesis. https://grothendiecksga.com/read/sga6/en/0.html ; https://grothendiecksga.com/read/sga6/en/0-app-II.html

[^ref-f4424d116eb4]: Shane Kelly, K-Theory Lecture 2 Grothendieck–Riemann–Roch (University of Tokyo, 2025), Theorem 2.31 and Remark 2.32, PDF zero-index pp. 13–14, including the displayed curve calculation; lecture p. 1 gives the algebraically closed base-field convention. https://www.ms.u-tokyo.ac.jp/~kelly/Course2025KTheory/KTheoryLecture2.pdf