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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 ways of carrying a coherent-sheaf class along a proper map. In the version used here, f: X → Y is a proper morphism of smooth quasi-projective schemes over a field, and F is a coherent sheaf on X. One route forms the alternating class of the higher direct images of F in K-theory and then takes its Chern character. The other takes the Chern character on X and pushes the resulting characteristic class to Y in rational Chow groups. The two routes agree after multiplying by the appropriate Todd classes.[1]

In full, Td(T_Y) · ch_Y(f_![F]) = f_*^Chow(Td(T_X) · ch_X(F)) in A(Y) ⊗ Q. Here f_![F] = Σ_i (−1)^i [R^i f_*F]. The left pushforward is an alternating K-theoretic direct image; the right pushforward carries Chow cycles. SGA 6 prints f_* for both and explains the difference in prose. The Todd factor on the left is not inverted in this displayed form.[1]

Structural Signature

  • Proper map and smooth setting — theorem hypothesis. f: X → Y has the smooth quasi-projective source and target required for the bounded Chow-theoretic statement.[1]
  • Coherent-sheaf class — input. F supplies a K-class on X to be compared after direct image.[1]
  • Alternating higher direct image — first route. f_![F] includes every R^i f_*F with sign (−1)^i; ordinary f_*F alone need not carry the needed Euler-type information.[1][2]
  • Chern character — comparison map. It converts K-classes to rational Chow characteristic classes so that the routes can meet in one target group.[1][2]
  • Todd correction — constitutive adjustment. Td(T_X) and Td(T_Y), or the equivalent virtual relative Todd factor, correct the otherwise noncommuting pushforwards.[1]
  • Chow pushforward and equality — second route and output. Cycle pushforward of the corrected class on X equals the corrected Chern character of the K-theoretic direct image on Y.[1]

The formula also has the relative form ch_Y(f_![F]) = f_*^Chow(ch_X(F) · Td(T_X − f^*T_Y)). It follows by moving the invertible Td(T_Y) factor through the Chow pushforward with the projection formula. The inverse downstairs in this rearrangement is pulled back into the virtual relative class T_X − f^*T_Y; it is not an extra inverse inserted into the literal first equation.[1]

What It Is Not

It is not an equality of two operators both called f_* on the same group: one direct image acts on K-classes and the other on rational Chow classes. It is not a formula for h^0 by itself. For a curve mapped to a point, the K-theoretic direct image yields an alternating Euler characteristic, h^0 − h^1; finding an individual cohomology dimension can need more information.[1][2]

The version here does not automatically cover an arbitrary nonproper map, singular scheme, moduli stack, or compactified family. Generalizations require their own hypotheses and sources. The seed's reference to a Hodge-bundle computation on a moduli space is therefore not used as an unqualified example. A merely similar commuting-square diagram without these K, Chow, and Todd roles is not this theorem.[1]

Scope of Application

Use this entry when a proper morphism of smooth quasi-projective schemes over a field and a coherent sheaf meet the stated hypotheses, and the question is how the sheaf's alternating direct image relates to characteristic classes after Chow pushforward. Record the morphism, sheaf, base field, chosen Chow convention, and whether the target is a point or another scheme. If using an extension to singular, stack, or other settings, name the extension rather than transferring this bounded statement silently.[1]

The two mapped cases below use narrower source conventions. Kelly's curve-to-point calculation is given over an algebraically closed field. The cited SGA 6 Appendix II closed-immersion corollary is stated under characteristic zero, so its worked projective-hyperplane specialization is taken over the complex numbers. These limits attach to the cited case derivations, not to a claim that the general formula only exists in those settings.[1][2]

Clarity

Write the two routes with different labels. Starting with [F] on X, the K route is [F] → f_![F] → ch_Y(f_![F]). The Chow route is [F] → ch_X(F) → Td(T_X)ch_X(F) → f_*^Chow(…). The theorem says that the Chow result equals Td(T_Y)ch_Y(f_![F]). The typed routes prevent the common mistake of treating the two pushforwards as the same map.[1]

The relative form uses T_f = T_X − f^*T_Y. For a smooth morphism it is the tangent bundle along the fibers; for a regular immersion it is the negative normal-bundle class. Thus the correction changes its concrete expression between the two examples even though the abstract comparison remains the same.[1]

Manages Complexity

The theorem turns a problem about an alternating collection of higher direct images into an intersection-theoretic calculation with characteristic classes, when those Chow classes are accessible. In the curve-to-point case, it condenses the cohomological Euler characteristic of a line bundle into degree and genus. In the immersion case, the normal-bundle Todd factor explains the degree-two term that an uncorrected Chern-character pushforward would miss.[1][2]

This compression has a limit: it does not recover each R^i f_*F separately from their alternating K-class, nor does it make a difficult Chow computation automatic. The claim is an equality under hypotheses, not a promise that either side is computationally easy in every application.[1][2]

Abstract Reasoning

First check the map's properness and the stated smooth quasi-projective setting. Form the coherent-sheaf K-class and its alternating higher direct image. Take Chern character on both sides, compute the Todd classes of the tangent bundles, and use Chow pushforward for the characteristic class on X. Compare both sides in A(Y) ⊗ Q with the Todd factors in their literal positions.[1]

When the relative expression is simpler, replace the two absolute Todd factors by Td(T_X − f^*T_Y) upstairs. Check this transformation with the projection formula, especially the pullback on T_Y. At a point target, the pushforward extracts a degree; at a regular immersion, the normal bundle enters with an inverse Todd class.[1][2]

Knowledge Transfer

The curve projection and the hyperplane embedding share the same six roles: bounded proper map, sheaf K-class, alternating direct image, Chern character, Todd correction, and Chow pushforward equality. The first sends a curve to a point and produces a number. The second sends a divisor into a surface and produces a class with codimension-one and codimension-two terms. This variation shows why the theorem is a relative comparison, not merely the classical curve formula.[1][2]

The general mathematical fact that this is a proved statement belongs to the live Formal Theorem genus. Its K-class and alternating direct-image route independently presuppose live K-theory. The formula's corrected comparison can suggest a broad structural analogy, but its identity remains tied to characteristic classes, Chow groups, and algebraic-geometric hypotheses; neither example warrants a new Prime.

Examples

A smooth projective curve mapped to a point

Let C be a smooth projective curve of genus g over an algebraically closed field, f: C → Spec(k), and L ≅ O(D) a line bundle. The K pushforward has rank χ(C,L) = h^0(C,L) − h^1(C,L). On a curve, ch(L) = 1 + D and Td(T_C) = 1 − K_C/2, where deg K_C = 2g − 2. Chow pushforward to the point takes the degree-one part, giving deg D − (2g − 2)/2 = deg D + 1 − g. This is Kelly's explicit specialization of the theorem.[2]

Mapped back: the map is C→Spec(k); the K-class is [L]; the alternating direct image is [H^0(C,L)] − [H^1(C,L)]; the Chern character is 1+D; the Todd correction is 1−K_C/2 with target Todd class 1; the Chow output is deg D+1−g. It gives the Euler characteristic, not h^0 alone.[2]

A regular hyperplane immersion

Work over C. Let i: H ≅ P¹ ↪ P² be a hyperplane and take F = O_H; let h be the hyperplane class on P². The short exact sequence for the divisor gives [i_*O_H] = [O_P²] − [O_P²(−1)], hence ch(i_!O_H) = 1 − e^(−h) = h − h²/2 in A(P²) ⊗ Q. The normal bundle is N = O_H(1), and its inverse Todd class is 1 − h|_H/2 on H. Chow pushforward yields i_*(1 − h|_H/2) = h − h²/2, matching the K route. This is a worked specialization of SGA 6's characteristic-zero immersion formula, not an example quoted verbatim from its text.[1]

Mapped back: the map is a proper regular immersion; the K-class is [O_H]; the alternating direct image is [i_*O_H] with no higher terms in this case; the Chern character is 1−e^(−h); the Todd correction is Td(N)^(−1); the Chow output is h−h²/2. The −h²/2 term makes the correction visible.[1]

Structural Tensions

The sourced theorem is an exact equality under explicit hypotheses. The initial candidate pressures—omitting Todd factors for a shorter formula, applying the formula beyond its hypotheses, and asking an Euler characteristic for an individual cohomology dimension—are correctness or scope limits, not opposed valid aims inside the theorem. This entry therefore records no intrinsic structural trade-off. The diagnostic questions are whether the two pushforwards are correctly typed, the Todd factors are in the stated positions, and the desired output is an alternating class or an individual cohomology group.[1][2]

Structural–Framed Character

Evaluative weight: the equality is a mathematical claim, not a policy or aesthetic ranking. Human-practice dependence: mathematicians choose classes, maps, and a formulation, but the identity is not made true by that choice. Institutional origin: the named result belongs to the historical development of algebraic geometry; its validity is the proved statement under its hypotheses. Vocabulary travel: “Todd correction” or “commuting routes” in another field does not transfer the K/Chow identity. Import versus recognition: recognize the theorem by the typed characteristic-class equation, not by whether a writer uses its name.[1][2]

The comparison has a formal, repeatable layout, yet its mathematical content depends on coherent sheaves, proper maps, Chern character, Chow groups, and Todd classes. Its character: a domain-specific formal theorem with a K-theoretic prerequisite, not a substrate-independent Prime.

Structural Core vs. Domain Accent

The portable skeleton is two typed routes from an input class to a shared target, made equal by a correction. Here the actual theorem specifies every type and correction: an alternating coherent-sheaf class, Chern character, rational Chow pushforward, and Todd classes under proper and smooth hypotheses. A curve-to-point projection and a regular immersion instantiate this same equality with different relative tangent classes; neither justifies stripping away those algebraic-geometric conditions.[1][2]

The proof-status role reaches live Formal System through the approved Formal Theorem parent. The K-class route reaches live Invariance through K-theory's existing dependency. Neither live Prime names the corrected two-route skeleton itself. That generic skeleton is a future-Prime question requiring independent cross-domain admission; this entry establishes only the algebraic-geometric theorem and proposes no direct Prime edge.

This entry presupposes K-theory and is a kind of Formal theorem.

Formal theorem is an approved strict subsumption parent: GRR is a proved statement, while that genus contains many other theorems. K-theory is an independent strict composition/presupposes parent: the equation needs an alternating K/G-class direct image, although the theorem is not a subtype of K-theory. The two edges reflect distinct proof-status and constitutive-input axes.[1]

Grothendieck group is not added merely because of the shared name. Its live identity is universal completion of a commutative monoid; the coherent-sheaf G₀ class here is defined using exact-sequence relations. A direct edge would need its own identity-level proof. Generic Prime notions such as transformation or invariance do not alone state this theorem's K/Chow/Todd comparison.

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 Riemann–Roch for a curve is the point-target specialization, not the whole relative statement. Hirzebruch–Riemann–Roch also concerns a point target for a higher-dimensional smooth variety; the Grothendieck form permits a general proper map between the stated smooth schemes. Ordinary direct image of a sheaf omits higher terms unless they vanish. Chow pushforward is a different operation from K-theoretic direct image. An uncorrected square generally fails, and the target Todd factor is not inverted in equation (1.1). A singular or stack formulation needs its own extension and hypotheses.[1][2]

References

[1] 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 registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w ↩x ↩y ↩z ↩27

[2] 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 registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n