Grothendieck–Hirzebruch–Riemann–Roch theorem

Grothendieck–Hirzebruch–Riemann–Roch theorem

In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces.

Riemann–Roch type theorems relate Euler characteristics of the cohomology of a vector bundle with their topological degrees, or more generally their characteristic classes in (co)homology or algebraic analogues thereof. The classical Riemann–Roch theorem does this for curves and line bundles, whereas the Hirzebruch–Riemann–Roch theorem generalises this to vector bundles over manifolds. The Grothendieck–Hirzebruch–Riemann–Roch theorem sets both theorems in a relative situation of a morphism between two manifolds (or more general schemes) and changes the theorem from a statement about a single bundle, to one applying to chain complexes of sheaves.

The theorem has been very influential, not least for the development of the Atiyah–Singer index theorem. Conversely, complex analytic analogues of the Grothendieck–Hirzebruch–Riemann–Roch theorem can be proved using the families index theorem. Alexander Grothendieck, its author, was rumored to have finished the proof around 1956 but did not publish his theorem because he was not satisfied with it. Instead Armand Borel and Jean-Pierre Serre, wrote up and published Grothendieck's preliminary (as he saw it) proof.


Let "X" be a smooth quasi-projective scheme over a field. Under these assumptions, the Grothendieck group


of bounded complexes of coherent sheaves is canonically isomorphic to the Grothendieck group of bounded complexes of finite-rank vector bundles. Using this isomorphism, consider the Chern character


(a rational combination of Chern classes) as a functorial transformation

:mbox{ch}: K_0(X) o A(X, {Bbb Q})


:A_d(X,{Bbb Q}),

is the Chow group of cycles on "X" of dimension "d" modulo rational equivalence, tensored with the rational numbers. In case "X" is defined over the complex numbers, the latter group maps to the topological cohomology group

:H^{2 mathrm{dim}(X) - 2d}(X, {Bbb Q}).

Now consider a proper morphism

:f: X o Y,

between smooth quasi-projective schemes and a bounded complex of sheaves {mathcal F^ull}.

The Grothendieck–Riemann–Roch theorem relates the push forward maps

:f_{mbox{! = sum (-1)^i R^i f_*: K_0(X) o K_0(Y)

and the pushforward

:f_* colon A(X) o A(Y),,

by the formula

: mbox{ch}(f_{mbox{!{mathcal F}^ull)mbox{td}(Y) = f_* (mbox{ch}({mathcal F}^ull) mbox{td}(X) ). Here td("X") is the Todd genus of (the tangent bundle of) "X". Thus the theorem gives a precise measure for the lack of commutativity of taking the push forwards in the above senses and the chern character and shows that the needed correction factors depends on "X" and "Y" only. In fact, since the Todd genus is functorial and multiplicative in exact sequences, we can rewrite the Grothendieck Hirzebruch Riemann Roch formula to

: mbox{ch}(f_{mbox{!{mathcal F}^ull) = f_* (mbox{ch}({mathcal F}^cdot) mbox{td}(T_f) ).

where T_f is the relative tangent sheaf of "f". This is often useful in applications, for example if "f" is a locally trivial fibration.

Generalising and specialising

Generalisations of the theorem can be made to the non-smooth case by considering a proper generalisation of the combination ch(—)td("X") and to the non-proper case by considering cohomology with compact support. The arithmetic Riemann–Roch theorem extends the Grothendieck–Riemann–Roch theorem to arithmetic schemes.

The Hirzebruch–Riemann–Roch theorem is (essentially) the special case where "Y" is a point and the field is the field of complex numbers.


Grothendieck's version of the Riemann–Roch theorem was originally conveyed in a letter to Serre around 1956–7. It was made public at the initial Bonn Arbeitstagung, in 1957. Serre and Armand Borel subsequently organized a seminar at Princeton to understand it. The final published paper was in effect the Borel–Serre exposition.

The significance of Grothendieck's approach rests on several points. First, Grothendieck changed the statement itself: the theorem was, at the time, understood to be a theorem about a variety, whereas after Grothendieck, it was known to essentially be understood as a theorem about a morphism between varieties. In short, he applied a strong categorical approach to a hard piece of analysis. Moreover, Grothendieck introduced K-groups, as discussed above, which paved the way for algebraic K theory.


* | year=1958 | journal=Bulletin de la Société Mathématique de France | issn=0037-9484 | volume=86 | pages=97–136
* | year=1998

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Hirzebruch–Riemann–Roch theorem — In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch s 1954 result contributing to the Riemann–Roch problem for complex algebraic varieties of all dimensions. It… …   Wikipedia

  • Riemann–Roch theorem — In mathematics, specifically in complex analysis and algebraic geometry, the Riemann–Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. It relates… …   Wikipedia

  • Riemann-Roch theorem for smooth manifolds — In mathematics, a Riemann Roch theorem for smooth manifolds is a version of results such as the Hirzebruch Riemann Roch theorem or Grothendieck Riemann Roch theorem (GRR) without a hypothesis making the smooth manifolds involved carry a complex… …   Wikipedia

  • Riemann-Roch-Theorem — Der Satz von Riemann Roch (nach dem Mathematiker Bernhard Riemann und seinem Schüler Gustav Roch) ist eine zentrale Aussage der Theorie kompakter riemannscher Flächen. Er gibt an, wie viele linear unabhängige meromorphe Funktionen mit… …   Deutsch Wikipedia

  • Alexandre Grothendieck — en 1970. Naissance 28 mars 1928 Berlin ( …   Wikipédia en Français

  • Alexander Grothendieck — User:Geometry guy/InfoboxAlexander Grothendieck (born March 28, 1928 in Berlin, Germany) is considered to be one of the greatest mathematicians of the 20th century. He made major contributions to: algebraic topology, algebraic geometry, number… …   Wikipedia

  • Gustav Roch — Pour les articles homonymes, voir Roch. Gustav Roch Gustav Roch Naissance 9 décembre 1 …   Wikipédia en Français

  • Liste de théorèmes — par ordre alphabétique. Pour l établissement de l ordre alphabétique, il a été convenu ce qui suit : Si le nom du théorème comprend des noms de mathématiciens ou de physiciens, on se base sur le premier nom propre cité. Si le nom du théorème …   Wikipédia en Français

  • List of mathematics articles (G) — NOTOC G G₂ G delta space G networks Gδ set G structure G test G127 G2 manifold G2 structure Gabor atom Gabor filter Gabor transform Gabor Wigner transform Gabow s algorithm Gabriel graph Gabriel s Horn Gain graph Gain group Galerkin method… …   Wikipedia

  • Todd class — In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is… …   Wikipedia