- Grothendieck–Hirzebruch–Riemann–Roch theorem
In

mathematics , specifically inalgebraic geometry , the**Grothendieck–Riemann–Roch theorem**is a far-reaching result oncoherent cohomology . It is a generalisation of theHirzebruch–Riemann–Roch theorem , aboutcomplex manifold s, which is itself a generalisation of the classicalRiemann–Roch theorem forline bundle s oncompact Riemann surface s.Riemann–Roch type theorems relate

Euler characteristic s of thecohomology of avector bundle with theirtopological degree s, 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 amorphism between two manifolds (or more general schemes) and changes the theorem from a statement about a single bundle, to one applying tochain complex es 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 thefamilies 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. InsteadArmand Borel andJean-Pierre Serre , wrote up and published Grothendieck's preliminary (as he saw it) proof.**Formulation**Let "X" be a

smooth quasi-projective scheme over a field. Under these assumptions, theGrothendieck group :$K\_0(X),$

of

bounded complex es of coherent sheaves is canonically isomorphic to the Grothendieck group of bounded complexes of finite-rank vector bundles. Using this isomorphism, consider theChern character :$mbox\{ch\},$

(a rational combination of

Chern classes ) as afunctor ial transformation:$mbox\{ch\}:\; K\_0(X)\; o\; A(X,\; \{Bbb\; Q\})$

where

:$A\_d(X,\{Bbb\; Q\}),$

is the Chow group of cycles on "X" of dimension "d" modulo rational equivalence, tensored with the

rational number s. In case "X" is defined over thecomplex number s, the latter group maps to the topologicalcohomology 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 (thetangent 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 inexact sequence s, 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 alocally 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 . Thearithmetic Riemann–Roch theorem extends the Grothendieck–Riemann–Roch theorem toarithmetic scheme s.The

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

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

* | year=1998

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