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| 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 manifold]]s, which is itself a generalisation of the classical [[Riemann–Roch theorem]] for [[line bundle]]s on [[compact Riemann surface]]s.
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| Riemann–Roch type theorems relate [[Euler characteristic]]s of the [[cohomology]] of a [[vector bundle]] with their [[topological 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–Riemann–Roch theorem sets both theorems in a relative situation of a [[morphism]] between two manifolds (or more general [[scheme (mathematics)|schemes]]) and changes the theorem from a statement about a single bundle, to one applying to [[chain complex]]es of [[sheaf (mathematics)|sheaves]].
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| The theorem has been very influential, not least for the development of the [[Atiyah–Singer index theorem]]. Conversely, [[complex analysis|complex analytic]] analogues of the Grothendieck–Riemann–Roch theorem can be proved using the index theorem for families. [[Alexander Grothendieck]] gave a first proof in a 1957 manuscript, later published.<ref>A. Grothendieck. Classes de faisceaux et théorème de Riemann-Roch (1957). Published in SGA 6, Springer-Verlag (1971), 20-71.</ref> [[Armand Borel]] and [[Jean-Pierre Serre]] wrote up and published Grothendieck's proof in 1958.<ref>A. Borel and J.-P. Serre. Bull. Soc. Math. France 86 (1958), 97-136.</ref> Later, Grothendieck and his collaborators simplified and generalized the proof.<ref>SGA 6, Springer-Verlag (1971).</ref>
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| ==Formulation==
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| Let ''X'' be a [[smooth scheme|smooth]] [[quasi-projective scheme]] over a [[Field (mathematics)|field]]. Under these assumptions, the [[Grothendieck group]]
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| :<math>K_0(X)\,</math>
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| of [[bounded complex]]es of [[coherent sheaf|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 [[functor]]ial transformation | |
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| :<math>\mbox{ch} \colon K_0(X) \to A(X, {\Bbb Q}),</math> | |
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| where
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| :<math>A_d(X,{\Bbb Q})\,</math>
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| is the [[Chow ring|Chow group]] of cycles on ''X'' of dimension ''d'' modulo [[Chow ring#Rational equivalence|rational equivalence]], [[tensor product|tensor]]ed with the [[rational number]]s. In case ''X'' is defined over the [[complex number]]s, the latter group maps to the topological [[cohomology group]]
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| :<math>H^{2 \mathrm{dim}(X) - 2d}(X, {\Bbb Q}).</math>
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| Now consider a [[proper morphism]]
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| :<math>f \colon X \to Y\,</math> | |
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| between smooth quasi-projective schemes and a bounded complex of sheaves <math>{\mathcal F^\bull}.</math>
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| The '''Grothendieck–Riemann–Roch theorem''' relates the pushforward map
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| :<math>f_{\mbox{!}} = \sum (-1)^i R^i f_* \colon K_0(X) \to K_0(Y)</math>
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| and the pushforward
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| :<math>f_* \colon A(X) \to A(Y),\,</math>
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| by the formula
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| :<math> \mbox{ch}(f_{\mbox{!}}{\mathcal F}^\bull)\mbox{td}(Y) = f_* (\mbox{ch}({\mathcal F}^\bull) \mbox{td}(X) ). </math> | |
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| 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 depend on ''X'' and ''Y'' only. In fact, since the Todd genus is functorial and multiplicative in [[exact sequence]]s, we can rewrite the Grothendieck–Riemann–Roch formula as
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| :<math> \mbox{ch}(f_{\mbox{!}}{\mathcal F}^\bull) = f_* (\mbox{ch}({\mathcal F}^\bull) \mbox{td}(T_f) ),</math> | |
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| where ''T''<sub>''f''</sub> is the relative tangent sheaf of ''f'', defined as the element ''TX'' − ''f''<sup>*</sup>''TY'' in ''K''<sub>0</sub>(''X''). For example, when ''f'' is a [[smooth morphism]], ''T''<sub>''f''</sub> is simply a vector bundle, known as the tangent bundle along the fibers of ''f''.
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| ==Generalising and specialising==
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| Generalisations of the theorem can be made to the non-smooth case by considering an appropriate generalisation of the combination ch(—)td(''X'') and to the non-proper case by considering [[cohomology with compact support]].
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| The [[arithmetic Riemann–Roch theorem]] extends the Grothendieck–Riemann–Roch theorem to [[arithmetic scheme]]s.
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| The [[Hirzebruch–Riemann–Roch theorem]] is (essentially) the special case where ''Y'' is a point and the field is the field of complex numbers.
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| == History ==
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| [[Alexander Grothendieck]]'s version of the Riemann–Roch theorem was originally conveyed in a letter to [[Jean-Pierre 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.
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| 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 [[algebraic variety|variety]], whereas Grothendieck saw it as a theorem about a morphism between varieties. By finding the right generalization, the proof became simpler while the conclusion became more general. In short, Grothendieck applied a strong [[category theory|categorical]] approach to a hard piece of [[Mathematical analysis|analysis]]. Moreover, Grothendieck introduced [[Algebraic K-theory|K-groups]], as discussed above, which paved the way for [[algebraic K-theory]].
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{Citation | last1=Borel | first1=Armand | author1-link=Armand Borel | last2=Serre | first2=Jean-Pierre | author2-link=Jean-Pierre Serre | title=Le théorème de Riemann–Roch | mr=0116022 | year=1958 | journal=Bulletin de la Société Mathématique de France | volume=86 | pages=97–136 | issn=0037-9484 | language=French }}
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| * {{Citation | last1=Fulton | first1=William | author1-link=William Fulton (mathematician) | title=Intersection theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=3-540-62046-X | mr=1644323 | year=1998 | zbl=0885.14002 | edition=2nd | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | volume=2 }}
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| *{{cite book
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| | last = Berthelot
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| | first = Pierre
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| | authorlink = Pierre Berthelot (mathematician)
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| | coauthors = [[Alexandre Grothendieck]], [[Luc Illusie]], eds.
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| | title = Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics '''225''')
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| | year = 1971
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| | publisher = [[Springer Science+Business Media|Springer-Verlag]]
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| | location = Berlin; New York
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| | language = French
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| | pages = xii+700
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| | nopp = true
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| |doi=10.1007/BFb0066283
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| |isbn= 978-3-540-05647-8
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| }}
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| ==External links==
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| * The [http://mathoverflow.net/questions/63095/how-does-one-understand-grr-grothendieck-riemann-roch thread] "how does one understand GRR? (Grothendieck Riemann Roch)" on [[MathOverflow]].
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| {{DEFAULTSORT:Grothendieck-Hirzebruch-Riemann-Roch theorem}}
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| [[Category:Topological methods of algebraic geometry]]
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| [[Category:Theorems in algebraic geometry]]
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