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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. | |||
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]]. | |||
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> | |||
==Formulation== | |||
Let ''X'' be a [[smooth scheme|smooth]] [[quasi-projective scheme]] over a [[Field (mathematics)|field]]. Under these assumptions, the [[Grothendieck group]] | |||
:<math>K_0(X)\,</math> | |||
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 | |||
:<math>\mbox{ch} \colon K_0(X) \to A(X, {\Bbb Q}),</math> | |||
where | |||
:<math>A_d(X,{\Bbb Q})\,</math> | |||
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]] | |||
:<math>H^{2 \mathrm{dim}(X) - 2d}(X, {\Bbb Q}).</math> | |||
Now consider a [[proper morphism]] | |||
:<math>f \colon X \to Y\,</math> | |||
between smooth quasi-projective schemes and a bounded complex of sheaves <math>{\mathcal F^\bull}.</math> | |||
The '''Grothendieck–Riemann–Roch theorem''' relates the pushforward map | |||
:<math>f_{\mbox{!}} = \sum (-1)^i R^i f_* \colon K_0(X) \to K_0(Y)</math> | |||
and the pushforward | |||
:<math>f_* \colon A(X) \to A(Y),\,</math> | |||
by the formula | |||
:<math> \mbox{ch}(f_{\mbox{!}}{\mathcal F}^\bull)\mbox{td}(Y) = f_* (\mbox{ch}({\mathcal F}^\bull) \mbox{td}(X) ). </math> | |||
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 | |||
:<math> \mbox{ch}(f_{\mbox{!}}{\mathcal F}^\bull) = f_* (\mbox{ch}({\mathcal F}^\bull) \mbox{td}(T_f) ),</math> | |||
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''. | |||
==Generalising and specialising== | |||
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]]. | |||
The [[arithmetic Riemann–Roch theorem]] extends the Grothendieck–Riemann–Roch theorem to [[arithmetic 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 == | |||
[[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. | |||
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]]. | |||
== | ==Notes== | ||
{{reflist}} | |||
==References== | |||
* {{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 }} | |||
* {{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 }} | |||
*{{cite book | |||
| last = Berthelot | |||
| first = Pierre | |||
| authorlink = Pierre Berthelot (mathematician) | |||
| coauthors = [[Alexandre Grothendieck]], [[Luc Illusie]], eds. | |||
| 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''') | |||
| year = 1971 | |||
| publisher = [[Springer Science+Business Media|Springer-Verlag]] | |||
| location = Berlin; New York | |||
| language = French | |||
| pages = xii+700 | |||
| nopp = true | |||
|doi=10.1007/BFb0066283 | |||
|isbn= 978-3-540-05647-8 | |||
}} | |||
== | ==External links== | ||
* 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]]. | |||
{{DEFAULTSORT:Grothendieck-Hirzebruch-Riemann-Roch theorem}} | |||
[[Category:Topological methods of algebraic geometry]] | |||
[[Category:Theorems in algebraic geometry]] | |||
Revision as of 10:25, 22 January 2014
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–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–Riemann–Roch theorem can be proved using the index theorem for families. Alexander Grothendieck gave a first proof in a 1957 manuscript, later published.[1] Armand Borel and Jean-Pierre Serre wrote up and published Grothendieck's proof in 1958.[2] Later, Grothendieck and his collaborators simplified and generalized the proof.[3]
Formulation
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
where
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
Now consider a proper morphism
between smooth quasi-projective schemes and a bounded complex of sheaves
The Grothendieck–Riemann–Roch theorem relates the pushforward map
and the pushforward
by the formula
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 sequences, we can rewrite the Grothendieck–Riemann–Roch formula as
where Tf is the relative tangent sheaf of f, defined as the element TX − f*TY in K0(X). For example, when f is a smooth morphism, Tf is simply a vector bundle, known as the tangent bundle along the fibers of f.
Generalising and specialising
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.
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.
History
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.
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 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 categorical approach to a hard piece of analysis. Moreover, Grothendieck introduced K-groups, as discussed above, which paved the way for algebraic K-theory.
Notes
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References
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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
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External links
- The thread "how does one understand GRR? (Grothendieck Riemann Roch)" on MathOverflow.