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| The '''Riemann–Roch theorem''' is an important tool in [[mathematics]], specifically in [[complex analysis]] and [[algebraic geometry]], for the computation of the dimension of the space of [[meromorphic function]]s with prescribed zeroes and allowed [[pole (complex analysis)|poles]]. It relates the complex analysis of a connected [[Compact space|compact]] [[Riemann surface]] with the surface's purely topological [[genus (mathematics)|genus]] ''g'', in a way that can be carried over into purely algebraic settings.
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| Initially proved as '''Riemann's inequality''' by {{harvtxt|Riemann|1857}}, the theorem reached its definitive form for Riemann surfaces after work of [[Bernhard Riemann|Riemann]]'s short-lived student {{harvs|txt|authorlink=Gustav Roch|first=Gustav|last=Roch|year=1865}}. It was later generalized to [[algebraic curve]]s, to higher-dimensional [[algebraic variety|varieties]] and beyond.
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| == Preliminary notions==
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| [[File:Triple torus illustration.png|right|thumb|A Riemann surface of genus 3.]]
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| A [[Riemann surface]] ''X'' is a [[topological space]] that is locally homeomorphic to an open subset of '''C''', the set of complex numbers. In addition, the [[transition map]]s between these open subsets are required to be [[holomorphic]]. The latter condition allows to transfer the notions and methods of [[complex analysis]] dealing with holo- and [[meromorphic function]]s on '''C''' to the surface ''X''. For the purposes of the Riemann–Roch theorem, the surface ''X'' is always assumed to be [[compact topological space|compact]]. Colloquially speaking, the [[Genus_(mathematics)|genus]] ''g'' of a Riemann surface is its number of handles; for example the genus of the Riemann surface shown at the right is three. More precisely, the genus defined as half of the first [[Betti number]], i.e., half of the '''C'''-dimension of the first [[singular homology]] group H<sub>1</sub>(''X'', '''C''') with complex coefficients. The genus [[Classification theorem|classifies]] compact Riemann surfaces [[up to]] [[homeomorphism]], i.e., two such surfaces are homeomorphic (but not necessarily [[diffeomorphic]]) if and only if their genus is the same. Therefore, the genus is an important topological invariant of a Riemann surface. On the other hand, [[Hodge theory]] shows that the genus coincides with the ('''C'''-)dimension of the space of holomorphic one-forms on ''X'', so the genus also encodes complex-analytic information about the Riemann surface.<ref>Griffith, Harris, p. 116, 117</ref>
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| A [[Divisor (algebraic geometry)#Weil divisor|divisor]] ''D'' is an element of the [[free abelian group]] on the points of the surface. Equivalently, a divisor is a finite linear combination of points of the surface with integer coefficients.
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| Any meromorphic function ''f'' gives rise to a divisor denoted (''f'') defined as
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| :<math>(f):=\sum_{z_\nu \in R(f)} s_\nu z_\nu</math>
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| where ''R''(''f'') is the set of all zeroes and poles of ''f'', and ''s<sub>ν</sub>'' is given by
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| :<math>s_\nu :=\begin{cases} a & \text{if } z_\nu \text{ is a zero of order }a \\
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| -a & \text{if } z_\nu \text{ is a pole of order }a. \end{cases}</math>
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| The set ''R''(''f'') is known to be finite; this is a consequence of ''X'' being compact and the fact that the zeros of a (non-zero) holomorphic function do not have an [[accumulation point]]. Therefore, (''f'') is well-defined. Any divisor of this form is called [[principal divisor]]. Two divisors that differ by a principal divisor are called [[linearly equivalent]]. The divisor of a meromorphic [[1-form]] is defined similarly. A divisor of a global meromorphic 1-form is called the [[canonical divisor]] (usually denoted ''K''). Any two meromorphic 1-forms will yield linearly equivalent divisors, so the canonical divisor is uniquely determined up to linear equivalence (hence "the" canonical divisor).
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| The symbol ''deg''(''D'') denotes the ''degree'' (occasionally also called index) of the divisor ''D'', i.e. the sum of the coefficients occurring in ''D''. It can be shown that the divisor of a global meromorphic function always has degree 0, so the degree of the divisor depends only on the linear equivalence class.
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| The number ''l''(''D'') is the quantity that is of primary interest: the [[Dimension (vector space)|dimension]] (over '''[[complex number|C]]''') of the vector space of meromorphic functions ''h'' on the surface, such that all the coefficients of (''h'') + ''D'' are non-negative. Intuitively, we can think of this as being all meromorphic functions whose poles at every point are no worse than the corresponding coefficient in ''D''; if the coefficient in ''D'' at ''z'' is negative, then we require that ''h'' has a zero of at least that [[Multiplicity (mathematics)#Multiplicity of a root of a polynomial|multiplicity]] at ''z'' – if the coefficient in ''D'' is positive, ''h'' can have a pole of at most that order. The vector spaces for linearly equivalent divisors are naturally isomorphic through multiplication with the global meromorphic function (which is well-defined up to a scalar).
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| == Statement of the theorem ==
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| The Riemann–Roch for a compact Riemann surface of genus ''g'' with canonical divisor ''K'' states
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| :''l''(''D'') − ''l''(''K'' − ''D'') = ''deg''(''D'') − ''g'' + 1.
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| Typically, the number ''l''(''D'') is the one of interest, while ''l''(''K'' − ''D'') is thought of as a correction term (also called index of speciality<ref>Stichtenoth p.22</ref><ref>Mukai pp.295-297</ref>) so the theorem may be roughly paraphrased by saying
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| :''dimension'' − ''correction'' = ''degree'' − ''g'' + 1.
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| The correction term ''l''(''K'' − ''D'') is always non-negative, so that
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| :''l''(D) ≥ ''deg''(''D'') − ''g'' + 1.
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| This is called ''Riemann's inequality''. ''Roch's part'' of the statement is the description of the possible difference between the sides of the inequality. On a general Riemann surface of genus ''g'', K has degree 2''g'' − 2, independently of the meromorphic form chosen to represent the divisor. This follows from putting ''D'' = 0 in the theorem. In particular, as long as ''D'' has degree at least 2''g'' − 1, the correction term is 0, so that
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| :''l''(D) = ''deg''(''D'') − ''g'' + 1.
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| The theorem will now be illustrated for surfaces of low genus. There are also a number other closely related theorems: an equivalent formulation of this theorem using [[line bundle]]s and a generalization of the theorem to [[algebraic curve]]s.
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| ===Examples===
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| The theorem will be illustrated by picking a point ''P'' on the surface in question and regarding the sequence of numbers
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| :''l''(''n'' · ''P''), ''n'' ≥ 0
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| i.e., the dimension of the space of functions that are holomorphic everywhere except at ''P'' where the function is allowed to have a pole of order at most ''n''. For ''n'' = 0, the functions are thus required to be [[entire function|entire]], i.e., holomorphic on the whole surface ''X''. By [[Liouville's theorem (complex analysis)|Liouville's theorem]], such a function is necessarily constant. Therefore ''l''(0) = 1. In general, the sequence ''l''(''n'' · ''P'') is an increasing sequence.
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| ====Genus zero====
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| The [[Riemann sphere]] (also called [[complex projective line]]) is [[simply-connected]] and hence its first singular homology is zero. In particular its genus is zero. The sphere can be covered by two copies of '''C''', with [[transition map]] being given by
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| :<math>\mathbf C^\times \ni z \mapsto 1/z.</math>
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| Therefore the form ω = d''z'' on one copy of '''C''' extends to a meromorphic form on the Riemann sphere: it has a double pole at infinity, since
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| :<math>d(1/z) = -\frac 1{z^2} dz.</math>
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| Thus, its divisor ''K'' := div(ω) = −2''P'' (where ''P'' is the point at infinty).
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| Therefore, the theorem says that the sequence ''l''(''n'' · ''P'') reads
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| :1, 2, 3, ... .
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| This sequence can also be read off from the theory of [[partial fraction]]s. Conversely if this sequence starts this way, then ''g'' must be zero.
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| ====Genus one====
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| [[File:Torus cycles.png|right|thumb|A torus.]]
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| The next case is a Riemann surface of genus ''g'' = 1, such as a [[torus]] '''C''' / Λ, where Λ is a two-dimensional [[lattice (group)|lattice]] (a group isomorphic to '''Z'''<sup>2</sup>). Its genus is one: its first singular homology group is freely generated by two loops, as shown in the illustration at the right. The standard complex coordinate ''z'' on '''C''' yields a one-form ω = d''z'' on ''X'' that is everywhere holomorphic, i.e., has no poles at all. Therefore, ''K'', the divisor of ω is zero.
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| On this surface, this sequence is
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| :1, 1, 2, 3, 4, 5 ... ;
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| and this characterises the case ''g'' = 1. Indeed, for ''D'' = 0, ''l''(''K'' − ''D'') = ''l''(0) = 1, as was mentioned above. For ''D'' = ''nP'' with ''n'' > 0, the degree of ''K'' − ''D'' is strictly negative, so that the correction term is 0. The sequence of dimensions can also be derived from the theory of [[elliptic function]]s.
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| ====Genus two and beyond====
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| For ''g'' = 2, the sequence mentioned above is
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| :1, 1, ?, 2, 3, ... .
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| It is shown from this that the ? term of degree 2 is either 1 or 2, of course depending on the point. It can be proven that in any genus 2 curve there are exactly six points whose sequences are 1, 1, 2, 2, ... and the rest of the points have the generic sequence 1, 1, 1, 2, ... In particular, a genus 2 curve is a [[hyperelliptic curve]]. For ''g'' > 2 it is always true that at most points the sequence starts with ''g+1'' ones and there are finitely many points with other sequences (see [[Weierstrass point]]s).
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| ===Riemann-Roch for line bundles===
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| Using the close correspondence between divisors and [[holomorphic line bundle]]s on a Riemann surface, the theorem can also be stated in a different, yet equivalent way: let ''L'' be a holomorphic line bundle on ''X''. Let <math>H^0(X,L)</math> denote the space of holomorphic sections of ''L''. This space will be finite-dimensional; its dimension is denoted <math> h^0(X,L)</math>. Let ''K'' denote the [[canonical bundle]] on ''X''. Then, the Riemann–Roch theorem states that
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| :<math>h^0(X,L)-h^0(X,L^{-1}\otimes K)=\textrm{deg}(L)+1-g.</math>
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| The theorem of the previous section is the special case of when ''L'' is a [[point bundle]]. The theorem can be applied to show that there are ''g'' holomorphic sections of ''K'', or [[one-form]]s, on ''X''. Taking ''L'' to be the trivial bundle, <math> h^0(X,L)=1</math> since the only holomorphic functions on ''X'' are constants. The degree of ''L'' is zero, and <math>L^{-1}</math> is the trivial bundle. Thus,
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| :<math>1-h^0(X,K)=1-g.</math>
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| Therefore, <math>h^0(X,K)=g</math>, proving that there are ''g'' holomorphic one-forms.
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| ===Riemann-Roch theorem for algebraic curves===
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| Every item in the above formulation of the Riemann–Roch theorem for divisors on Riemann surfaces has an analogue in [[algebraic geometry]]. The analogue of a Riemann surface is a [[non-singular variety|non-singular]] [[algebraic curve]] ''C'' over a field ''k''. The difference in terminology (curve vs. surface) is because the dimension of a Riemann surface as a real [[manifold]] is two, but one as a complex manifold. The compactness of a Riemann surface is paralleled by the condition that the algebraic curve be [[complete variety|complete]], which is equivalent to being [[projective variety|projective]]. Over a general field ''k'', there is no good notion of singular (co)homology. The so-called [[geometric genus]] is defined as
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| :<math>g(C) := dim_k \Gamma(C, \Omega^1_C)</math> | |
| i.e., as the dimension of the space of globally defined (algebraic) one-forms (see [[Kähler differential]]). Finally, meromorphic functions on a Riemann surface are locally represented as fractions of holomorphic functions. Hence they are replaced by [[rational function]]s which are locally fractions of [[regular function]]s. Thus, writing ''l''(''D'') for the dimension (over ''k'') of the space of rational functions on the curve whose poles at every point are not worse than the corresponding coefficient in ''D'', the very same formula as above holds:
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| :''l''(''D'') − ''l''(''K'' − ''D'') = ''deg''(''D'') − ''g'' + 1.
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| where ''C'' is a projective non-singular algebraic curve over an [[algebraically closed field]] ''k''. In fact, the same formula holds for projective curves over any field, except that the degree of a divisor needs to take into account [[multiplicity (mathematics)|multiplicities]] coming from the possible extensions of the base field and the [[residue field]]s of the points supporting the divisor.<ref>{{Citation | last1=Liu | first1=Qing | author1-link=Qing Liu | title=Algebraic Geometry and Arithmetic Curves | publisher=[[Oxford University Press]] | isbn=978-0-19-850284-5 | year=2002}}, Section 7.3</ref> Finally, for a proper curve over an [[Artinian ring]], the Euler characteristic of the line bundle associated to a divisor is given by the degree of the divisor (appropriately defined) plus the Euler characteristic of the structural sheaf <math>\mathcal O</math>.<ref>* {{Citation | last1=Altman | first1=Allen | last2=Kleiman | first2=Steven | author2-link=Steven Kleiman | title=Introduction to Grothendieck duality theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics, Vol. 146 | year=1970}}, Theorem VIII.1.4., p. 164</ref>
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| The smoothness assumption in the theorem can be relaxed, as well: for a (projective) curve over an algebraically closed field, all of whose local rings are [[Gorenstein ring]]s, the same statement as above holds, provided that the geometric genus as defined above is replaced by the [[arithmetic genus]] ''g''<sub>a</sub>, defined as
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| :<math>g_a := dim_k H^1(C, \mathcal O_C).</math><ref>{{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Generalized divisors on Gorenstein curves and a theorem of Noether | url=http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.kjm/1250520873 | year=1986 | journal=Journal of Mathematics of Kyoto University | issn=0023-608X | volume=26 | issue=3 | pages=375–386}}</ref>
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| (For smooth curves, the geometric genus agrees with the arithmetic one.) The theorem has also been extended to general singular curves (and higher-dimensional varieties).<ref>{{Citation | last1=Baum | first1=Paul | last2=Fulton | first2=William | author2-link=William Fulton (mathematician) | last3=MacPherson | first3=Robert | author3-link=Robert MacPherson (mathematician) | title=Riemann-Roch for singular varieties | year=1975 | journal=[[Publications Mathématiques de l'IHÉS]] | issn=1618-1913 | issue=45 | pages=101–145}}</ref>
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| == Proof ==
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| The statement for algebraic curves can be proved using [[Serre duality]]. The integer ''I(D)'' is the dimension of the space of global sections of the [[line bundle]] <math>\mathcal L(D)</math> associated to ''D'' (''cf.'' [[Cartier divisor]]). In terms of [[sheaf cohomology]], we therefore have <math>I (D) = \mathrm {dim} H^0 (X, \mathcal L(D))</math>, and likewise <math>I (\mathcal K_X - D) = \mathrm {dim} H^0 (X, \omega_X \otimes \mathcal L(D)^\vee) </math>. But Serre duality for non-singular projective varieties in the particular case of a curve states that <math>H^0 (X, \omega_X \otimes \mathcal L(D)^\vee)</math> is isomorphic to the dual <math>\simeq H^1 (X, \mathcal L (D))^\vee</math>. The left hand side thus equals the [[Euler characteristic]] of the divisor ''D''. When ''D = 0 '', we find the Euler characteristic for the structure sheaf ''ie'' <math>1-g</math> by definition. To prove the theorem for general divisor, one can then proceed by adding points one by one to the divisor and taking some off and ensure that the Euler characteristic transforms accordingly to the right hand side.
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| The theorem for compact Riemann surfaces can be deduced from the algebraic version using [[Chow's theorem]] and the [[GAGA]] principle: in fact, every compact Riemann surface is defined by algebraic equations in some complex projective space. | |
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| == Applications ==
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| An irreducible plane algebraic curve of degree ''d'' has (''d''-1)(''d''-2)/2-''g'' singularities, when properly counted. It follows that, if a curve has (''d''-1)(''d''-2)/2 different singularities, it is a [[rational curve]] and, thus, admits a rational parameterization.
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| The [[Riemann–Hurwitz formula]] concerning (ramified) maps between Riemann surfaces or algebraic curves is a consequence of the Riemann–Roch theorem.
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| [[Clifford's theorem on special divisors]] is also a consequence of the Riemann–Roch theorem. It states that for a special divisor (i.e., such that ''l''(''K'' − ''D'') > 0) satisfying ''l''(''D'') > 0, the following inequality holds:<ref>{{Citation | last1=Fulton | first1=William | author1-link=William Fulton (mathematician) | title=Algebraic curves | url=http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf | publisher=[[Addison-Wesley]] | series=Advanced Book Classics | isbn=978-0-201-51010-2 | year=1989}}, p. 109</ref>
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| :<math>l(D) \leq \frac{deg D}2+1.</math>
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| == Generalizations of the Riemann-Roch Theorem ==
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| The '''Riemann–Roch theorem for curves''' was proved for Riemann surfaces by Riemann and Roch in the 1850s and for algebraic curves by [[Friedrich Karl Schmidt]] in 1931 as he was working on [[perfect field]]s of [[Characteristic (algebra)|finite characteristic]]. Under the [http://www.rzuser.uni-heidelberg.de/~ci3/manu.html#RH hand of [[Peter Roquette]] ]:
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| <blockquote>The first main achievement of F. K. Schmidt is the discovery that the classical | |
| theorem of Riemann-Roch on compact Riemann surfaces can be transferred
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| to function fields with finite base field. Actually, his proof of the Riemann-Roch
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| theorem works for arbitrary perfect base fields, not necessarily finite.</blockquote>
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| It is foundational in the sense that the subsequent theory for curves tries to refine the information it yields (for example in the [[Brill–Noether theory]]).
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| There are versions in higher dimensions (for the appropriate notion of [[divisor (algebraic geometry)|divisor]], or [[line bundle]]). Their general formulation depends on splitting the theorem into two parts. One, which would now be called [[Serre duality]], interprets the ''l''(''K'' − ''D'') term as a dimension of a first [[sheaf cohomology]] group; with ''l''(''D'') the dimension of a zeroth cohomology group, or space of sections, the left-hand side of the theorem becomes an [[Euler characteristic]], and the right-hand side a computation of it as a ''degree'' corrected according to the topology of the Riemann surface.
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| In [[algebraic geometry]] of dimension two such a formula was found by the [[Italian school of algebraic geometry|geometers of the Italian school]]; a [[Riemann–Roch theorem for surfaces]] was proved (there are several versions, with the first possibly being due to [[Max Noether]]). So matters rested before about 1950.
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| :{{main|Riemann–Roch theorem for surfaces}}
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| An ''n''-dimensional generalisation, the [[Hirzebruch–Riemann–Roch theorem]], was found and proved by [[Friedrich Hirzebruch]], as an application of [[characteristic class]]es in [[algebraic topology]]; he was much influenced by the work of [[Kunihiko Kodaira]]. At about the same time [[Jean-Pierre Serre]] was giving the general form of Serre duality, as we now know it.
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| [[Alexander Grothendieck]] proved a far-reaching generalization in 1957, now known as the [[Grothendieck–Riemann–Roch theorem]]. His work reinterprets Riemann–Roch not as a theorem about a variety, but about a morphism between two varieties. The details of the proofs were published by Borel-Serre in 1958.
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| Finally a general version was found in [[algebraic topology]], too. These developments were essentially all carried out between 1950 and 1960. After that the [[Atiyah–Singer index theorem]] opened another route to generalization.
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| What results is that the Euler characteristic (of a [[coherent sheaf]]) is something reasonably computable. If one is interested, as is usually the case, in just one summand within the alternating sum, further arguments such as [[vanishing theorem]]s must be brought to bear.
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| == Notes ==
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| {{reflist}}
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| ==References==
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| * Borel, Armand & Serre, Jean-Pierre (1958), Le théorème de Riemann-Roch, d'après Grothendieck, Bull.S.M.F. 86 (1958), 97-136.
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| * {{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Harris | first2=Joseph | author2-link=Joe Harris (mathematician) | title=Principles of algebraic geometry | publisher=[[John Wiley & Sons]] | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 | id={{MathSciNet | id = 1288523}} | year=1994}}
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| * Grothendieck, Alexander, et al. (1966/67), Théorie des Intersections et Théorème de Riemann-Roch (SGA 6), LNM 225, Springer-Verlag, 1971.
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| * {{cite book | last=Fulton | first=William | authorlink=William Fulton (mathematician) | title=Algebraic Curves | series=Mathematics Lecture Note Series | publisher=W.A. Benjamin | year=1974 | isbn=0-8053-3081-4{{Please check ISBN|reason=Check digit (4) does not correspond to calculated figure.}} }}, available online at [http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf]
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| * {{Cite book | last1=Jost | first1=Jürgen | title=Compact Riemann Surfaces | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-33065-3 | year=2006 | postscript=<!--None-->}}, see pages 208–219 for the proof in the complex situation. Note that Jost uses slightly different notation.
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| * {{Cite book | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=[[Algebraic Geometry (book)|Algebraic Geometry]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | oclc=13348052 | id={{MathSciNet | id = 0463157}} | year=1977 | postscript=<!--None-->}}, contains the statement for curves over an algebraically closed field. See section IV.1.
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| * {{springer|title=Riemann-Roch theorem|id=p/r081980}}
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| * {{Cite book | last1=Hirzebruch | first1=Friedrich | author1-link=Friedrich Hirzebruch | title=Topological methods in algebraic geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Classics in Mathematics | isbn=978-3-540-58663-0 | id={{MathSciNet | id = 1335917}} | year=1995 | postscript=<!--None-->}}. A good general modern reference.
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| * {{cite book | author=Shigeru Mukai | authorlink=Shigeru Mukai | coauthors=William Oxbury (translator) | title=An Introduction to Invariants and Moduli | series=Cambridge studies in advanced mathematics | volume=81 | year=2003 | isbn=0-521-80906-1 | publisher=Cambridge University Press | location=New York }}
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| * ''Vector bundles on Compact Riemann Surfaces'', M.S. Narasimhan, p. 5-6.
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| *{{Cite journal|first=Bernhard|last=Riemann|year=1857|title=Theorie der Abel'schen Functionen|journal=Journal für die reine und angewandte Mathematik|volume=54|pages=115–155|url=http://resolver.sub.uni-goettingen.de/purl?PPN243919689_0054|ref=harv|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->}}
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| *{{Cite journal|last=Roch|first=Gustav|author-link=Gustav Roch|year=1865|title=Ueber die Anzahl der willkurlichen Constanten in algebraischen Functionen|journal=Journal für die reine und angewandte Mathematik|volume=64|pages=372–376|url=http://resolver.sub.uni-goettingen.de/purl?PPN243919689_0064|ref=harv|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->}}
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| *{{citation|first=Friedrich Karl|last=Schmidt|author-link=Friedrich Karl Schmidt|year=1931|title=Analytische Zahlentheorie in Körpern der Charakteristik ''p''|journal=[[Mathematische Zeitschrift]]|volume=33|pages=1–32|url=http://digreg.mathguide.de/cgi-bin/ssgfi/anzeige.pl?db=reg&ci=MathZ&id=ART&sd=y1931v33p?&nr=076522&ew=SSGFI|zbl=0001.05401|doi=10.1007/BF01174341}}
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| * {{cite book | last= Stichtenoth | first=Henning|title=Algebraic Function Fields and Codes | publisher=Springer-Verlag | year= 1993 | isbn=3-540-56489-6 }}
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| *[http://www.math.ucdavis.edu/~kapovich/ Misha Kapovich], [http://www.math.ucdavis.edu/%7Ekapovich/RS/RiemannRoch.pdf ''The Riemann–Roch Theorem] (lecture note) an elementary introduction
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| * J. Gray, [http://www.emis.de/journals/DMJDMV/xvol-icm/19/Gray.MAN.ps.gz The ''Riemann-Roch theorem and Geometry, 1854-1914''.]
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| *[http://mathoverflow.net/questions/55454/is-there-a-riemann-roch-for-smooth-projective-curves-over-an-arbitrary-field/55471#55471 Is there a Riemann-Roch for smooth projective curves over an arbitrary field?] on [[MathOverflow]]
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