In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the Grothendieck–Riemann–Roch theorem is a far-reaching result on
coherent cohomology. It is a generalisation of the
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 generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algebra ...
, about
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a com ...
s, which is itself a generalisation of the classical
Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
for
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
s on
compact Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versio ...
s.
Riemann–Roch type theorems relate
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
s of the
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
of a
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
with their
topological degree In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely connected to fixed-point theory. When one solution ...
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
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
between two manifolds (or more general
schemes) and changes the theorem from a statement about a single bundle, to one applying to
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
es of
sheaves.
The theorem has been very influential, not least for the development of the
Atiyah–Singer index theorem. Conversely,
complex analytic
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...
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.
Armand Borel
Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in alg ...
and
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
wrote up and published Grothendieck's proof in 1958. Later, Grothendieck and his collaborators simplified and generalized the proof.
Formulation
Let ''X'' be a
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
quasi-projective scheme In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used ...
over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
. Under these assumptions, the
Grothendieck group
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic i ...
of
bounded complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
es of
coherent sheaves
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
is canonically isomorphic to the Grothendieck group of bounded complexes of finite-rank vector bundles. Using this isomorphism, consider the
Chern character
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
(a rational combination of
Chern classes
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
) as a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
ial transformation:
:
where
is the
Chow group
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties ( ...
of cycles on ''X'' of dimension ''d'' modulo
rational equivalence
Rationality is the Quality (philosophy), quality of being guided by or based on reasons. In this regard, a person Action (philosophy), acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong e ...
,
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
ed with the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s. In case ''X'' is defined over the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, the latter group maps to the topological
cohomology group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
:
:
Now consider a
proper morphism In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.
Some authors call a proper variety over a field ''k'' a complete variety. For example, every projective variety over a field ...
between smooth quasi-projective schemes and a bounded complex of sheaves
on
The Grothendieck–Riemann–Roch theorem relates the pushforward map
:
(alternating sum of
higher direct images) and the pushforward
:
by the formula
:
Here
is the
Todd genus 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 encounter ...
of (the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
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
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
s, we can rewrite the Grothendieck–Riemann–Roch formula as
:
where
is the relative tangent sheaf of ''f'', defined as the element
in
. For example, when ''f'' is a
smooth morphism In algebraic geometry, a morphism f:X \to S between schemes is said to be smooth if
*(i) it is locally of finite presentation
*(ii) it is flat, and
*(iii) for every geometric point \overline \to S the fiber X_ = X \times_S is regular.
(iii) means ...
,
is simply a vector bundle, known as the tangent bundle along the fibers of ''f''.
Using
''A''1-homotopy theory, the Grothendieck–Riemann–Roch theorem has been extended by to the situation where ''f'' is a
proper map
In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.
Definition
There are several competing definit ...
between two smooth schemes.
Generalising and specialising
Generalisations of the theorem can be made to the non-smooth case by considering an appropriate generalisation of the combination
and to the non-proper case by considering
cohomology with compact support In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support.
Singular cohomology with compact support
Let X be a topological space. Then
:\d ...
.
The
arithmetic Riemann–Roch theorem In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions.
Background
The main motivation behind Arakelov geometry is th ...
extends the Grothendieck–Riemann–Roch theorem to
arithmetic scheme In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions.
Background
The main motivation behind Arakelov geometry is th ...
s.
The
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 generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algebra ...
is (essentially) the special case where ''Y'' is a point and the field is the field of complex numbers.
A version of Riemann–Roch theorem for oriented cohomology theories was proven by Ivan Panin and Alexander Smirnov. It is concerned with multiplicative operations between algebraic oriented cohomology theories (such as
algebraic cobordism In mathematics, algebraic cobordism is an analogue of complex cobordism for smooth quasi-projective schemes over a field. It was introduced by .
An oriented cohomology theory on the category of smooth quasi-projective schemes Sm over a field ''k'' ...
). The Grothendieck-Riemann-Roch is a particular case of this result, and the Chern character comes up naturally in this setting.
Examples
Vector bundles on a curve
A vector bundle
of rank
and degree
(defined as the degree of its determinant; or equivalently the degree of its first Chern class) on a smooth projective curve over a field
has a formula similar to
Riemann–Roch for line bundles. If we take
and
a point, then the Grothendieck–Riemann–Roch formula can be read as
:
hence,
:
This formula also holds for coherent sheaves of rank
and degree
.
Smooth proper maps
One of the advantages of the Grothendieck–Riemann–Roch formula is it can be interpreted as a relative version of the Hirzebruch–Riemann–Roch formula. For example, a smooth morphism
has fibers which are all equi-dimensional (and isomorphic as topological spaces when base changing to
). This fact is useful in moduli-theory when considering a moduli space
parameterizing smooth proper spaces. For example,
David Mumford
David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded t ...
used this formula to deduce relationships of the Chow ring on the
moduli space of algebraic curves
In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending ...
.
Moduli of curves
For the moduli stack of genus
curves (and no marked points)
there is a universal curve
where
(is the moduli stack of curves of genus
and one marked point. Then, he defines the tautological classes
:
where
and
is the relative dualizing sheaf. Note the fiber of
over a point
this is the dualizing sheaf
. He was able to find relations between the
and
describing the
in terms of a sum of
(corollary 6.2) on the chow ring
of the smooth locus using Grothendieck–Riemann–Roch. Because
is a smooth
Deligne–Mumford stack
In algebraic geometry, a Deligne–Mumford stack is a stack ''F'' such that
Pierre Deligne and David Mumford introduced this notion in 1969 when they proved that moduli spaces of stable curves of fixed arithmetic genus are proper smooth Deligne ...
, he considered a covering by a scheme
which presents