Grothendieck–Riemann–Roch theorem


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 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 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. Armand Borel and Jean-Pierre Serre 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 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 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 on
The Grothendieck–Riemann–Roch theorem relates the pushforward map
and the pushforward
by the formula
Here is the Todd genus 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 is the relative tangent sheaf of f, defined as the element in. For example, when f is a smooth morphism, is simply a vector bundle, known as the tangent bundle along the fibers of f.
Using A1-homotopy theory, the Grothendieck–Riemann–Roch theorem has been extended by to the situation where f is a proper map 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.
The arithmetic Riemann–Roch theorem extends the Grothendieck–Riemann–Roch theorem to arithmetic schemes.
The Hirzebruch–Riemann–Roch theorem is the special case where Y is a point and the field is the field of complex numbers.
The 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. The Grothendieck-Riemann-Roch is a particular case of it, and the Chern character comes up naturally in that setting.

Examples

Vector bundles on a curve

A vector bundle of rank and degree 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.

Degree of a vector bundle on a curve

Notice the degree of a vector bundle on a curve it included in the formula. Since, the degree of could be taken as

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. This fact is useful in moduli-theory when considering a moduli space parameterizing smooth proper spaces. For example, David Mumford used this formula to deduce relationships of the Chow ring on the moduli space of algebraic curves.

Moduli of curves

For the moduli stack of genus curves there is a universal curve where on the chow ring of the smooth locus using Grothendieck-Riemann-Roch. Because is a smooth Deligne–Mumford stack, he considered a covering by a scheme which presents for some finite group. He uses Grothendieck-Riemann-Roch on to get
Because
this gives the formula
The computation of can then be reduced even further. In even dimensions,
Also, on dimension 1,
where is a class on the boundary. In the case and on the smooth locus there are the relations
which can be deduced by analyzing the Chern character of.

Closed embedding

Closed embeddings have a description using the Grothendieck-Riemann-Roch formula as well, showing another non-trivial case where the formula holds. For a smooth variety of dimension and a subvariety of codimension, there is the formula
Using the short exact sequence
there is the formula
for the ideal sheaf since.

Applications

Quasi-projectivity of moduli spaces

Grothendieck-Riemann-Roch can be used in proving that a coarse moduli space, such as the moduli space of pointed algebraic curves, admits an embedding into a projective space, hence is a quasi-projective variety. This can be accomplished by looking at canonically associated sheaves on and studying the degree of associated line bundles. For instance, has the family of curves
with sections
corresponding to the marked points. Since each fiber has the canonical bundle, there are the associated line bundles
and.
It turns out that
is an ample line bundlepg 209, hence the coarse moduli space is quasi-projective.

History

's version of the Riemann–Roch theorem was originally conveyed in a letter to Jean-Pierre Serre around 1956–1957. It was made public at the initial Bonn Arbeitstagung, in 1957. Serre and Armand Borel subsequently organized a seminar at Princeton University 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.