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.
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.
of bounded complexes of coherent sheaves is canonically isomorphic to the Grothendieck group of bounded complexes of finite-rank vector bundles.
between smooth quasi-projective schemes and a bounded complex of sheaves
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
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 Navarro & Navarro (2017) to the situation where f is a proper map between two smooth schemes.
and to the non-proper case by considering cohomology with compact support.
A version of Riemann–Roch theorem for oriented cohomology theories was proven by Ivan Panin and Alexander Smirnov.
The Grothendieck-Riemann-Roch is a particular case of this result, and the Chern character comes up naturally in this setting.
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
For example, David Mumford used this formula to deduce relationships of the Chow ring on the moduli space of algebraic curves.
is a smooth Deligne–Mumford stack, he considered a covering by a scheme
there are the relations which can be deduced by analyzing the Chern character of
, such as the moduli space of pointed algebraic curves
, admits an embedding into a projective space, hence is a quasi-projective variety.
[9] has the family of curves with sections corresponding to the marked points.
It turns out that is an ample line bundle[9]pg 209, hence the coarse moduli space
Alexander Grothendieck'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.
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.