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, 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 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. 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. See also Kawasaki's Riemann–Roch formula Notes References External links The Grothendieck-Riemann-Roch