- Formulation due to Baum, Fulton and MacPherson
- The Riemann–Roch theorem for Deligne–Mumford stacks
- Notes
- References
- See also
- External links
(The article discusses various generalizations of the Riemann–Roch theorem.)
Formulation due to Baum, Fulton and MacPherson
We need a few notations: 
are functors on the category C of schemes separated and locally of finite type over the base field k with proper morphisms such that
is the Grothendieck group of coherent sheaves on X,
is the rational Chow group of X,- for each proper morphism f,
are the direct image and push-forward along f, respectively.
Also, if 
is a local complete intersection morphism; i.e., it factors as a closed regular embedding 
into a smooth scheme P followed by a smooth morphism 
, then let
be the class in the Grothendieck group of vector bundles on X; it is independent of the factorization and is called the virtual tangent bundle of f.
Then the Riemann–Roch theorem amounts to the construction of a unique natural transformation:[1]
between the functors such that
- for
, where we wrote 
and 
is the Todd class of the virtual tangent sheaf.
The Riemann–Roch theorem for Deligne–Mumford stacks
Aside from algebraic spaces, no straightforward generalization is possible for stacks. The complication already appears in the orbifold case (Kawasaki's Riemann–Roch).
One of the significant applications of the theorem is that it allows one to define a virtual fundamental class in terms of the K-theoretic virtual fundamental class. More precisely,
Notes
References
- {{cite arxiv|last=Edidin|first=Dan|date=2012-05-21|title=Riemann-Roch for Deligne-Mumford stacks|eprint=1205.4742|class=math.AG}}
- {{Citation | title=Intersection theory | publisher=Springer-Verlag | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-62046-4 | mr=1644323 | year=1998 | volume=2 | edition=2nd | author=William Fulton}}
- {{cite arxiv|last=Toen|first=B.|date=1998-03-17|title=Riemann-Roch Theorems for Deligne-Mumford Stacks|eprint=math/9803076}}
- {{Cite arxiv|last=Bertrand|first=Toen|date=1999-08-18|title=K-theory and cohomology of algebraic stacks: Riemann-Roch theorems, D-modules and GAGA theorems|eprint=math/9908097}}
- {{cite arxiv|last=Lowrey|first=Parker|last2=Schürg|first2=Timo|date=2012-08-30|title=Grothendieck-Riemann-Roch for derived schemes|eprint=1208.6325|class=math.AG}}
See also
- Kawasaki's Riemann–Roch formula
External links
- https://mathoverflow.net/questions/25218/why-is-riemann-roch-for-stacks-so-hard