Sheaf on an algebraic stack
In algebraic geometry, a quasi-coherent sheaf on an algebraic stack is a generalization of a quasi-coherent sheaf on a scheme. The most concrete description is that it is a data that consists of, for each a scheme S in the base category and in, a quasi-coherent sheaf on S together with maps implementing the compatibility conditions among 's.
For a Deligne–Mumford stack, there is a simpler description in terms of a presentation : a quasi-coherent sheaf on is one obtained by descending a quasi-coherent sheaf on U. A quasi-coherent sheaf on a Deligne–Mumford stack generalizes an orbibundle.
Constructible sheaves can also be defined on an algebraic stack and they appear as coefficients of cohomology of a stack.
Definition
The following definition isLet be a category fibered in groupoids over the category of schemes of finite type over a field with the structure functor p. Then a quasi-coherent sheaf on is the data consisting of:
- for each object, a quasi-coherent sheaf on the scheme,
- for each morphism in and in the base category, an isomorphism
- :
Examples
- The Hodge bundle on the moduli stack of algebraic curves of fixed genus.