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 (in a sense).

Constructible sheaves (e.g., as ℓ-adic sheaves) can also be defined on an algebraic stack and they appear as coefficients of cohomology of a stack.

Definition

The following definition is

Let 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:

1. for each object , a quasi-coherent sheaf on the scheme ,
2. for each morphism in and in the base category, an isomorphism
3. :

satisfying the cocycle condition: for each pair ,

: equals .

(cf. equivariant sheaf.)

Examples

• The Hodge bundle on the moduli stack of algebraic curves of fixed genus.

ℓ-adic formalism

The ℓ-adic formalism (theory of ℓ-adic sheaves) extends to algebraic stacks.

See also

• Hopf algebroid - encodes the data of quasi-coherent sheaves on a prestack presentable as a groupoid internal to affine schemes (or projective schemes using graded Hopf algebroids)

Notes

References

• Editorial note: This paper corrects a mistake in Laumon and Moret-Bailly's Champs algébriques.

External links

• https://mathoverflow.net/questions/69035/the-category-of-l-adic-sheaves
• http://math.stanford.edu/~conrad/Weil2seminar/Notes/L16.pdf Adic Formalism, Part 2 Brian Lawrence March 1, 2017