Sheaf on an algebraic stack

☆ Save On Wikipedia ↗

In algebraic geometry, a quasi-coherent sheaf on an algebraic stack X {\displaystyle {\mathfrak {X}}} {\displaystyle {\mathfrak {X}}} 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 ξ {\displaystyle \xi } {\displaystyle \xi } in X ( S ) {\displaystyle {\mathfrak {X}}(S)} {\displaystyle {\mathfrak {X}}(S)}, a quasi-coherent sheaf F ξ {\displaystyle F_{\xi }} {\displaystyle F_{\xi }} on S together with maps implementing the compatibility conditions among F ξ {\displaystyle F_{\xi }} {\displaystyle F_{\xi }}'s.

For a Deligne–Mumford stack, there is a simpler description in terms of a presentation U → X {\displaystyle U\to {\mathfrak {X}}} {\displaystyle U\to {\mathfrak {X}}}: a quasi-coherent sheaf on X {\displaystyle {\mathfrak {X}}} {\displaystyle {\mathfrak {X}}} is one obtained by descending a quasi-coherent sheaf on U.[1] 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 (Arbarello, Cornalba & Griffiths 2011, Ch. XIII., Definition 2.1.)

Let X {\displaystyle {\mathfrak {X}}} {\displaystyle {\mathfrak {X}}} 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 X {\displaystyle {\mathfrak {X}}} {\displaystyle {\mathfrak {X}}} is the data consisting of:

  1. for each object ξ {\displaystyle \xi } {\displaystyle \xi }, a quasi-coherent sheaf F ξ {\displaystyle F_{\xi }} {\displaystyle F_{\xi }} on the scheme p ( ξ ) {\displaystyle p(\xi )} {\displaystyle p(\xi )},
  2. for each morphism H : ξ → η {\displaystyle H:\xi \to \eta } {\displaystyle H:\xi \to \eta } in X {\displaystyle {\mathfrak {X}}} {\displaystyle {\mathfrak {X}}} and h = p ( H ) : p ( ξ ) → p ( η ) {\displaystyle h=p(H):p(\xi )\to p(\eta )} {\displaystyle h=p(H):p(\xi )\to p(\eta )} in the base category, an isomorphism
    ρ H : h ∗ ( F η ) → ≃ F ξ {\displaystyle \rho _{H}:h^{*}(F_{\eta }){\overset {\simeq }{\to }}F_{\xi }} {\displaystyle \rho _{H}:h^{*}(F_{\eta }){\overset {\simeq }{\to }}F_{\xi }}
satisfying the cocycle condition: for each pair H 1 : ξ 1 → ξ 2 , H 2 : ξ 2 → ξ 3 {\displaystyle H_{1}:\xi _{1}\to \xi _{2},H_{2}:\xi _{2}\to \xi _{3}} {\displaystyle H_{1}:\xi _{1}\to \xi _{2},H_{2}:\xi _{2}\to \xi _{3}},
h 1 ∗ h 2 ∗ F ξ 3 → h 1 ∗ ( ρ H 2 ) h 1 ∗ F ξ 2 → ρ H 1 F ξ 1 {\displaystyle h_{1}^{*}h_{2}^{*}F_{\xi _{3}}{\overset {h_{1}^{*}(\rho _{H_{2}})}{\to }}h_{1}^{*}F_{\xi _{2}}{\overset {\rho _{H_{1}}}{\to }}F_{\xi _{1}}} {\displaystyle h_{1}^{*}h_{2}^{*}F_{\xi _{3}}{\overset {h_{1}^{*}(\rho _{H_{2}})}{\to }}h_{1}^{*}F_{\xi _{2}}{\overset {\rho _{H_{1}}}{\to }}F_{\xi _{1}}} equals h 1 ∗ h 2 ∗ F ξ 3 = ∼ ( h 2 ∘ h 1 ) ∗ F ξ 3 → ρ H 2 ∘ H 1 F ξ 1 {\displaystyle h_{1}^{*}h_{2}^{*}F_{\xi _{3}}{\overset {\sim }{=}}(h_{2}\circ h_{1})^{*}F_{\xi _{3}}{\overset {\rho _{H_{2}\circ H_{1}}}{\to }}F_{\xi _{1}}} {\displaystyle h_{1}^{*}h_{2}^{*}F_{\xi _{3}}{\overset {\sim }{=}}(h_{2}\circ h_{1})^{*}F_{\xi _{3}}{\overset {\rho _{H_{2}\circ H_{1}}}{\to }}F_{\xi _{1}}}.

(cf. equivariant sheaf.)

Examples

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