Quasi-coherent sheaf

Definition

Let $X$ be a scheme. A quasi-coherent sheaf on $X$ is a sheaf $\mathcal F$ of $\mathcal O_X$ -modules such that on every affine open $U=\operatorname{Spec}A$ , the restriction $\mathcal F|_U$ is isomorphic to the sheaf $\widetilde M$ associated to an $A$ -module $M$ .

The category is denoted $\operatorname{QCoh}(X)$ . In derived algebraic geometry the same notation often denotes a stable derived category rather than its ordinary abelian heart.

Examples

The structure sheaf $\mathcal O_X$ , vector bundles, and sheaves associated to modules on an affine scheme are quasi-coherent. Coherent sheaves impose additional finiteness conditions; see coherent sheaf.

Geometric-Langlands warning

The early schematic slogan

D\text{-}\operatorname{mod}(\operatorname{Bun}_G) \simeq \operatorname{QCoh}(\operatorname{LocSys}_{\widehat G})

is correct for tori after suitable conventions but is too small on the spectral side for general reductive $G$ . The corrected category uses ind-coherent sheaves with nilpotent singular support.

References

Alexander Grothendieck, “Éléments de géométrie algébrique I,” Publications Mathématiques de l’IHÉS 4 (1960). DOI.