Modules over a Scheme

1 Basic Properties Definition 1. Let X be a scheme. We denote the category of sheaves of OX -modules by OXMod or Mod(X). The full subcategories of qausi-coherent and coherent modules are denoted by Qco(X) and Coh(X) respectively. Mod(X) is a grothendieck abelian category, and it follows from (AC,Lemma 39) and (H, II 5.7) that Qco(X) is an abelian subcategory of Mod(X). If X is noetherian, then Coh(X) is also an abelian subcategory. The fact that Qco(X) is an abelian subcategory of Mod(X) means that the following operations preserve the quasi-coherent property: • If φ : F −→ G is a morphism of quasi-coherent sheaves and H is a quasi-coherent submodule of G , then φ−1H is a quasi-coherent submodule of F . • If F is a quasi-coherent sheaf and G1, . . . ,Gn are quasi-coherent submodules then the finite union ∑ i Gi is quasi-coherent (we mean the categorical union in Mod(X), but it follows that this submodule is also the union in Qco(X)). To see this, realise the union as the image of a morphism out of a finite coproduct. • Finite limits and colimits of quasi-coherent modules are quasi-coherent.