Quasi-Coherent Sheaves Over a Diagram of Schemes

It suffices to prove that the induced sequence (7.2) is exact. To verify this, it suffices to prove that the sequence is exact after taking the section at 4 ,h) j,V) — i,U)) G Zar(A,)/(L [/) with V = Spec 5 being affine. We have a commutative diagram 

Definition 7.4. We say that M G Mod(X.) is coherent if it is equivariant and locally coherent. We denote the full subcategory of Mod(X,) consisting of coherent objects by Coh(A,). 

1 The full subcategory Lqc(X.) of Mod(X ) consisting of locally quasi-coherent objects is a plump subcategory. 

2 If Ai is equivariant resp. locally quasi-coherent, quMsi-coherent), then so is Ai ° .  

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Quasi-Coherent Sheaves Over a Diagram of Schemes... 

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