Comparison of Local Ext Sheaves

Proof It is easy to see that we may assume that F is a single coherent sheaf, and G is a single injective object of Qch(X ). To prove this case, it suffices to show that 

) Qch(X ) Qch(X.) has a faithful exact left adjoint p, there exists some injective object I of Qch(X ) such that G is a direct summand of (p,) I. We may assume that G = (p.) I. As 

Let G — I be a K-injective resolution in Mod(X,) such that I is bounded below. Let C be the mapping cone of this. Since ( )o has an exact left adjoint, Gq Iq is a A -injective resolution in A (Mod(Xo)). So it suffices to show that Homy (Fo, Cq) is exact. As each term of F is equivariant, this complex is isomorphic to Homy, (F, C)o, which is exact by the lemma.  

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