Sheafified Duality—Preliminary Form

Prooff Using (2.6.2) and (3.2.3), and checking all the requisite commu-tativities, one shows for fixed F G Dqc(U) that the composite duality map 

One of our goals is to prove this Corollary under considerably weaker hypotheses—see (4.4.2) below. For this purpose we need some facts about pseudo-coherence, reviewed in the following section. 

In the rest of these exercises, assume all schemes to be quasi-compact and separated, so that by (3.9.6), j induces an equivalence D(A  

In particular, if / is an open immersion then there is a functorial isomorphism 

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