# Six operations for (quasi)-coherent sheaves

Can someone point me to a reference with an overview of what Grothendieck's six operations formalism looks like for schemes and (quasi)-coherent sheaves (or derived category objects with (quasi)-coherent cohomology sheaves)? Do I have to read Residues and Duality? I'm particularly curious about what the two shriek functors look like. Are there distinguished triangles associated to a closed immersion and its open complement? What kind of theorems about commutation of pushforwards with pullbacks are true?

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I think that you can find a clear treatment of this, and the answers to your questions, in the book of Cisinski and Deglise (arxiv.org/abs/0912.2110v3). –  Adeel Mar 23 '13 at 4:30
There is no lower shriek functor in the category of quasi-coherent sheaves, unless the morphism is proper. See the appendix by Deligne to Hartshorne's "Residues and Duality" (you have to expand the category to get a good lower shriek). –  Damian Rössler Mar 23 '13 at 9:22
@DamianRössler: what do you mean by "you have to expand the category"? –  bananastack 1 hour ago