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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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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 Mar 6 '15 at 23:05

Well, you could read SGA. But my two favorite sources for this material are here for an abstract treatment that doesn't (as far as I remember) talk specifically about quasi-coherent sheaves, and here for a considerably longer but readable treatement that does.

(More precisely, the first reference has a section that basically takes various properties of quasi-coherent sheaves as axioms and proceeds from there. If you're willing to accept these axioms without working through all the geometry, that's probably the reference you're looking for.)

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Lipman's notes (now published in Springer LN 1960) is an excellent source for the theory. As Damian said in a previous comment there is no hope to define a "lower shrieck" basically because extension by zero from an open subset destroys quasi-coherence. – Leo Alonso Mar 23 '13 at 17:51

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