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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4$\begingroup$ 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). $\endgroup$– Damian RösslerCommented Mar 23, 2013 at 9:22
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1$\begingroup$ @DamianRössler: what do you mean by "you have to expand the category"? $\endgroup$– bananastackCommented Mar 6, 2015 at 23:05
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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.)
answered Mar 23, 2013 at 4:31
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3$\begingroup$ 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. $\endgroup$ Commented Mar 23, 2013 at 17:51