Timeline for Derived functor
Current License: CC BY-SA 3.0
8 events
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| Aug 7, 2014 at 3:54 | history | edited | David White | CC BY-SA 3.0 |
Fixed typo, minor spacing edits
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| Nov 30, 2010 at 12:30 | comment | added | Leo Alonso | I don't think so. The treatments I know of unbounded complexes (Spaltenstein, Bökstedt-Neeman, Franke, myself with Jeremías and Souto, Serpé... to name a few) work mostly over Grothendieck abelian categories or sheaf categories regardless of any base field. The trouble might arise if you want to look at DG modules over a DG algebra but even in this case you may use Brown representability due to the existence of compact generators. So, perhaps, the short answer is "no". | |
| Nov 30, 2010 at 12:18 | comment | added | Harry Gindi | One last question: Are unbounded chain complexes badly behaved in positive characteristic? I heard something like that, but I've never heard an explanation of why. | |
| Nov 30, 2010 at 11:40 | comment | added | Leo Alonso | You are right, and me too, I guess. In a few words, $D^{\leq 0}$ is closed monoidal but not triangulated (just suspended in the sense of Keller-Vossieck), while $D^{-}$ is triangulated but not closed monoidal. In either case, boundedness causes some kind of trouble. Simplicial (sheaves of) modules are OK, but not for duality, or these kind of results. | |
| Nov 30, 2010 at 11:20 | comment | added | Harry Gindi | Dear Leo, is it not true that the monoidal structure on simplicial abelian groups and simplicial modules is actually closed (and also that this category is complete-cocomplete as well as homotopy-complete-cocomplete)? I could have sworn that it was. Is there an issue when considering simplicial abelian sheaves (or simplicial $\cal O_X$-modules) that I'm overlooking? Thanks. | |
| Nov 30, 2010 at 11:06 | comment | added | Leo Alonso | If you are seriously interested in cohomology, Grothendieck duality, and these kind of things, the the closed monoidal structure is extremely useful. Compare the statement of the internal tensor-hom adjunction in classical references like Hartshorne's with the modern approach that you can find in Lipman's book. In topology, is as considering non-connective spectra. You gain the existence of arbitrary coproducts, and as a consequence the possibility of a sereies of constructions like, e.g. Brown representability. | |
| Nov 30, 2010 at 10:38 | comment | added | Harry Gindi | Is there really any reason to consider unbounded chain complexes (outside of homotopy theory)? Toën-Vezzosi restrict to bdd chain complexes and are still able to develop a pretty comprehensive theory of derived algebraic geometry. Bdd-below (or above) chain complexes have a nice homotopy theory by the Dold-Kan correspondence, so I'm curious about whether or not we really need the added complexity. Also, $K(A)$ should probably be defined to be be "Chain complexes modulo homotopy" or "triangulated", since at least in homotopy theory, "derived category" and "homotopy category" are synonymous. | |
| Nov 30, 2010 at 10:23 | history | answered | Leo Alonso | CC BY-SA 2.5 |