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I am trying to learn about simplicial commutative rings, and would be grateful if one can help with some basic facts about them. Basically, I would like to understand how to do homological algebra over a simplicial ring.

  1. Let $A$ be a simplicial commutative ring. Is the category of simplicial modules over $A$ abelian?
  2. If so, does its derived category exist?
  3. Does any simplicial module have a projective resolution? a flat resolution? an injective resolution?

  4. Consider the associated DG-algebra $N(A)$. Is there any relation between the derived category of DG-modules and the derived category I was hoping exists in 3 above?

  5. Any references that discuss these basic issues?

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    $\begingroup$ 1. Not a question. 2. Yes. 3. Yes. 4. Yes. However, these are somehow the wrong questions to ask – simplicial modules already have a built-in notion of weak equivalence, and this is ignored if you treat it as an abelian category. It is akin to taking the category of chain complexes as an abelian category in its own right. $\endgroup$
    – Zhen Lin
    Commented Sep 25, 2015 at 14:44

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Yes, there are many references that discuss this. The first is Quillen's Homotopical Algebra. Chapter II contains much of what you're asking about, especially II.4 and II.6

For a more modern version, see Schwede-Shipley Algebras and Modules in Monoidal Model Categories. Section 5 is all examples and contains precise references in Quillen's work above.

This covers your (2), (3), and (6). I agree with Zhen Lin that (4) is the wrong question to ask, since the structure as an abelian category is less important than the model structure if you are trying to study the homotopy category from (3). For (5), check out the Dold-Kan theorem. A nice write-up is in Goerss-Schemmerhorn Model Categories and Simplicial Methods 4.1. This also includes the model structure on simplicial R modules and what resolutions look like for simplicial R modules, answering your (4) in the correct homotopically meaningful sense. It is clear from 4.1 that $N$ is both an equivalence of categories (if by DG we mean non-negatively graded) and a Quillen equivalence, so the homotopy theories agree, and this answers your (5)

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    $\begingroup$ Thank you for the answer, but there is something very basic I still don't understand: How do I take Hom (or RHom ) between two simplicial modules? I mean, if I take two bounded above complexes, then the Hom between them will be bounded below, right? so in the DG-setting, this is OK. But if in the simplicial world I only have bounded above complexes, what happens to $RHom$? $\endgroup$
    – user78856
    Commented Sep 28, 2015 at 8:54
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    $\begingroup$ You have to take the appropriate truncation, of course. So, for instance, simplicial RHom between "discrete" modules is "discrete", in contrast to the unbounded setting. $\endgroup$
    – Zhen Lin
    Commented Sep 28, 2015 at 8:58
  • $\begingroup$ @Zhen Lin, thanks for the comments. Any reference that discuss this issue? $\endgroup$
    – user78856
    Commented Sep 28, 2015 at 9:00
  • $\begingroup$ Not that I'm aware of, but I haven't looked especially hard. That particular example is an easy calculation, at any rate. $\endgroup$
    – Zhen Lin
    Commented Sep 28, 2015 at 9:01
  • $\begingroup$ I don't know of a reference that discusses this. I looked in Weibel's book on homological algebra, but no luck. I checked Hirschhorn too but couldn't find it. At this point, if it's not in Goerss-Schemmerhorn then I'd probably go back to Quillen, or chapter 8 of Jardine's local homotopy theory, or maybe one of the Dwyer-Kan papers. Due to teaching and a visitor to our department I probably won't have a chance to respond again till next weekend, but maybe the references here will help. $\endgroup$
    – David White
    Commented Sep 28, 2015 at 13:06

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