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I have some seemingly basic questions about $N$-periodic derived categories to which I have not found answers in any of the usual places.

Let $R$ be a ring, and let $D(R)_{\mathbb Z/N\mathbb Z}$ denote the $N$-periodic derived category of $R$-modules. In other words, we consider the category of $N$-periodic complexes of $R$-modules, pass the homotopy category, and then invert quasi-isomorphisms.

Really I just want to get a handle on how to work with such categories, but here are a few specific questions:

  1. Is there a model structure on $N$-periodic complexes which allows us to do basic things like calculate hom sets in $D(R)_{\mathbb Z/N\mathbb Z}$ via cofibrant/fibrant replacement?

  2. What replaces the usual "induction on degree" arguments which are so ubiquitous in homological algebra? (e.g. for proving lifting properties)

  3. Is what I wrote above even the "correct" definition of the $N$-periodic derived category?

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    $\begingroup$ Is it not OK to just work in the usual Z-graded but unbounded derived category, and then pass to fixed points under shift-by-N? $\endgroup$
    – Vivek Shende
    Commented Mar 19, 2016 at 3:38
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    $\begingroup$ This might be the way to go, but I'd still be interested in a good reference going over the basic construction. For instance, how does one calculate morphisms in the resulting category? $\endgroup$
    – John Pardon
    Commented Mar 19, 2016 at 3:47
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    $\begingroup$ You can think of $N$-periodic complexes as modules over the differential graded algebra $R[x^{\pm 1}]$, where the differential is trivial and $x$ lies in degree $N$. This gives an answer to $(1)$ (there's a model structure on modules over any differential graded algebra, where the fibrations and weak equivalences are maps that are fibrations and weak equivalences on the underlying chain complex). $\endgroup$
    – Jacob Lurie
    Commented Mar 19, 2016 at 14:41
  • $\begingroup$ Bernhard Keller has written a foundational article on orbit categories: math.uiuc.edu/documenta/vol-10/17.pdf $\endgroup$
    – F. C.
    Commented Mar 19, 2016 at 18:23

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