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I am wondering: Are there any general theorems or principles relating the theory of Z-graded dg objects and the theory of Z/2Z-graded dg objects? I am mainly interested in dg algebras, dg Lie algebras, and dg categories over fields of characteristic zero.

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    $\begingroup$ Z/2-dgas are equivalent to algebras over the commutative Z-dga $\mathbb{Z}[u^{\pm 1}]$, where $|u| = 2$. Based on this you have a number of standard constructions associated to a map of commutative dgas $R \to S$, such as tensoring up (equivalent to what Steve Huntsman proposed), forgetting (which gives 2-periodic Z-dgas), and various derived adjunctions which simplify because the range is flat over the domain. Is this something like what you are interested in? $\endgroup$
    – Tyler Lawson
    Commented Jul 28, 2010 at 2:39
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    $\begingroup$ Putting a bounty on this question, hoping that Tyler (or someone else) can elaborate... $\endgroup$
    – Kevin H. Lin
    Commented Aug 13, 2010 at 9:49
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    $\begingroup$ This is discussed, at least for dg-categories, in section 4 of Dyckerhoff's thesis: front.math.ucdavis.edu/0904.4713 where it is needed to understand LG models associated to isolated hypersurface singularities. It elaborates on Tyler's comment. $\endgroup$
    – Matthew Ballard
    Commented Aug 13, 2010 at 14:53
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    $\begingroup$ Propsition 3.1 in the paper arxiv.org/abs/1004.0687 by Dyckerhoff and Murfet gives explicit examples of some of the constructions mentioned by Tyler. $\endgroup$
    – Jesse Burke
    Commented Aug 13, 2010 at 15:50
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    $\begingroup$ A more general question concerning the behavior of (C)DG-categories and (C)DG-modules with respect to a change of the grading group is discussed in Section 2.5 of our new preprint arxiv.org/abs/1010.0982 $\endgroup$
    – Leonid Positselski
    Commented Oct 6, 2010 at 8:51

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To me, the main difference between the Z-graded and Z/2-graded cases is that the former allows certain simplifying boundedness restrictions, which in the latter do not seem to make sense. Typically, one considers a nonpositive (in the cohomological grading) Z-graded dg-algebra and dg-modules bounded above or below over it, as appropriate. The case of connected, simply connected (in the sense of cochains, not just cohomology) nonnegative dg-algebra is similar.

The typical simplification achieved under such restrictions is that a dg-module whose underlying graded module is projective is always homotopy projective. Also, the two ways of defining the differential derived functors (by taking infinite direct sums or products along the diagonals) become equivalent, since the sums/products are actually finite.

When one has to consider dg-algebras that do not satisfy the above kind of restrictions and/or unbounded dg-modules, the Z-graded situation is not any simpler than, and not much different from, the Z/2-graded situation.

References: 1. Keller "Deriving DG-categories"; 2. Husemoller, Moore, Stasheff "Differential homological algebra and homogeneous spaces".

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