I am wondering: Are there any general theorems or principles relating the theory of Zgraded dg objects and the theory of Z/2Zgraded dg objects? I am mainly interested in dg algebras, dg Lie algebras, and dg categories over fields of characteristic zero.
To me, the main difference between the Zgraded and Z/2graded 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) Zgraded dgalgebra and dgmodules bounded above or below over it, as appropriate. The case of connected, simply connected (in the sense of cochains, not just cohomology) nonnegative dgalgebra is similar. The typical simplification achieved under such restrictions is that a dgmodule 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 dgalgebras that do not satisfy the above kind of restrictions and/or unbounded dgmodules, the Zgraded situation is not any simpler than, and not much different from, the Z/2graded situation. References: 1. Keller "Deriving DGcategories"; 2. Husemoller, Moore, Stasheff "Differential homological algebra and homogeneous spaces". 


$\mathbb{Z}[u^{\pm 1}]$
, 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 2periodic Zdgas), and various derived adjunctions which simplify because the range is flat over the domain. Is this something like what you are interested in? – Tyler Lawson Jul 28 '10 at 2:39