TITLE: Graded analogues of theorems in commutative algebra
QUESTION [22 upvotes]: Many theorems in commutative algebra hold true in a ($\mathbb{Z}$-)graded context. More precisely, we can take any theorem in commutative algebra and replace every occurrence of the word

commutative ring by commutative graded ring (without the sign for commutativity)
module by graded module
element by homogeneous element
ideal by homogeneous ideal (i.e. ideal generated by homogeneous elements)

This results in further substitutions, e.g. a $^\ast$local ring is a graded ring with a unique maximal homogeneous ideal, we get a notion of graded depth etc. After all these substitutions we can ask whether the theorem is still true. 
One book that does some steps in this direction is Cohen-Macaulay Rings by Bruns and Herzog, especially Section 1.5. For example, they have as Exercise 1.5.24 the following graded analogue of the Nakayama lemma: 

Let $(R,\mathfrak{m})$ be a $^\ast$local ring, $M$ be a finitely generated graded $R$-module and $N$ a graded submodule. Assume $M = N + \mathfrak{m}M$. Then $M = N$. 

Moreover, a student of mine recently showed that the graded analogue of Lazard's theorem (a module is flat if and only if it is a filtered colimit of free modules) is also true. 
Usually one proves this kind of theorems essentially by a combination of two techniques: 

Copy the ungraded proof and just substitute ungraded for graded concepts in the manner sketched above.
If one is annoyed by the length of the resulting argument, use some shortcuts by some translations between ungraded and graded. (E.g. a noetherian graded ring that is graded Cohen-Macaulay is also ungraded Cohen-Macaulay.)

Sometimes one can also be lucky and the statement is suitably algebro-geometry that one can argue geometrically with the stack $[\operatorname{Spec} R/\mathbb{G}_m]$ for a graded ring $R$, using that a $\mathbb{Z}$-grading corresponds to a $\mathbb{G}_m$-action. 
In any case, my question is the following: 

Is there any class of statements, where one knows automatically that the graded analogue is true if the original statement in ungraded commutative algebra is true, without going through the whole proof?

I am not sure whether one can hope here for a model-theoretic approach as I know almost nothing about model theory, but any such statement could save a lot of work in proving graded analogues of known theorems.

REPLY [2 votes]: Since the OP is asking for some "class of statements" which can be readily transferred from the ungraded to the graded case, let me outline a couple of thoughts, which are not really tied to neither  the commutative case, nor the $\mathbb{Z}$-gradation itself: 

On the one hand, if $R$ is a $k$-algebra and $G$ is a group, then a $G$-gradation on $R$ can be equivalently viewed either as a $(kG)^*$-action or as a $kG$-coaction (where $kG$ is the group hopf algebra and $(kG)^*$ its dual hopf algebra). So, the graded algebra $R$ is alternatively viewed as an algebra in the category of $(kG)^*$-modules or an algebra in the category of $kG$-comodules.  In this sense, the statements and the properties which are "naturally" transferred between the graded and the ungraded case are those which are in some sense "compatible" or "invariant" with respect to the $kG$-coaction (or the $(kG)^*$-action). I do not know if it sounds like an abuse of terminology to speak about  


properties in the category of $(kG)^*$-modules or properties in the category of $kG$-comodules

on the other hand, the Category $R_{gr}$-$mod$ of graded modules over the $G$-graded ring $R$, is equivalent (actually i think it is isomorphic in most cases) to a closed subcategory of modules (in the ungraded sense) over the smash product algebra $R\sharp kG$. (Here the smash product is understood in the sense of the one associated to a Hopf module algebra).
In this sense, we could say that the statements and properties which one might expect to "naturally transfer" over between graded and ungraded, are  

the ones which are preserved under the smash product between $R$ and the group hopf algebra $kG$.   

(In some sense, this implies that it might be meaningful to consider properties which "transfer" between the $R$-modules and the $R\sharp kG$-modules). 

One might reasonably expect that  

all those statements that depend only on the structure of abelian category on the category of graded modules   

mentioned in Fred Rohrer's answer fall into the classes mentioned above, however i can't tell for sure on the exact overlap between these classes.