Graduated analogues of theorems in commutative algebra

Many theorems in commutative algebra apply in one ($ mathbb {Z} $-) graduated context. More specifically, we can use any sentence in commutative algebra and replace any occurrence of the word

  • Commutative ring by commutative grading ring (unsigned for commutativity)
  • Module through graduated module
  • Element by homogeneous element
  • Ideal by homogeneous ideal (that is, ideally generated by homogeneous elements)

This leads to further substitutions, eg. on $ ast $Local ring is a graded ring with a unique maximum homogeneous ideal, we get an idea of ​​graded depth, etc. After all these substitutions we can ask if the sentence is still true.

A book that takes a few steps in this direction is Cohen-Macaulay rings Bruns and Herzog, especially section 1.5. For example, in Exercise 1.5.24 they have the following graded analogue of the Nakayama lemma:

To let $ (R, mathfrak {m}) $ be a $ ast $local ring, $ M $ to be a finitely graded one $ R $Module and $ N $ a graduated submodule. Accept $ M = N + mathfrak {m} M $, Then $ M = N $,

A student of mine has recently shown that the graduated analogue of Lazard's theorem (a module is flat even if it is a filtered colimit of free modules) is also true.

Normally, this kind of theorem is essentially proved by a combination of two techniques:

  1. Copy the ungraded proof and replace graded terms in the manner outlined above.
  2. If you get annoyed about the length of the resulting argument, use some abbreviations for some translations between unrated and graded. (For example, a noetheric ring graded after Cohen-Macaulay is also unnamed Cohen-Macaulay.)

Sometimes you can be lucky, and the statement is suitably an algebra geometry that you can argue geometrically with the stack $[Spec R/mathbb{G}_m]$ for a stepped ring $ R $with this a $ mathbb {Z} $-grading corresponds to one $ mathbb {G} _m $-Action.

In any case, my question is this:

Is there a class of statements that automatically recognizes that the graded analog is true if the original statement in the unclassified commutative algebra is true without going through all the proof?

I am not sure if one can hope for a model-theoretical approach here, since I know almost nothing about model theory, but such a statement could save a lot of work in the detection of graded analogues of known theorems.