Cfrgtkky

15 8 $begingroup$ 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 fo...