A Note on Graded Rings and Modules

TL;DR

This paper highlights a technical aspect of graded rings and modules often used in proving the Artin-Rees lemma, emphasizing its importance and potential for simplifying proofs of significant results in commutative algebra.

Contribution

It demonstrates that a commonly used but underappreciated technical step can be central to proving major theorems, including generalizations by Cartier and Tate.

Findings

Technical step is crucial in proving the Artin-Rees lemma

Simplifies proofs of key theorems in commutative algebra

Enables generalizations of results by Cartier and Tate

Abstract

In this note, we consider a situation that is generally used as an intermediate technical step in proving the Artin-Rees lemma but otherwise is not much discussed in introductory accounts of commutative algebra. I hope to show in this note that such technical step deserves more recognition and emphasis in any introduction to commutative algebra because it can be used to prove some significant results in a straight-forward manner, including a generalization of a theorem by Pierre Cartier and John Tate.