Length and Multiplicities in Graded Commutative Algebra

TL;DR

This paper reviews key concepts of length and multiplicity in graded commutative algebra, emphasizing their applications in algebraic topology and providing detailed proofs often considered folklore.

Contribution

It offers a comprehensive exposition of length and multiplicity in graded commutative algebra with detailed proofs and applications in topology.

Findings

Includes proofs of folklore results in graded algebra

Connects algebraic concepts to topological properties

Provides detailed computations relevant to algebraic topology

Abstract

This paper is a review of concepts from graded commutative algebra with specific attention given to length and multiplicity. The author's motivation for this paper comes from the study of equivariant cohomology in algebraic topology where the graded commutative algebra of the module is intimately connected to topological properties of the space (as shown by Quillen). Results (and their proofs) which are often left as exercises, or considered 'folklore' in the commutative algebra community are included in this paper, as are references to relevant applications in topology. As such, this paper is aimed at algebraic topologists and geometers looking for a detailed exposition of length and multiplicity computations in graded commutative algebra.