Spectra for commutative algebraists
John P C Greenlees · Contemporary mathematics - American Mathematical Society · 2007
The article is designed to explain to commutative algebraists what spectra are, why they were originally defined, and how they can be useful for commutative algebra.Contents 0. Introduction. 1 1.Motivation via the derived category.2 2. Why consider spectra? 4 3. How to construct spectra (Step 1).6 4. The smash product (Step 2).11 5. Brave new rings.17 6.Some algebraic uses of ring spectra.19 7. Local ring spectra.21 References 24 0. Introduction.This article grew out of a short series of talks given as part of the MSRI emphasis year on commutative algebra.The purpose is to explain to commutative algebraists what spectra (in the sense of homotopy theory) are, why they were originally defined, and how they can be useful for commutative algebra.An account focusing on applications in commutative algebra rather than foundations can be found in [13], and an introduction to the methods of proof can be found in another article in the present volume [14].Historically, it was only after several refinements that spectra sufficiently rigid for the algebraic applications were defined.We will follow a similar path, so it