The smash product for derived categories in stable homotopy theory

TL;DR

This paper explores how increasing levels of algebraic structure on ring spectra in stable homotopy theory influence the properties of their derived categories, particularly the monoidal and symmetric structures.

Contribution

It establishes the relationship between E_n structures on ring spectra and the resulting monoidal, braided, and symmetric structures on their derived categories.

Findings

E_2 structure induces a monoidal product on the derived category.

E_3 structure provides a braiding on the monoidal product.

E_4 extension results in a symmetric monoidal structure.

Abstract

An E_1 (or A-infinity) ring spectrum R has a derived category of modules D_R. An E_2 structure on R endows D_R with a monoidal product. An E_3 structure on R endows the monoidal product with a braiding. If the E_3 structure extends to an E_4 structure then the braided monoidal product is symmetric monoidal.