Classical obstructions and S-algebras

Abstract

A bstract . Classical obstruction theory can be applied to the problem of finding an S -algebra structure, or a commutative S -algebra structure, on a ring spectrum. It is shown that there is no obstruction to upgrading the homotopy unit in the ring spectrum to a strict unit in the S -algebra. INTRODUCTION A ring spectrum is a spectrum E equipped with a homotopy-associative multiplication map μ : E ∧ E → E which has a two-sided homotopy unit η : S → E. It is a commutative ring spectrum if μ is homotopic to μτ, where τ interchanges factors in E ∧ E . Thus the (commutative) ring spectra are the (commutative) monoids in the stable homotopy category. We should like to replace the multiplication μ by a strictly associative multiplication map in the general case; and by a strictly associative and commutative multiplication map in the case of a commutative ring spectrum. (The object E may be replaced by a weakly equivalent object in the process.) These are notions at the point set or model category level, and they make sense if the model category which we are using has a symmetric monoidal smash product. We work in the category of S -modules, which has this property. It would be possible to adapt the theory, making necessary modifications, to other symmetric monoidal model categories for stable homotopy theory or to other contexts such as differential graded objects in an abelian category.