ABSTRACT

This chapter deals with deformations of commutative algebras. Higher algebra is the study of algebraic structures that arise in the setting of higher category theory. Higher algebra generalizes ordinary algebra, or algebra in the setting of ordinary category theory. Ordinary algebra is set based, meaning that it is carried out in the language of ordinary categories. Higher categorical algebra is truly homotopical and not just homological in nature, meaning that many of its most important objects simply do not exist within the world of chain complexes or derived categories. The portion of the theory that can be formulated in these terms is differential graded algebra, the abstract study of which employs the language of differential graded categories. The chapter briefly reviews the theory of localization of ring spectra. An associative ring spectrum is an algebra object of the monoidal ∞-category of spectra. A commutative ring spectrum is a commutative algebra object of the symmetric monoidal ∞-category of spectra.