Localization is a well known method for simplifying problems. For example, given a Homology theory \(H\) on spaces, say singular homology with coefficient \(R\), then localization of a topological space \(X\) with respect to \(H\) amounts to extracting "the part of \(X\) homology \(H\) can see". Hence localization only focus on the information we care about and simplifies spaces accordingly. Besides spaces, in homotopy theory people also care about structured ring spectra, which are generalized notion of spaces with additional algebraic structure. One might wonder whether the localization methods for spaces also work for structured ring spectra.
In this talk, we will review how Bousfield proved that for topological spaces, localization with respect to homology always exist. Then we will discuss how to adapt this argument to the category of structured ring spectra and prove an analogous existence theorem. This is joint work with John E Harper.