Commutative ring objects in pro-categories and generalized Moore spectra (Q384687)

This paper revolves around two central ideas in modern stable homotopy theory. First there is the notion of a type \(n\) finite spectrum for a given prime \(p\) and in turn the notion of \(K(n)\)-local spectra where \(K(n)\) is the height \(n\) Morava \(K\)-theory for \(p\). The second notion is that of a highly homotopically coherent ring spectrum, in this case an \(E_\infty\) ring spectrum.NEWLINENEWLINEOver the past two decades or so, a considerable body of theory has revolved around these two ideas, and in this paper a significant advance is made in bringing them together in a way that many experts have long believed should be possible. On the one hand there is a well known sense in which \(K(n)\)-localisation is effected by taking a homotopy limit over a tower involving smashing an \(E(n)\)-local spectrum with generalised type \(n\) Moore spectra. On the other hand, the known vanishing of André-Quillen type obstruction groups for obstructions for \(E_\infty\) ring structures on Lubin-Tate spectra follows from the fact that various algebraic objects are limits of certain closely related towers.NEWLINENEWLINEThe main result of this paper shows that a tower of generalised type \(n\) Moore spectra \(\{M_I\}_I\) whose homotopy limit is the \(p\)-complete sphere admits the structure of an \(E_\infty\) algebra in the category of pro-spectra. Remarkably, none of the individual spectra \(M_I\) can ever be an \(E_\infty\) ring spectrum.NEWLINENEWLINEAlthough the result is not so hard to state, there are considerable technical results required to prove it and the paper works out the necessary details on the model categorical aspects as well as operads in pro-categories.