Finiteness of compositions of localizations via fracture diagrams

In the last decades the study of the stable homotopy category made considerable progress due to the chromatic approach. The core idea of this programme is to decompose said category in simpler pieces which we can effectively understand and then recompose them together to reconstruct the global picture. This operation is accomplished via the Bousfield localizations with respect to the spectra E(n) and K(n), called respectively Johnson-Wilson spectrum and Morava K-theory. These two objects have crucial properties which encapsulate the information of complex orientation of spectra at height lesser or equal to n. Having established the importance of these localization functors, it is not difficult to understand that we want to consider also their compositions and that some kind of regularity in these situations is desirable. There are classical results going in this direction: for example Ravenel showed that the composition of the localization with respect to K(n) with the localization with respect to K(m) is zero whenever n>m. Also, he proved that we have an equality between Bousfield class of E(n) and the class of the wedge of the spectra K(i) for i between 0 and n. This should be read as some version of our intention of gluing back information after we decomposed it in smaller pieces. This work aims to answer the following question: if we fix n as upper bound of the chromatic height, and consider localizations coming from wedges of K(i)'s, for i lesser or equal to n, are their compositions finitely many up to isomorphism? Not only we will provide a positive answer, but we will formulate it in an axiomatic framework which will allow us to propose the proof for any collection of localizations satisfying properties similar to the ones illustrated above. One of the key points of the proof is that we can reduce the composition of two iterated localizations to the combinatorics of a finite poset which models how they arise as homotopy limits of diagrams involving simple localizations.