In this talk, I will try to take “a baby step” towards getting a chromatic picture of the $H\mathbb{F}_{p}$ based synthetic spectra. I will start by describing an interesting collection of finite localizations of $H\mathbb{F}_{p}$ based synthetic spectra. These can be thought of, in a precise sense, as generalizations of the finite localizations of spectra. One of these localizations will allow us to identify a nice class of finite ($H\mathbb{F}_{p}$ based synthetic) spectra. Then, with an aim of doing things in a systematic way we will turn our attention towards the study of non-nilpotent self-maps of this nice class of finite spectra. I will use spectra with somewhat unusual properties, introduced by Palmieri, to detect these self-maps. I will show the existence of these self-maps for appropriate finite spectra and show that these maps have the expected properties. Using these self-maps I will construct generalized Moore spectra. I will end by showing how these generalized Moore Spectra can be used to define a few more finite localizations, hence providing us with a better chromatic picture of $H\mathbb{F}_{p}$ based synthetic spectra.
