Question

Is it known whether or not the endomorphism rings (or ring spectra) of the localized sphere spectra in $L_nSp$ and $L_{K(n)}Sp$ are Noetherian or not? Are they well understood? Perhaps, in the vein of making it an algebraic stable homotopy category, we should instead look at $[L_{K(n)}F(n),L_{K(n)}F(n)]$ for some finite, type-n spectrum. If we do indeed have that either one of these rings is Noetherian, can we use Hovey, Palmieri and Strickland's ideas about Noetherian stable homotopy categories to better understand these rings or the localized categories (though in some sense they are sort of simple categories)? Are the ideals of these rings connected at all to the localizing subcategories of the respective stable homotopy categories, or maybe to collections of tensor ideals?

Answer 1

Firstly, using the universal property of localisation we see that $[L_nS,L_nS]_*=\pi_*(L_nS)$ and similarly for $L_{K(n)}$, so we can just talk about the homotopy rings.

Next, $\pi_*(L_1S)$ is closely related to the image of $J$ and is described completely in papers by Bousfield and Ravenel. There is a copy of $\mathbb{Q}/\mathbb{Z}_{(p)}=\mathbb{Z}/p^\infty$ in degree $-2$, so the annihilators of powers of $p$ give a strictly increasing chain of ideals, proving that $\pi_*(L_1S)$ is not Noetherian. In $\pi_*(L_{K(1)}S)$ there is no $\mathbb{Z}/p^\infty$ but there are copies of $\mathbb{Z}/p^k$ for all finite $k$ so again the ring is not Noetherian. At height $2$ there are voluminous calculations by Shimomura and coauthors, and Behrens has recently explained an easier way to organise the answers, although they are still very complicated. The same line of argument shows that they are very far from being Noetherian. At larger heights I do not think that anything very significant is known.