Let $p$ be a prime and $h \in \mathbb N$ a height.

Question 1: Does there exist a compact $T(h)$-local spectrum $A$ with a unital multiplication making $\pi_\ast A$ is a Noetherian ring?

A priori it's unclear -- before telescopic localization, one needs infinitely many nilpotent generators to generate the homotopy of a nonzero finite spectrum. So the hope is that localizing something very big yields something which is not so big.

The answer is trivially yes when $h = 0$. It is also yes when $h=1$ [1]. In fact, $\pi_\ast(\mathbb S/p)$ is periodic of order 8 or $2(p-1)$, and finite for each $\ast$. These properties are generic (i.e. objects with these properties are closed under extensions and retracts), so they are shared by all compact $T(1)$-local spectra, and imply Noetherianness when the spectrum in question is a ring. This leads to

Question 2: Let $X$ be a compact $T(h)$-local spectrum (with $h \geq 1$). Note that $X$ is periodic. Is $\pi_0(X)$ finite?

An affirmative answer to Question 2 would imply an affirmative answer to Question 1.

[1] In The localization of spectra with respect to homology, Thm 4.1, Bousfield reports these results of calculations of Mahowald (when $p=2$) and Miller (when $p$ is odd).

Let $p$ be a prime and $h \in \mathbb N$ a height.

Question 1: Does there exist a compact $T(h)$-local spectrum $A$ with a unital multiplication making $\pi_\ast A$ a Noetherian ring?

A priori it's unclear -- before telescopic localization, one needs infinitely many nilpotent generators to generate the homotopy of a nonzero finite spectrum. So the hope is that localizing something very big yields something which is not so big.

The answer is trivially yes when $h = 0$. It is also yes when $h=1$ [1]. In fact, $\pi_\ast(\mathbb S/p)$ is periodic of order 8 or $2(p-1)$, and finite for each $\ast$. These properties are generic (i.e. objects with these properties are closed under extensions and retracts), so they are shared by all compact $T(1)$-local spectra, and imply Noetherianness when the spectrum in question is a ring. This leads to

Question 2: Let $X$ be a compact $T(h)$-local spectrum (with $h \geq 1$). Note that $X$ is periodic. Is $\pi_0(X)$ finite?

An affirmative answer to Question 2 would imply an affirmative answer to Question 1.

[1] In The localization of spectra with respect to homology, Thm 4.1, Bousfield reports these results of calculations of Mahowald (when $p=2$) and Miller (when $p$ is odd).