By nullification with respect to $K(A,n)$, I mean the Bousfield localization $L_{A,n}$ where a spectrum $E$ is $L_{A,n}$-local if and only if $\tilde E^\ast(K(A,n)) = 0$.
Question: Is there a good description of which spectra are $L_{A,n}$-local? Is there a good description of the localization functor $L_{A,n}$?
Notes:
I'd be happy to understand the case $A = \mathbb Z/p$ when $E$ is also $p$-local.
For example, $K(n)$-local spectra are $L_{\mathbb Z/p,m}$-local for $m \geq n+1$.
If $E$ is $L_{A,n}$-local, then $E$ is also $L_{A,n+1}$-local. This follows from the Zabrodsky lemma. As a consequence, $HA$ is $L_{A,n}$-acyclic for all $n$.
It's a fact (due to Bousfield I think) that $K(\mathbb Z/p, n)$ is always either $E$-local or $E$-acyclic (as a space -- note that $E$-acyclicicity the same whether we consider $K(\mathbb Z/p,n)$ as a space or suspension spectruem, but $E$-locality may be different I think).
And I believe any nontrivial homological localization of $p$-local spectra [EDIT: assuming it kills some suspension spectrum -- e.g. not the harmonic localization!] must kill some $K(\mathbb Z/p,n)$.
So I'm asking about the "minimally nontrivial" localizations of $p$-local spectra.This follows from the theorem of Bousfield that if a space $X$ is $E$-acyclic, then $K(\pi_n(X), n)$ is $E$-acyclic, and then you can show that if some $K(A,n)$ is $E$-acyclic, so is $K(\mathbb Z/p, n)$ for some $p$.One form of the Sullivan conjecture (proved by Miller says that in the unstable case, every finite-dimensional complex is $L_{A,n}^{unst}$-local for every finite $A$ and $n \geq 1$.
A related theorem of Lee says that in the stable case, every finite spectrum is $L_{A,n}$-local for every finite $A$ and $n \geq 2$. Unlike the unstable case, I don't think this extends to finite-dimensional spectra -- e.g. I think it already fails for any infinite wedge of equidimensional spheres. Certainly it fails even for the finite case at $n=1$ by the Segal conjecture (proved by Carlsson).
Nick Kuhn points out in his answer below that in the unstable case, Neisendorfer showed the following:
Let $X$ be the $p$-completion of a simply-connected finite space with $\pi_2(X)$ finite. Then for all $m \in \mathbb N$, we have $L_{\mathbb Z/p, 1}^{unst} \tau_{\geq m} X = X$.
I believe this extends to say that if $X$ is the $p$-completion of an $n$-connected finite space with $\pi_{n+1}(X)$ finite, then for all $m$ we have $L_{\mathbb Z/p, n}^{unst} \tau_{\geq m} X = X$.
The stable analog of Neisendorfer's theorem then says that if $L$ is a localization of $p$-complete spectra which kills $H\mathbb Z/p$, then for any spectrum $E$, $\tau_\geq n E \to E$ is an $L$-equivalence for all $n \in \mathbb N$.
