# What is nullification with respect to an Eilenberg-MacLane space?

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 $$n \in \mathbb N$$.

• What about MU-localization? Isn't this an example of a nontrivial localization which kills no $K(\mathbb{Z}/p,n)$? – Saal Hardali Apr 24 at 6:32
• @SaalHardali I thought MU-localization was the same as $H\mathbb Z$-localization, which is to say the identity... – Tim Campion Apr 24 at 12:16
• Maybe I'm forgetting some hypotheses for this to be true... – Tim Campion Apr 24 at 12:22
• $H\mathbb{Z}$-localization is not the identity. (For example it kills K-theory). – Dylan Wilson Apr 24 at 13:53
• I would like to add that I gave $MU$ instead of the simpler $H\mathbb{Z}$ because its also not true that localization w.r.t. all the morava $K$-theories (i.e. the harmonic localization) is the same as $MU$-localization which indicates in some sense that there are baziilions of non-trivial loclization of this sort. – Saal Hardali Apr 24 at 14:06