It is a theorem of Neil Strickland's that the category of harmonic spectra (i.e. the category of $p$-localized spectra localized at the infinite wedge of Morava K-theories) has no small objects. That is to say that there is not a single spectrum $X$ in the category of harmonic spectra such that $[X,\bigvee X_i]=\bigoplus[X,X_i]$. Take a finite spectrum in this category (for instance the sphere spectrum $\mathbb{S}$). Does anyone have an example of a coproduct in the category of harmonic spectra for which the above is NOT true? While Strickland's proof makes sense, it seems completely amazing to me. What HAPPENS to the sphere and the cells of finite spectra when we localize? It seems that if something were small to begin with, i.e. the above homotopy groups factored globally, killing some spectra shouldn't change this! Note that the same question can be asked of the $BP$-local category.

Are there conditions (other than finiteness) that we can put on the coproduct to make the statement true when the object on the left is finite (i.e. in the thick subcategory generated by the sphere)?


I believe that any example of a direct sum of harmonic spectra which isn't itself harmonic should answer the question. Namely, if the $X_i, i \in \mathbb{N}$ are harmonic and $\bigoplus X_i$ isn't, then the map $$ \bigoplus [S, X_i] \to [S, L_{\mathrm{harm}}\bigoplus X_i]$$ isn't an isomorphism, because $\bigoplus X_i \to L_{\mathrm{harm}} (\bigoplus X_i)$ isn't an equivalence.

When the class of $E$-local spectra is closed under direct sums, then the $E$-local sphere is in fact compact and the $E$-localization functor commutes with (homotopy) colimits; then the localization functor is called smashing. (The $E_n$-localization functors are smashing, as are the finite versions of them.) Conversely, if the $E$-local sphere is compact, then homotopy groups in the $E$-local category commute with direct sums, which means that direct sums in the $E$-local category are the same as direct sums in spectra (i.e., $E$-localization is smashing).

So the point is that harmonic localization is not smashing. In fact, every finite spectrum (more generally, every suspension spectrum) is harmonic, and every spectrum is a homotopy colimit of finite spectra, so it suffices to produce a single example of a spectrum which isn't harmonic. An example is $H \mathbb{Z}/2$, which has no rational homotopy and no $K(n)$-homology for $n \geq 1$ (essentially because a formal group law can't have height $n$ and height $\infty$ at once).

To be a little more explicit, you could find a directed system $X_1 \to X_2 \to \dots$ of finite (harmonic) spectra whose homotopy colimit is $H \mathbb{Z}/2$. Then the map $\bigoplus X_i \to L_{\mathrm{harm}} \bigoplus X_i$ is not an equivalence (proof: the homotopy colimit of the directed system -- the mapping telescope -- is in the thick subcategory generated by the direct sum, and $\varinjlim X_i $ is not harmonic).

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    $\begingroup$ I don't understand the first paragrah. Why should it be so? $\endgroup$ – Fernando Muro Jan 8 '13 at 7:48
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    $\begingroup$ @Fernando: In the maps on homotopy groups $\oplus [S^n,X_i] \to [S^n, \oplus X_i] \to [S^n, L(\oplus X_i)]$, the first map is always an isomorphism but the second can't be because the map isn't a weak equivalence. $\endgroup$ – Tyler Lawson Jan 8 '13 at 13:34
  • $\begingroup$ @Akhil, @Tyler, here's something that confuses me though. $L$, as a localization functor, preserves (htpy) colimits (HPS p. 42). So, $L(\vee X_i)=\vee L(X_i)=\vee X_i$, hence the last part should be $L$-local? What am I missing? $\endgroup$ – Jonathan Beardsley Jan 8 '13 at 21:47
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    $\begingroup$ Yes, homotopy colimits in a localization always require reapplying the localization functor. $\endgroup$ – Akhil Mathew Jan 8 '13 at 22:32
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    $\begingroup$ Yeah @Fernando, the proof that there are not small objects is in an appendix of Hovey and Strickland's paper on localizations at Morava K-theories, the exact title of which escapes me at the moment. The basic idea is, the colocalizing category of $E$-local objects has small objects if and only if $\langle E\rangle\geq\langle F(n)\rangle$ for some finites type-n spectrum $F(n)$. He then proves that if $\langle E\rangle\geq\langle F(n)\rangle$ for some $n$ then $L_EI\neq 0$, where $I$ is the Brown-Comenetz dual of the sphere. However, localizing at $BP$, $I$, $HF_p$ or the harmonic spectrum.. $\endgroup$ – Jonathan Beardsley Jan 10 '13 at 3:16

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