@Dedalus Could you e-mail me the solution?

@TimCampion if by localize you mean in the sense of a reflective subcategory then isnt it true that the $K(n)$-local category doesn't contain any such nontivial subcategory?

@HarryGindi Definitely!

@TimCampion What does translation-invariant mean? Note that for any non-principal ultrafilter on $\mathbb{Z}$, either the set of odd or even numbers is large, and exactly one of them.

@PiotrPstrągowski Yeah, I just realized this... I'm confused now, because I could have sworn I'd at least heard the term "translation invariant ultrafilter" before... maybe it's actually translation-invariant finitely additive measures or something...

I thought some such condition would be needed to satisfy 2/3 though...

@SaalHardali That's a good point too.

@ufabao i remember being confused about this too, and can't remember if i ever resolved it. however, i don't think the Ind should do anything too fancy, as e.g. it contains the original thing as a full subcategory, so a compact k-module and an endomorphism should be giving an object of the kernel. also, your description of the kernel is slightly incorrect: it's not just that the endomorphism is noninvertible, it must in fact be nilpotent.