In trying to understand the higher algebraic geometry of the stable homotopy category, one thing I've come across repeatedly is the claim that one should only consider the Balmer spectrum of a tt-category whose objects are all compact. One argument for this is that a certain fundamental result (Theorem 2.14 of [this paper](https://arxiv.org/pdf/1912.08963.pdf)) depends on dualizability; but dualizability is not equivalent to compactness, or even stronger/weaker than compactness. Moreover, the result in question only becomes problematic because there will be non-radical ideals in the non-rigid case, and it's essentially a tt-version of the Nullstellensatz. I don't have any issue with non-radical ideals, myself, and in fact they're quite important for deformation theory, so I don't find this argument very convincing.


The reason this has come up is that I've been comparing and contrasting some localizations and completions. One famous result, for example, is that the category of *perfect* complexes over a quasicompact quasiseparated scheme $X$ has Balmer spectrum $X$. However, over a Noetherian ring $R$ (I'll take $R=\mathbb{Z}$ for concreteness), the same is true for the full derived category if we replace thick subcategories by localizing subcategories. The difference is that the irreducible thick subcategories of $D(\mathbb{Z})$ (which are to be thought of as the "residue class fields" at each prime tt-ideal) are the categories of (derived) $p$-*complete* complexes, whereas the irreducible localizing subcategories of $D^{\text{perf}}(\mathbb{Z})$ are the categories of $p$-*local* complexes. (Of course, p-completion doesn't live within the category of perfect complexes.)


This distinction arises, in particular, when we try to compare the tt-geometry of $D(\mathbb{Z})$ and the stable homotopy category. I recently [asked](https://mathoverflow.net/questions/431711/what-is-the-balmer-spectrum-of-the-p-complete-stable-homotopy-category) about the Balmer spectrum of $\operatorname{Sp}\_p^{\wedge}$, since I haven't been able to find any information about this but wanted to compare it to $D(\mathbb{Z})$ as is done (slightly less formally) in Barthel and Beaudry's chapter of the Handbook. I was told, once again, that it's problematic to apply the Balmer spectrum construction here; but, as discussed above, I don't see why that is.


Hence my question: does the Balmer spectrum really fail to describe the AG of stable symmetric monoidal infinity-categories containing non-compact objects? In particular, are there any specific examples of important results failing in this context?


EDIT: As Brian Shin pointed out to me, the most obvious difference is that one needs to replace thick subcategories with localizing subcategories.



How does the Balmer spectrum fail to describe the algebraic geometry of categories of non-compact objects?
After some discussion with Brian and reading the papers referenced by Balmer's survey, I realized what's going on here. The problem is not that the theory fails for "big" categories. Rather, it's that there are two possible theories: taking the thick primes of $C^{\omega}$, and taking the smashing primes of $C$. While we can exhibit the former as a retract of the latter (intersect a smashing prime with the compact objects/take the smashing subcategory generated by a thick prime), they are not in general equivalent. In fact, for the stable homotopy category, the question of whether they're the same is actually equivalent to the Telescope Conjecture. (Here the smashing primes are K(n)-localization, and the thick primes are finite localization.)


