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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) 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 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.

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    $\begingroup$ Isn't the problem more about smallness than compactness ? If you hand me an abstract category $C$, compactness is more or less meaningless as I can always view the objects of $C$ as compact in $Ind(C)$. I think smallness might be a bigger problem with respect to some of the fundamental results of tt-geometry. Also, if you're working in the "big" case, thick $\otimes$-ideals are typically not the "correct" thing to look at, you usually want to also assume that your ideals are closed under arbitrary coproducts $\endgroup$ Nov 1, 2022 at 21:52
  • $\begingroup$ (note that Remark 2.18 in the linked survey of Balmer's indicates that smallness plays a role, as well as his mention of infinite coproducts) $\endgroup$ Nov 1, 2022 at 21:55
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    $\begingroup$ I'd just say that the answer depends on whether you'd like to think about a complex of $\Bbb Z/4$'s with differential being multiplication by 2 as a zero object, or not. (more or less, whether you think about the category of acyclic complexes as a generalized Serre class, or as an approximation to possible "Bousfield class"-like structure). Balmer spectrum is useful to define a notion generalising support of cohomology for perfect complexes on qcqs scheme; some useful localisations of unbounded complexes fail in being either or both "qc" or "qs". Usual homotopy category fails in second regard. $\endgroup$
    – Denis T
    Nov 1, 2022 at 21:57
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    $\begingroup$ Because colimits behave badly in unbounded category, most results using checking conditions on stalks in some way either become wrong, or start requiring workarounds. $\endgroup$
    – Denis T
    Nov 1, 2022 at 22:03

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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.)

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  • $\begingroup$ What do you mean by thick primes in compact objects? Compact objects might not span a symmetric monoidal full subcategory. $\endgroup$
    – Z. M
    Nov 4, 2022 at 5:29
  • $\begingroup$ @Z.M they may not in general, but they do in the case of interest; namely, the case where C is generated by a small set of compact rigid objects. $\endgroup$ Nov 4, 2022 at 19:26
  • $\begingroup$ But you are interested in examples such as the category of p-complete objects, right? In this case, the unit is not compact. $\endgroup$
    – Z. M
    Nov 4, 2022 at 20:25
  • $\begingroup$ @Z.M True, but I think that this is just evidence that the p-complete category is itself problematic. I'm satisfied with the explanation that this is due to the fact that p-completion is not smashing. $\endgroup$ Nov 4, 2022 at 23:47

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