Homotopy Theory

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Clark Barwick

00:23

I couldn't have said it better. @RuneHaugseng, I owe you an email.

1 hour later…

Tomer Schlank

01:35

So I heard many times pepole claim that "the experts believe the telescope conjecture to be false" and I aware of the paper by Mahowald, Ravenel and Shick, which give some computational reason. But I was still wondering:

1) looking at the paper it seems the objection is that the telescope conjecture will force many "a-priory-unexpected" differentials, but other then being unexpected a-priory if I really want to believe the conjecture is there any thing so weird about them a posteriori?

2) Is there another\more conceptual reason to be a telesceptic?

Clark Barwick

01:49

So: there are two ways to describe open subsets of an affine derived scheme Spec R (where R is a E∞ ring): 1. (let's just call these "open subsets") maps Spec S → Spec R that induce a fully faithful Mod(S) → Mod(R); 2. (let's call these "quasicompact open subsets") maps Spec S → Spec R that induce a fully faithful Perf(S) → Perf(R). Now the telescope conjecture for Spec R says that any open subset is quasicompact open.

By the thick subcategory theorem, the quasicompact opens of Spec of the p-local sphere are classified: they're Spec of the telescopic localizations. If the telescope conjecture is true, these are all the opens. If it is false, then for each n, we have a quasicompact open, a sort of "maximally non-quasicompact" open, and potentially a whole bunch of stuff in between.

(I'm not sure that "quasicompact" is really the right word. Let me see if I can think of something more intelligent.)

Tomer Schlank

02:04

@ClarkBarwick: So the point is, that we think that the sphere spectrum should be very non-noetherian?

Clark Barwick

02:15

If we disbelieve the telescope conjecture, I guess so!

(I suppose "perfect open" would be a better expression than "quasicompact open.")

Tomer Schlank

Clark, what you wrote makes me more of a believer then a sceptic , why would the sphere spectrum be so badly behaved?

Clark Barwick

02:34

Well, it is terribly non-regular.

Clark Barwick

02:48

I'm not trying to turn you into a telebeliever. I'm in no position to contradict the telescepticism of my computational overlords. But here's something else strange. The best thing we've got to understand Spec(S^0) is Spec(MU), which is a torsor over Spec(S^0) for the flat group scheme Spec(S[BU]). But the telescope conjecture is true for Spec(MU). So inverse image of any open under Spec(MU) → Spec(S^0) is perfect in the sense above.

Clark Barwick

03:07

(I bet I'm going to get something backwards in the following. Sorry.) So we have Spec S_{E(n)} ⊂ Spec S_{T(n)} ⊂ Spec S^0, where S_{E(n)} is the E(n)-localization and S_{T(n)} is the T(n)-localization. (And if L is any smashing localization with LK(n) = K(n) and LK(n+1)=0, then Spec S_{E(n)} ⊂ Spec LS ⊂ Spec S_{T(n)}.)

I think I'm right in saying that under p: Spec(MU) → Spec(S^0), the image of the inverse image of Spec S_{T(n)} must be contained in Spec S_{E(n)}. The telesceptic who thinks that all of these open sets should "count" might find this uncomfortable, since p is meant to be a kind of quotient map.

13 hours later…

Clark Barwick

16:38

So (and I'm worried that I'm hogging the airwaves to talk to myself here) ... I think the argument I gave implies that the telesceptic must accept one of the following about the nature of DAG: (1) the Zariski topology in DAG should consist of only perfect open sets; or (2) the map p is not really a quotient map in DAG after all. Personally, I think that (2) is a psychologically worse state of affairs.

1 hour later…

Qiaochu Yuan