Its not so clear to me that the entire moduli stack of fornal groups is the correct thing to consider. The smashing localization in spectra are i bijection the moduli stacks of formal groups of bounded height. The only way to get the entire moduli space is by having MU which is a weird miracle and not really well.motivated from a purely categorical perspective. For example theres this gap between being MU-local and being Harmonic.

Also its not clear to me what chromatic homotopy theory has to say about dissonant spectra (if at all). Perhaps the best hope is to get the category of harmonic spectra as the global sections of some sheaf of categories over M_fg i don't know. Just sharing a bit my thoughts about this.

Also there's the point at infinity in M_fg which corresponds to HF_p which behaves very differently than morava K-theories (corresponding to the points of finite height). For example HF_p-local spectra is a pretty mysterious category as far as I know.

"smashing localizations are in bijection..." this sentence assumes the telescopic conjecture ^^

@SaalHardali The telescope conjecture and dissonant spectra are just "echoes" of the non-noetherianness of the sphere spectrum. This is a well-understood subject in classical algebraic geometry, nothing mysterious

Precisely, for any ring R thick ⊗-ideals of Perf_R are in bijections with subsets of Spec R closed under specialization. To extend this to D(R)=Ind(Perf_R) you need some Noetherianness assumptions.

Similarly, the thick subcategory theorem just says that thick ⊗-ideals of Sp^{\omega} are in bijection with the subsets of (the underlying space of) M_{fg} closed under specialization. That's why we think that the "underlying space" of $\mathrm{Spec}\mathbb{S}$ should be M_{fg}, at least in some form

The telescope conjecture is trying to understand how this relates to thick ⊗-ideals in Ind(Sp^\omega)=Sp, but of course this behaviour can, in general, be wild even for classical rings

@DenisNardin I get what you're saying. Still I think in some sense a more hopeful thing to consider is $\mathcal{M}_fg^{\le n}$ for fixed $n$ and maybe $\mathcal{M}_fg$ should only be considered as an ind topos or something of the sort. Anyway my thoughts on this are really not that valuable anyways. I just find this topic very interesting so I can't resist to express my opinion.

If I don't misunderstand something, in On the (co)homology of commutative rings, Quillen related Hochschild cohomology to a kind of cohomology theory on a category, which is related to the tangent $\infty$-category formalism. I wonder whether we have a dual of this, namely relating HH to something about the tangent $\infty$-category of an $\infty$-category?

@FrankScience Doesn't Quillen's approach work for both homology and cohomology?

It does in the commutative case

Searching google I found a paper of Buchewitz from 2006 introducing the HH analogue of the cotangent complex, which he calls the "Derived Hochschild Complex" of a morphism

There's also a newer paper by John Francis for E_n algebras math.northwestern.edu/~jnkf/writ/cotangentcomplex.pdf

@HarryGindi It seems to me that this is also about Hochschild Cohomology.

@HarryGindi It is mysterious for me that Quillen's approach works for homology but gives something different from Hochschild homology.

I don't know whether we already have results to relate Hochschild homology (or its topological variants) to deformations of an algebra of certain type.

To me it is a bit mysterious why Hochschild cohomology should be related to deformations. I don't think there's any reason to expect Hochschild homology to have any deformation iterpretation

@FrankScience The Buchewitz paper is also about hochschild homology

You can see, he defines both HH^* and HH_* in the paper in terms of the derived hochschild complex

Another question seemingly loosely related: do we have genuine geometric intuition when we talk about "free loop space" of, say, $\operatorname{Spec}(R)$ in DAG (this is another description of HH for commutative rings).

@FrankScience Sure, that's exactly the pullback of the diagonal along itself, like in topology

In DAG it has a more "infinitesimal" nature (because of how homotopy limits work there), but it's essentially the same thing

@HarryGindi Do you have typos? I come up with the result Your search - derived hochschild complex "buchewitz" - did not match any documents.

derived Hochschild whatever is usually called Shukla, I don't know if this is what you're talking about

totally misspelled buchweitz lmao

@FrankScience arxiv.org/pdf/math/0606593.pdf

!purge 20 HarryGindi

Anyway, I wonder if it is exactly the cotangent complex for the category of noncommutative algebras using the tangent infinity category stuff

@FrankScience what do you mean by "genuine geometric intuition"?

HH^1=Exal, right?

@ReubenStern It is very vague. I learn from Lurie that, it should be considered an object "combinatorial" rather than "geometric".

@HarryGindi If I recall correctly there's a shift. Quillen theory is completely equivalent to using the tangent ∞-category

@DenisNardin Could you please make it somewhat precise about the equivalence? I can see the similarity. For example, we work with the infty-category of simplicial commutative rings.

@FrankScience Sorry, do you mean the equivalence between Quillen's theory and Lurie's?

Yes

This seems easy

Well, Lurie defines the cotangent complex as the adjoint of the forgetful functor $Sp(sRing_{/A})→sRing_{/A}$. Quillen does pretty much the same thing with a Quillen adjunction between the derived category of A-modules and algebras over A.

HH^2 classifies square zero algebra extensions

So we need to prove that the derived category of A-modules is equivalent to Sp(sRing_{/A}) over sRing_{/A}

And for fairness, I shouldn't say "Lurie's" theory, since the theory predates him by quite a bit

It seems to me that Lurie's books only define something similar to what you wrote for topological cotangent complex.

Nah, it works for any presentable ∞-category. Of course you can take E_∞-algebras, but you can also take sRing

The warning is that the cotangent complexes you get are different (but that shouldn't be a surprise)

Well, I just looked at math.harvard.edu/~lurie/papers/SAG-rootfile.pdf

page 1715, section 25.3.3

It seems to me there is a difference between tangent $\infty$-category of something similar to what you wrote and the simplicially derived version.

He claims that this will lead to a different concept of square-zero extensions

Yeah you are right. I've found a paper by Schwede that studies the situation in detail

math.uni-bonn.de/people/schwede/stable.pdf

By the way

about Hochschild cohomology

I think the result you want is theorem 5.2 in that paper

But I need to digest the notation

It now seems clearer that Quillen identifies the simplicial "derived" version of Der(A,M) with HH(A,M) for a simplicial noncommutative ring A and a bi-module M.

Hochschild cohomology HH^*(A,M), I meant

Seemingly this is how HH^* related to deformations.

Yes, that is the reason. I just feel it's weird because it shouldn't have anything to do with the free loop space. Yet it does

But for HH_* you don't have anything like this

Sorry

I reread Quillen's paragraph

It seems that there is an alteration

It seems more complicated than a shift.

Isn't proposition 3.6 in Quillen's paper the precise statement you are after?

Yes

Oh, the point is that the degree 0 part is wrong

And we don't know whether there is a comparison map

I mean, I look at the complex of Hochschild cohomology.

We don't know whether the twistings are different even in higher degrees from the result per se.

I need to check the preceding computations instead of just looking at the result.

@DenisNardin I just reread Quillen's proof. From his computation, it seems to me that the cohomology he defined is a shift of cofiber of HH^•(A,M) to A, in general. I cannot relate his LAb(B) to Hochschild homology.