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am splitting off from Adams spectral sequence
for section 1.2.2 in Higher Algebra.
I generalized the setting of the spectral sequence of a filtered stable homotopy type to an arbitrary homological functor $\pi_\bullet$. As far as I can see nothing is gained by assuming that the functor $\pi_\bullet$ comes from a t-structure, and in practice we’re often interested in the spectral sequence obtained by applying more general homological functors, e.g. $[X,-]$ where $X$ is some compact object.
Proposition 1 was missing some kind of Mittag-Leffler hypothesis, as mentioned in Dylan’s note. It would be nice to figure out exactly what that hypothesis is. The dual setting considered in Higher Algebra is simpler because it’s OK to assume that a homological functor $\pi_\bullet$ preserves sequential colimits, but, in practice, it’s not OK to assume that it preserves all sequential limits. I assume the Mittag-Leffler condition identifies exactly which limits need to be preserved.
Okay, thanks! And thanks for joining in.
I have added a link to Mittag-Leffler condition, though I guess that should have some more comments attached to it.
Could you point to further resources? Where is that Mittag-Leffler condition thing written out?
I think the following are sufficient hypotheses for prop. 1 (I’m just inspecting the proof in Higher Algebra):
Let $\mathcal{C}$ be a stable ∞-category and $\pi_\bullet :\mathcal{C}\to\mathcal{A}$ a homological functor where $\mathcal{A}$ is an abelian category with sequential limits. Suppose that
Then the spectral sequence converges strongly.
So, at least when $\mathcal{A}$ is the category of abelian groups, the Mittag-Leffler condition should say that the $lim^1\pi_\bullet$ of the diagrams in (1) vanish.
A reference where a necessary AND sufficient condition is given for the weak convergence of such spectral sequences is Hilton and Stammbach’s Homological Algebra (Theorem 7.5). They work with chain complexes, but the arguments work more generally.
Thanks! I’m offline now, but i’ll come back to this tomorrow. Maybe you’d enjoy putting what you just wrote into the entry?
Yes, I’ll update the page. I needed to convince myself that hypothesis (1) isn’t unreasonably strong. The hypothesis of Bousfield’s convergence theorem (Theorem 1.3.5 in Dylan’s notes or Theorem 6.6 in the original paper) is actually a sufficient condition for the spectral sequence to be “strongly Mittag-Leffler” in Bousfield’s terminology, which implies (1).
I beefed up the examples a bit and described more explicitly what the convergence of the spectral sequence means in this generality. I think condition (2) in Proposition 1 can be weakened further (e.g. for the Atiyah-Hirzebruch spectral sequence this condition amounts to the spectrum being connective, which is not actually necessary for strong convergence).
I think condition (2) in Proposition 1 can be weakened further (e.g. for the Atiyah-Hirzebruch spectral sequence this condition amounts to the spectrum being connective, which is not actually necessary for strong convergence).
My mistake, condition (2) in the case of the AHSS does not restrict the generality: it says that $[X,\tau_{\geq q} E]=0$ for $q\gg 0$, which is always true if $X$ is a finite spectrum.
Thanks! Much appreciated.
I have added the actual definition of the spectral sequence now. Copying from HA I stuck to the colimit version now.
Added another warning on the dual pictures somewhere, but needs more harmonizing. My battery is dying now, though…
I think everything is consistent now. I’ve dualized all the statements from HA.
Note that the indexing is different than the one in HA: $p$ is always the “suspension index” and $q$ is the “filtration index”. Can’t get any simpler than that, I think.
Thanks! Excellent.
One question: is the chromatic spectral sequence the Lurie spectral sequence for the chromatic tower?
Yes, I believe all three variants in Dylan’s notes are examples of the general construction (in stable categories of $BP_*BP$-comodules, $p$-local spectra, and quasi-coherent sheaves on the stack of $p$-local formal groups, respectively). So are the slice spectral sequences in equivariant and motivic homotopy theory. I’m not sure I know of any “stable” spectral sequence which is not a special case of this one…
Thanks!
This is all rather beautiful. I wish I had appreciated that earlier.
It’s also not very well publically recorded, or is it?
I have created a table-entry, for inclusion in other entries:
So far that has (just) three entries. I am not sure yet how to put in the other entries that you indicated.
The spectral sequence in quasicoherent sheaves given by the height filtration, what would it be called?
And conversely, the slice spectral sequences, what is the canonical name of the corresponding towers?
(I suppose I could look up some stuff and find this out, but you’d save me some valuable minutes if you have a brief answer for me…)
Indeed, it’s quite neat! The idea also works for filtered objects in unstable (∞,1)-categories, although the notion of homological functor becomes messier since we have to allow groups and pointed sets in low degrees.
I’m sure this was a well-known construction before HA, although I don’t know a reference.
The spectral sequence in quasicoherent sheaves given by the height filtration, what would it be called?
The table could include more generally the filtration on the category of sheaves (with value in a stable category) on a filtered stack $X$. The filtration itself I would call “filtration by support”, but I don’t know a good name for the spectral sequence.
And conversely, the slice spectral sequences, what is canonical name of the corresponding towers?
They are (unfortunately) called the slice filtration or slice tower.
@Marc -
the filtration on the category of sheaves (with value in a stable category) on a filtered stack $X$
Does this give an example of that case: A thick subcategory theorem for modules over certain ring spectra?
I don’t think it’s directly related: as I understand it, Akhil shows that for a certain class of stacks $X$, any thick subcategory of $QCoh(X)$ is of the form $QCoh_Z(X)$ for some closed substack $Z\subset X$.
I meant: does this give us filtrations to which your observation applies?
@DavidRoberts: I don’t think so. We would get such a filtration only if the stack in question were filtered by closed substacks (if we wanted, for some reason, to filter our category by thick subcategories). For example, Akhil’s result applies to the moduli of elliptic curves which, as far as I can tell, doesn’t really have an interesting (Zariski) filtration.
Also, Akhil shows more: he shows that for a certain class of stacks, we can detect thick subcategories in the category of modules over the global sections by looking at the category of quasi-coherent sheaves. But in any case it’s not immediately relevant here, as far as I can tell.
ok, thanks, was just wondering.
I added a little bit more explicit detail to the assertion here that a $\mathbb{Z}$-complex in $\mathcal{C}$ induces a chain complex in $Ho(\mathcal{C})$.
I have been editing a bit more at spectral sequence of a filtered stable homotopy type, trying to fill in more details of the proofs.
I am not done yet, but will have to interrupt now for a bit.
Some comments for past and potential future co-editors (in particular Marc):
for index sanity I would like to have the homological version, too, closely following Lurie. So I decided to eventually give both the homological and the cohomological version explicitly in the entry. Of course it is redundant, but for actually working with the entry it can be a relief to be able to just read off entries without having to think about what needs to be dualized.
In particular I start now with saying “this is a filtered object, this is a cofiltered object”.
But I am really not done yet and really have to dash off now for some other duty. This here is mainly to say: I’ll come back to editing this later this in a few hours and if anyone feels the urge to go editing himself, maybe I am asking: please wait until tomorrow, when I am finished!
for index sanity I would like to have the homological version, too, closely following Lurie. So I decided to eventually give both the homological and the cohomological version explicitly in the entry. Of course it is redundant, but for actually working with the entry it can be a relief to be able to just read off entries without having to think about what needs to be dualized.
In particular I start now with saying “this is a filtered object, this is a cofiltered object”.
This is good, filtered should have been cofiltered before.
I notice that you’re now using Lurie’s indexing in the filtered case, which is not dual to the simpler indexing I had set up in the cofiltered case (see post #11). I know Lurie’s indexing is the “standard” one, but as far as I can tell that standard indexing is only justified when it gives a first quadrant spectral sequence, to make it easy to see which differentials vanish. Anyway, the filtered and cofiltered cases should be harmonized one way or the other, but I think the natural indexing is more helpful in this generality.
Yes, that’s why I posted this warning. But I want it all in Lurie-convention for the moment, not to get myself and others all mixed up when comparing. I’ll try to add discussion/harmonization a little later.
Sorry, but I am retaining all what you did and nothing of your efforts will be wasted in the end. Just give me a bit more time with editing. I’ll get back to you a bit later…
I have now split up into two sub-sections:
In the first of these I have filled in essentially all the relevant proofs from HA (only missing is the proof that $\phi$ on p. 41 is surjective). I closely followed HA but here and there I think I added a tad more detail, also I believe I fixed one or two minor typos in the text. Of course very likely I also introduced new typos.
To the second of these I moved the discussion that used to be there prior to my latest intervention, hence the survey of the statements for cofiltered objects.
I have not yet (not yet!) edited that any further, not yet harmonoized further, except for adding a brief remark that this needs to be done. I still plan to do that, but probably not tonight.
Proposition 4 is missing the hypothesis that the functor $\pi$ should preserve some sequential colimits, cf. the dual statement.
There should be a note somewhere that the construction of the spectral sequence only uses the underlying triangulated category. For example, I’m pretty sure the Adams spectral sequence exists for $E$ any commutative monoid in $Ho(Spec)$. You need the higher structure to make the construction functorial though.
About the indexing: I think there can be no debate that the cofiltered case is simpler (compare for instance the expressions for the $E_1$ and $E_\infty$ pages). So the question is whether consistency with HA is more important. I would say “no”, but I really don’t feel strongly about it.
Okay, thanks (again). I have added that clause to prop 4 and added a remark as you suggest, now remark 7.
Concerning the indexing: I suppose I all agree with you on absolute idealistic grounds. Just for mundane purposes I stuck to HA because I wanted to write out the proofs in detail in the nLab page and laziness gave me the idea that this would be less of a pain if I just stuck to the HA indices for that purpose.
So I have no objections to your attitude, I guess I am just being lazy. You should either feel invited to edit yourself as you deem appropriate or else wait and trust (or hope) that I come back to this and adjust it.
Why does it take so much work to define the spectral sequence of a filtered stable homotopy type, with all this nonsense about $(\mathbb{Z},\le)^{\Delta[1]}$? It seems to me that there is a very easy construction (which is the same as the one I’m using in HoTT): each cofiber sequence $X_{p-1} \to X_p \to C_p$ gives a LES, and since the homotopy groups of each $X_p$ appear in two adjacent LES’s, we have an exact couple, with $D$ the homotopy groups of the $X_p$’s and $E$ the homotopy groups of the $C_p$’s.
Also, is it well-known that you can get a Leray-Serre-type SS from this construction as well? Given a fibration $F\xrightarrow{i} E\xrightarrow{p} B$ and a parametrized spectrum $J$ over $E$ (which could be something as simple as a constant Eilenberg-MacLane spectrum), consider the Postnikov tower of $p_* E$:
$p_* J \to \cdots \to (p_* J)_{n+1} \to (p_* J)_n \to \cdots$then apply the exact functor $r_*$, where $r:B\to 1$ is the projection, to get a cofiltered spectrum
$r_* p_* J \to \cdots \to r_*(p_* J)_{n+1} \to r_*(p_* J)_n \to \cdots$The homotopy groups of $r_* p_* J$ (the limit of the SS) are the (parametrized generalized) cohomology groups $J^*(E)$. The (co)fibers are $r_*$ of the (co)fibers of the $p_* J$ Postnikov tower; hence the first page is the (parametrized generalized) cohomology of $B$ with coefficients in the (co)fibers of the Postnikov tower. The latter can then be identified with the parametrized Eilenberg-MacLane spectra of the local systems associated to the homotopy groups of $F$. (This is perhaps a little more obvious in HoTT, but it shouldn’t be too hard classically either.)
Is it so much work? Isn’t it just a slightly over-pedantic way of book-keeping? I liked it because it was so simple.
But you have thought much more about it, and it would be nice if you added your perspective! Also, it would be nice to see how HoTT makes it yet more simple.
(I don’t have resources to come back to this at the moment, hopefully Marc sees this and gives you more useful reactions.)
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