Higher vanishing cycles

Question

The generalisation of the vanishing cycle formalism in SGA 7 is apparently since the 1970's an issue, Morava mentioned a connection with Bousfield localization. I find the Morava's remarks un-understandable - would someone explain?

Answer 1

My apologies if this is too much or too little; leave a comment and I can try and correct it. He's talking about a specific issue in homotopy theory that we'd like a better understanding of.

The stable homotopy category (implicitly localized at a prime p) has a stratification into "chromatic" layers, which correspond to a connection to formal group laws. We geometrically think of the stable homotopy category as some kind of category of sheaves on a moduli stack X which has a sequence of open substacks X(n) - these are the "E(n)-local categories", and there are Bousfield localization functors taking a general element M to its E(n)-localization L_E(n) M, which you can think of as restricting to the open substack. (A general Bousfield localization will take some notion of "equivalence" and construct a universal new category where those equivalences become isomorphisms, but in an appropriately derived way.)

The difference X(n) \ X(n-1) between two adjacent layers is a closed substack of X(n), which in our language is the "K(n)-local category". There is also a Bousfield localization functor that takes an element M to its K(n)-localization L_K(n) M. Bousfield localization is pretty general machinery and in the previous "open" situation it acted as restriction; in this "closed" situation it acts as a completion along the closed substack.

We have some general understanding of the K(n)-local categories. They act a lot like some kind of quotient stack of some Lubin-Tate space classifying deformations of a height n formal group law by the group scheme of automorphisms of said formal group law, which is the n'th Morava stabilizer group S_n. Geometrically we think about it as a point with a fairly large automorphism group (even though this is, of course, the wrong way to think about things). These are places where you can get dirty and do specific computations and examine one chromatic layer at a time.

There are two remaining pieces of data we need, then, to understand M itself from its localizations L_K(n) M: we need to understand how they are patched together into the E(n)-localizations, and we need to understand the limit of the L_E(n)M. The latter is a "chromatic convergence" question and not immediately relevant to the point under discussion.

In general there is a "patching" diagram, which is roughly something like the data you'd usually associate with a recollement. (My favorite reference for data in this kind of situation is Mazur's "Notes on etale cohomology of number fields".) We have a (homotopy) pullback diagram

LE(n) M  ->  LE(n-1) M
     |             |
     V             V
LK(n) M  ->  LE(n-1) LK(n) M

that tells us that a general E(n)-local object is reconstructed from a K(n)-local object (something concentrated on the closed stack), an E(n-1)-local object (concentrated on the open stack), and patching data (a map from the object on the open stack to the restriction of the complete object to the open stack). This roughly follows because the K(n)-localization of any E(n-1)-local object is trivial.

The functor that takes an object concentrated near the closed substack and restricts it (in a derived way) to the open substack is what Morava considers. Here, in the language of Bousfield localization, it is E(n-1)-localization applied to K(n)-local objects. What he seems to be proposing is that this general Bousfield localization setup should be one way of thinking about the vanishing cycles functors (and I concur with his dislike for the "vanishing" terminology) in which we can, in a fully derived way, view sheaves on a large stack as coming from patching data on an open-closed pair.

Just to close the loop, what we don't really understand at all in this picture is what this "trans-chromatic-layer" stuff really does. We have, for example, two stabilizer groups connected to formal group laws of adjacent heights, and we don't really understand what the specialization functor is really doing in this case.

Answer 2

I wouldn't go as far as to say I understand, but what I get from it is the following:
One can roughly consider the vanishing cycles functor as coming from gluing data for the recollement associated to the singular fibre. This is a very general piece of gadgetry (as is Bousfield localization) and one could view it as a general version of the vanishing cycles functor. If furthermore, one could not only cook up a version of the fundamental groupoid corresponding to homological fibre functors and their automorphisms but also associate information to open pieces and inclusions one could hope to view this in a reasonable way as the monodromy around the Bousfield localization giving rise to the vanishing cycles functor. This would allow one to define monodromy isomorphisms and the other bits and pieces one would like to have as part of the formalism (and might actually also be necessary to give the right definition of the vanishing cycles functor as well).

Maybe this is horribly wrong, but if it is hopefully it will motivate someone to correct me and I can remove this!