Jack Morava has a very pleasant introduction to some of the rationale behind chromatic homotopy theory and how algebraic geometry gets used to organize topological things, called Complex Cobordism and Algebraic Topology. It’s short, and it’s worth your time to read whether you’re an expert or a novice. I’d opened it again recently to help prepare for a talk I was to give, and I found a gem of a paragraph which I hadn’t really processed before. I’d like to try to share that with you now.
The story starts the usual way: there is a sense in which the homology groups can be thought of as a sheaf over the moduli space of formal group laws, equivariant against the action of the group of formal diffeomorphisms , thought of as change-of-coordinate maps. Equivalently, this can be packaged into saying that forms a sheaf over the moduli stack of formal groups. After -localization, the geometric points of this stack admit a nice description: the positive characteristic ones are indexed by “heights” , along with one stray rational point of “height 0”. There is a construction of a family of formal group laws, called the Honda formal group laws, which produce examples of each of these formal groups of finite height, and one can think of some of the action of as being “spent” on moving an arbitrary formal group law into one of these canonical forms. However, some of will remain — the part which gives automorphisms of the Honda formal group law — and this part is referred to as the “stabilizer group” .
E-theory comes in by attempting to restrict the sheaf to one of these geometric points. The inclusion of such a point is not typically a flat map (and so one would not expect restriction to produce another homology theory), but this can be corrected by instead considering the inclusion of its formal neighborhood. Then, an E-theory (up to the fuss I was raising in a previous post) is exactly given by the restriction of the bordism sheaf to the deformation space of one of these geometric points on the moduli of formal groups.
Good. Now let me tell you fewer generalities and more formulas. The Honda formal group law is cooked up so that its multiplication-by-p endomorphism, called its “-series”, is given by . Any formal group law in positive characteristic carries a Frobenius endomorphism , and so in the endomorphism ring of the Honda formal group law, there is a relation . It’s a theorem of Cartier that if the base field has sufficiently many roots of unity, then this is actually a complete description of the endomorphism ring: it is the maximal order of a division algebra
where the angle brackets denote a free noncommuting associative algebra. The stabilizer group can then be identified with the compositional units of the endomorphism ring: .
What about the point at infinite height? The notation is actually arranged so that you can sort of take limits in n: the -series becomes , and the endomorphism ring becomes
Much of the chromatic story is unable to be made explicit, but these two objects are quite familiar: the formal group law whose -series vanishes is precisely the additive formal group law . The endomorphisms of the additive formal group law are indeed exactly the power series concentrated in degrees which are powers of p, matching the description of the limiting endomorphism algebra. In terms of homology theories, the additive formal group law belongs to ordinary homology (as it is the only formal group law it can carry for degree limitations), and I’ve previously done a questionable job in justifying a connection between this endomorphism algebra and the Steenrod algebra.
Here, finally, is where Jack’s story becomes nonclassical. He says that the Morava E-theory of a finite spectrum gives a representation of the stabilizer group , and if we take these division algebras and quotient them by p, then we can pull back these E-theory representations to get a sequence of representations over the group
Then, he says, these should “limit” in an appropriate sense to the representation that ordinary mod- homology produces. This is a weird statement — the stabilizer groups themselves do not fit into any such sequence, and the limit appears to be illegitimate — but in the barest and fuzziest sense, it is sort of reasonable. The Atiyah–Hirzebruch spectral sequence computing the connective Morava -theory of a finite spectrum from its ordinary mod- homology will be affected by sparseness, and so for very large , we wouldn’t expect the differential behavior of the -AHSS to differ from the -AHSS.
I would posit that something more might be happening.
The height of a formal group law in positive characteristic has a multitude of different definitions. One of them is computed by finding the degree of the bottom nonzero term in the -series, which makes it plain that the characterization of the Honda formal group law given above is really a formal group law of height n. Another is that there is an integral equation expressing the logarithm associated to any formal group law :
For finite height formal group laws, this integral equation has no solution — eventually you end up having to integrate a monomial like . Indeed, the degree of the bottommost obstruction to this integral equation is in a degree of precisely that form, and that value is referred to as the “height”.
I bring this up as an attempt to find geometric justification for this quotient by p that Jack performs. By setting p equal to zero in the endomorphism algebra, we are asserting that the map has the action of the zero map, which can be minimally justified by setting itself to zero — i.e., restricting to the th order neighborhood on the formal group. On this neighborhood, the formal group law possesses a truncated logarithm which compares it to a truncated additive formal group law, and Jack is somehow asserting that as this neighborhood grows to encompass the whole formal object, the bordism sheaves “over the truncated objects” should fit together to recover the bordism sheaf over the additive object. (I’ve left “over the truncated objects” in quotes, because I don’t see how one can make this sane. The bordism sheaf doesn’t live “over the formal group” in any sense I can see.) But, if something like this were true, I would expect that the finiteness hypothesis could be relaxed to allow for bounded-below complexes, or at least suspension spectra. (Maybe it’s also worth pointing out that the E-theory of a space carries power operations, and in particular an action by . Possibly some construction like is relevant?) In any case, this is something I would like to know about.
This dishonest geometry also suggests another intriguing connection: there is a fast mnemonic for remembering the standard quotient algebras of the mod-2 Steenrod algebra. Since the dual Steenrod algebra itself corepresents automorphisms of the additive formal group law, these should corepresent a certain subset of them — a certain subgroup, even, since these are quotient Hopf algebras. Indeed, corepresents precisely the power series which vanish above order , together with further restrictions so that their inverses and compositions remain supported in this range. So, these algebras are another kind of “truncation” of the infinite height stabilizer group — possibly not quite the same, but certainly not far off from what Jack is talking about. It was of interest for a while to try to find spectra with the property that ; such spectra would behave as though they were taking mod-2 cohomology, modified so that its coalgebra of cooperations was (cf., the Adams spectral sequence method for computing the real -theory of spaces). This fell out, though, once the Hopf invariant one theorem was proven: it shows that there can be no such spectrum for . However, the spectra , , and all exist, and there is the following intriguing table:
I would like to think that the rightmost column of this table this isn’t an accident. In light of the above discussion with endomorphism algebras, is there some reason why this table should be true? Is there a natural candidate for subgroups of the stabilizer group which extend the E-theory side of this table, if not the side? Can partial theorems be proven about the nonexistent spectra, given these -local models? Can one see from the perspective of formal groups and some knowledge of the stabilizer groups this famous theorem that does not exist? It would also be nice to see the information flow the other way — to think of E-theories as repairing some specific failure in the ideology of that prevents it from existing and which they circumvent.
This doesn’t really fit anywhere, but Morava has another article, titled Forms of K-theory, which has another really unique perspective on things. Published in 1989 and written in 1973, he manages to construct -adic in a neighborhood of the cuspidal curve, along with a scattering of other really interesting ideas. It’s really cool to see some of the lesser known tributaries that led into chromatic homotopy theory, now that so much has become at least quasi-standard and worn smooth.
In this same preparation spree, I also rediscovered a talk by Mike Hopkins, From Spectra to Stacks, where he produces stacks associated to any ring spectrum rather than just those which are complex oriented. He does this by passing them through in a certain sense, and what’s interesting is this remark of his that isn’t special in this regard; any spectrum satisfying a few nicety properties will do. Morava, in Forms of -theory, remarks that it’s not clear why complex bordism should have such a rich tie to arithmetic geometry, and in some sense this Hopkins remark explains it: that tie is always there, but complex bordism is the first example of a spectrum where the algebra of and is free enough and ‘spread out’ enough that topology doesn’t have a chance to come in and piss all over the rug. Why there’s a connection between formal groups and topology, however, is no less of a mystery.
Anyway, these are also worth reading.