Is there a constructive description of type in the p-local stable homotopy category?

Question

The title pretty much sums it up - but let me give a little bit of background first.

In the p-local stable homotopy category (basically one localizes away the torsion spectra which are not p-torsion) the Morava K-theories are a non-negative integer indexed family of spectra (morally they are residue objects or "homological fibre functors") which classify the thick subcategories of SH^{(fin)_p the p-local stable homotopy category of finite spectra. This classification assigns to each p-local finite spectrum a type (namely the smallest thick subcategory it occurs in).

As far as I know the only definitions of type are directly in this way via the Morava K-theories or in terms of periodic self maps which are still really in terms of Morava K-theory.

Is there a more "constructive" definition of type? For instance can one determine the type of a p-local finite spectrum in terms of how bad the obstruction to it generating the whole of SH^{(fin}}_p is? Really I would be happy with any answer which was somewhat more constructive or to find out that the question is open/ridiculous.

Now let me explain somewhat the motivation for this question and how it came about. One can view the Morava K-theories (as I mentioned above) as residue objects/homological fibre functors (the term homological fibre functor has less algebraic geometry bias and sounds cooler) in the sense that their behaviour is analogous to residue fields of points in the derived category of quasi-coherent modules on a scheme (to be safe one should really take the derived category of O_X-modules with quasi-coherent cohomology) and with \kappa-modules in modular representation theory. Namely they all give tensor functors to some flavour of graded vector space category which classify thick subcategories.

The mod n Moore spectra are `Koszul objects' which we want to view as analogues of the usual Koszul complexes on a scheme and of Carlson modules in modular rep theory. In other words a Koszul object is a cone on some (possibly graded and maybe also twisted) element of the endomorphism ring of the tensor unit of our category.

Now one can associate some geometry to a biexact tensor product (by biexact tensor product I mean symmetric monoidal structure which is exact in each variable, there are no decency assumptions or extra axioms regarding compatibility with the triangulation required) on an essentially small triangulated category. It is possible to cook up a locally ringed space associated to such a category with tensor product (this is work of Paul Balmer). In the two algebraic cases, derived categories of schemes and stable categories in modular rep theory, one gets (with some mild hypothesis in the algebraic geometry case) back the scheme or recovers the projective support variety. This comes down in some sense to the fact that the Koszul objects determine the topology, or equivalently that they determine the thick subcategories in some sense.

This fails for the stable homotopy category of finite spectra (both globally and p-locally). The mod n Moore spectra are not enough - one needs the Morava K-theories. The locally ringed space one gets is not a scheme (nor an algebraic space). One can then ask if there is some "global" reason that this happens (even though it is not at all a surprise) other than the fact that the Morava K-theories are just there generating subcategories p-locally. This is motivated partially by trying to understand the failure of a certain comparison map to be injective (which I didn't mention - it would be interesting to have a good criterion for its injectivity and I currently only have quite hard to check ones) and to try to get some feel for what properties can cause the associated locally ringed space to fail to be algebraic (one really needs more examples computed for this and I haven't found the time yet unfortunately).

So basically I feel that if there were some definition of type that made clear the failure to be able to reduce the type by taking triangles and suspensions (or something other than just the residue objects being there) it might be quite enlightening. In particular, in general one wouldn't expect to produce an algebraic gadget from a topological triangulated category (in the sense of Schwede ). That is why I mentioned the extension problem as if topologicalness provided some global obstruction to Koszul objects being enough that would be very interesting.

That got very long - I hope it is interesting/useful.

Answer 1

My theory on this, which could be wrong, is that if the stable homotopy of the sphere (whatever the words "sphere" and "stable homotopy" mean in your situation) is Noetherian commutative, then type should be determined by prime ideals in the stable homotopy of the sphere.

For example

1. D(R). Stable homotopy is homology, sphere is R. If R is Noetherian commutative, the type of X is determined by ann H_* X. This is the Hopkins thick subcategory theorem.

2. Stmod(kG). This one is a little confusing, because stable homotopy is really Tate cohomology, which is never Noetherian. But group cohomology is Noetherian, and Stmod(kG) is just a localization of D(kG). The sphere is k. So we should expect the type of M to be determined by ann [k,M] in D(kG), which is something like the Benson-Carlson-Rickard theorem.

3. Stable homotopy category. Here the homotopy of the sphere is far, far from Noetherian, so you cannot expect the type to be determined by the homotopy of the sphere. There are "embedded primes"; prime ideals in the category not visible in the homotopy of the sphere. You kill p, and a new prime ideal pops up that was not previously visible.

Answer 2

By applying the thick subcategory theorem you can come up with lots of new definitions of "type", but it is somewhat unsatisfying because the fact that any of them are equivalent is very non-obvious. You can steal several from Ravenel's orange book. These include conditions like conditions on MU-homology or BP-homology of X that lift conditions on the K(n)-homology, or conditions on what kinds of endomorphisms exist in the (center of the) graded endomorphism ring of X, or conditions on the thick subcategory of the p-local stable homotopy category that X generates (it generates all spectra of type >= n), or on the slope of the smallest vanishing line of the Adams spectral sequence computing the homotopy groups of X, etc, etc.

I'm not sure what you mean by the potential factorizations you propose. For example, a spectrum Y of type greater than zero is annihilated by some power of p, and so any map from X factors through a cone X/p^n, and there are further factorizations through cones beyond that. However, to define what I mean by these further cones I need to say something about what a type-n self-map is in the first plane, and I'm not sure how to do that from first principles.

I hope to see more details and if I can say something more specific then I'll try.