In this post, we’ll prove Gross-Hopkins duality using the arithmetic geometry developed in the previous post. Recall the main theorem from last time:
Theorem (Gross-Hopkins): Once we pass to the generic fiber, the Gross-Hopkins map is -equivariant and etale.
Gross and Hopkins use this theorem to understand the canonical bundle of . Let denote the determinant given by the composite . If is a -equivariant sheaf on , denote by the same sheaf along with the -action twisted by , so that
Corollary: Let denote the sheaf of invariant differentials on the universal deformation of to , so that (where is the zero section). Then there is a -equivariant isomorphism of line bundles
Here’s a sketch of my attempt at a proof.
Proof Sketch. Gross and Hopkins prove (see Proposition 23.2 of their paper) that the canonical bundle of is, as a -equivariant bundle, given by . Since is etale on the generic fiber, we can pull this back to get the desired isomorphism. By Proposition 20.3 of their paper, we’re done.
Here’s why homotopy theorists care about this result. Like in any closed symmetric monoidal category, we can try to take the dual of an object. In general, this object won’t necessarily behave as we expect a dual to behave (i.e., to have evaluation and coevaluation maps that play nicely), but if the object is finite (say, compact), the notion of dualizability works well. We can play this game of dualizing objects in the category of spectra. One needs to be a little careful, because spectra form an -category, and one has to tiptoe their way through -category–land.
Just like in the category of abelian groups, we have Pontryagin duality in the category of spectra! In particular, there is a spectrum (called the Brown-Comenetz dualizing spectrum), analogous to — and built out of — the injective abelian group , such that . Note the reversal in the degree: this is a common characteristic of dualities in homotopy theory. There’s a canonical map . The fiber of this map is defined to be ; this is called the Anderson dualizing spectrum. We can now consider Anderson dual of a spectrum : this is the spectrum .
Gross-Hopkins duality provides a relationship between these two kinds of duality, namely Spanier-Whitehead duality and Brown-Comenetz/Anderson duality.
Theorem (Gross-Hopkins duality): Let denote the -local sphere. Then is -locally invertible. Moreover, if denotes Morava -theory (at height ) and denotes , then there’s a -equivariant isomorphism
This might seem like an unnecessarily complicated result to even be useful. It turns out that if , the formula in the above equation determines uniquely. In fact, one can write
where is a spectrum that exists only after -localization, called the “determinantal sphere”. If and , then is just the -sphere, but in general, it’s a very exotic.
We can use this formula to write down a relationship between Spanier-Whitehead and Brown-Comenetz duality:
To deduce this, we needed the statement of the completed -homology of ; this can be proven by using the Corollary from above: One uses the following facts to do so (a great, but technical, exposition is Strickland’s paper, titled Gross-Hopkins duality).
- There’s an equivalence , whatever the latter symbols mean.
- There’s another equivalence . If you’d like, take this to be the definition of , as we won’t ever be using the spectrum .
- There’s yet another equivalence . From these equivalences, we get:
By the corollary from above, it now suffices to show that . Recall that the reduced cohomology is defined as the kernel of , induced by the inclusion of the basepoint into . If is complex-oriented, so that (with ), then . This is exactly the maximal ideal of the formal group . Now, is exactly — precisely the relative cotangent space/module of invariant differentials ! Remembering that is complex-oriented, this tells us that .
Let’s revisit our equivalence
Notice the eerie similarity with Serre duality, which states that if is a projective variety of dimension over with dualizing sheaf , there is a canonical isomorphism
where is a coherent sheaf on . As far as I know, there’s no explanation for this similarity with Gross-Hopkins duality in homotopy theory. I think, though, that one can deduce the invertibility of from some general principles of Serre duality.
We need to understand when the structure sheaf (or some shift of it) of a derived stack is itself a dualizing complex. In this case, we say that is self-dual.
Dualizing sheaves are unique up to invertible objects (if and are two dualizing sheaves for , then ). Thus, self-duality implies that is invertible, where is the structure map.
For instance, if is a -ring such that is a regular Noetherian ring of finite Krull dimension, then is a dualizing complex for . This is Corollary 22.214.171.124 of Spectral Algebraic Geometry.
Let be an even periodic -ring. Suppose is a nonzero divisor. In general, the quotient is not an -ring. Suppose, though, that is another -ring. Then is a closed immersion, and . This is exactly the fiber of , which is . Inductively, it follows that if is an ideal generated by a regular sequence such that is an -ring, then is a closed immersion, and . Thus if is a dualizing sheaf for and is an ideal satisfying the above conditions, then is a dualizing sheaf for .
It follows from this discussion that any -quotient of a regular Noetherian -ring of finite Krull dimension is also self-dual.
We can deduce the invertibility of the Gross-Hopkins spectrum from this: it follows from the above discussion that is self-dual. Indeed, the above discussion shows that is self-dual. Since the map is etale and surjective, Remark 126.96.36.199 of Spectral Algebraic Geometry implies that is self-dual, which tells us that the Gross-Hopkins spectrum is (-locally) invertible.
A natural question to ask is the following. Can recent results by Barthel-Beaudry-Stojanoska on the Anderson self-duality of higher real K-theories be deduced from the discussion above? One step in this direction is the following.
If is any finite subgroup of , the map is an etale surjection. We know, by an argument similar to the proof of Theorem 1 in this post, that . Suppose is such that . Again utilizing Remark 188.8.131.52 of Spectral Algebraic Geometry, we learn that is self-dual, so that is in . It’d follow that is self-dual if we knew that is cyclic. This is known to be true in some cases; it’d be interesting to find necessary and sufficient conditions on that ensure that this group is cyclic.