Spectral cotangent space

I’m supposed to give a 10 minute talk on what I’ve been up to recently. 10 minute talks are pretty brutally short, though I think they’re better than 20 minute talks — those give you the illusion that you might be able to convey something, but you really just flatly can’t. With only 10 minutes you’re not kidding anyone.

Even so, I should still practice, and writing out what I want to say is a good place to start, and in turn a good blog post.

So, in chromatic homotopy theory a basic object of study is $E$-theory, which is a functor $\mathsf{Spectra} \to \mathsf{QCoh}(LT_n, \Gamma)$ from the category of spectra to the category of quasicoherent sheaves on the $n$th Lubin-Tate deformation space, equivariant against the action of the Morava stabilizer group $\Gamma$. This is some fancied up way of describing a homology theory $E_n^\vee$, taking values in $(E_n^\vee, E_n^\vee E_n)$-comodules. This functor factors through a localization factor to the category of $K(n)$-local spectra, and the whole of chromatic homotopy theory revolves around trying to understand $E$-theory and this localized category.

In the parent category of spectra, we are especially interested in spheres, and one characterization of spheres is that they are “invertible” with respect to the monoidal structure endowed by the smash product, in the sense that for each $n$ we have $\mathbb{S}^n \wedge \mathbb{S}^{-n} \simeq \mathbb{S}^0$, and this happens for no other spectra. The smash product descends to the $K(n)$-local category, where we can consider spectra $X$ with the same property: there is some $Y$ with $X \wedge Y = L_{K(n)} \mathbb{S}^0$. This group, denoted $\mathrm{Pic}_n$, turns out to be much more interesting — it is a p-adic analytic group. Understanding this group has turned out to be related to understanding the homotopy groups of the $K(n)$-local sphere, and so $\mathrm{Pic}_n$ is both hard to describe and greatly interesting.

There are also invertible objects in the target category of the $E$-theory functor: these are the equivariant line bundles. The Kunneth theorem shows that, in fact, the $E$-theory functor restricted to $\mathrm{Pic}_n$ factors through the subcategory of line bundles. We can also take the pullback of the corner whose edges are $K(n)$-local spectra and equivariant line bundles mapping to / including into equivariant sheaves on Lubin-Tate space — these are of course the $K(n)$-local spectra whose image under the E-theory functor is one of these line bundles. Hopkins, Mahowald, and Sadofsky show that the induced map from $\mathrm{Pic}_n$ to this pullback is an isomorphism (maybe this was known before them, but it seems to first appear there). This is a very useful detection theorem for when we’ve found an element of the Picard group, and so one thing we can do is to try to produce examples of line bundles first, and then lift these examples to spectra.

So, we want to build 1-dimensional free modules, essentially. One way you can get such a thing is to take a 1-dimensional smooth variety and build its cotangent sheaf — this gives a map $T^*$ from such things to 1-dimensional free modules. We can additionally suppose that we have some spectrum $X$ in mind so that $\mathrm{Spf}\;E_n^* X$ is such a 1-dimensional smooth formal variety. (An important thing here is that $X$ needs a diagonal map so that cohomology is ring-valued.) What we’re looking for, then, is a kind of interchange law — starting with X we can apply $\mathrm{Spf}\;E_n^*$ to get a variety and then $T^*$ to get the module, but we would like to have a functor $\tilde T^*$ which is spectrum-valued, and then $E_n^\vee \tilde T^* X = T^* \mathrm{Spf}\; E_n^* X$. This $\tilde T^*$ is what I’ve constructed.

So, with such a $\tilde T^*$, one can ask what good it is — we need to get some interesting things out, or we haven’t really accomplished our goal of learning more about $\mathrm{Pic}_n$. We’ll look at $\tilde T^* \Sigma^\infty_+ X$ for some input spaces $X$. When $X = \mathbb{C}\mathrm{P}^\infty$, it turns out we get $\mathbb{S}^2$. When $X = \mathbb{R}\mathrm{P}^\infty$, we get $\mathbb{S}^1$. The first interesting example comes from a calculation of Ravenel and Wilson (yes, that thing again): $X = \underline{\smash{H\mathbb{Z}/p^\infty}}_n$ has this property as well. In that case, it turns out that we get a spectrum whose image is the determinantally twisted line bundle, and so this is sometimes called the determinantal sphere.

The determinantal sphere appears in a few places — it’s really the only important example of an element of the $K(n)$-local Picard group that we know for all n. For instance, the Brown-Comenetz dual of the sphere, investigated by Hopkins and Gross, turns out to have this same property. So does a spectrum constructed by Goerss, Henn, Mahowald, and Rezk, in their paper on $\mathrm{Pic}_2$ at $p = 3$, though there’s some buzz about these two examples differing by a smash factor invisible to the E-theory functor (called an “exotic element”). There are also a lot of structural things that this construction points out! It’s all very interesting. But, in 10 minutes, that’s probably as much as I can hope to say — so that’s where I’ll quit.