The Chromatic Picture

Thick Subcategories, closed and open towers

Let $Sp$ dentoe $p$-local spectra for the remainder of this.

The thick subcategory theorem states that tensor ideals of $(S{p}^{fin}{)}^{\otimes }$ assemble into a downwards filtration

$$\cdots \hookrightarrow S{p}_{type\ge 2}\hookrightarrow S{p}_{type\ge 1}\hookrightarrow S{p}_{type\ge 0}:=S{p}^{fin}$$

We’ll refer to an object the full subcategory $S{p}_{type\ge n}$ as a finite spectrum of type $\ge n$.

The layers (verdier quotients) of these are to be interpreted as finite spectrum of type exactly $n$

$$S{p}_{type=n}:=S{p}_{type\ge n}/S{p}_{type\ge n+1}$$

(This is one characterization of $T(n)$-local spectra)

Bounding the type of a spectrum below is a “closed” procedure: the inclusion maps form the closed part of a recollement. This is a non-commutative version of completing a function along a closed subscheme. Hence, there’s a corresponding “open”-tower

$$S{p}_{type\ge n}\hookrightarrow Sp\hookleftarrow S{p}_{type\le n-1}$$

And the open tower of truncations from above is connected by maps in the other direction

$$S{p}_{type\le 0}\hookrightarrow S{p}_{type\le 1}\hookrightarrow S{p}_{type\le 2}\hookrightarrow \cdots$$

Geometric picture

The basic objects are:

• moduli of oriented spectral formal groups

$$\mathfrak{M}:=\left[\frac{Spec(MU)}{Aut(MU/\mathbb{S})}\right]$$

• moduli of oriented spectral formal groups up to height $n$

$${\mathfrak{M}}_{\le n}:=\left[\frac{Spec(E_n)}{Aut(E_n/\mathbb{S})}\right]$$

The ring spectrum map $MU\to E_n$ (This comes by construction of $E_n$, it’s a sort of a universal/Lubin-Tate orientation) induces a map ${\mathfrak{M}}_{\le }n\to \mathfrak{M}$. This will turn out to be an open map of stacks.

At height zero, we have

${\mathfrak{M}}_{\le 0}\simeq \left[\frac{Spec(H\mathbb{Q})}{Aut(\mathbb{H}Q/\mathbb{S})}\right]\simeq Spec(L_{H\mathbb{Q}}\mathbb{S})\simeq Spec(H\mathbb{Q})$

So one way to say what’s happening is that the “generic point” $(0):Spec(H\mathbb{Q})\to Spec(\mathbb{S})$ factors through the “affinization” $\mathfrak{M}\to Spec(\mathbb{S})$, which in turn gets interpolated by a sequence of open embeddings

$${\mathfrak{M}}_{\le 0}\hookrightarrow {\mathfrak{M}}_{\le 1}\hookrightarrow {\mathfrak{M}}_{\le 2}\hookrightarrow \cdots ,/\mathfrak{M}$$

Theorem (Gregoric)

There’s an equivalence of stable $\infty$-categories

$$QCoh({\mathfrak{M}}_{\le n})\simeq S{p}_{type\le n}$$

and pushing forward along the open embeddings correspond to the inclusion maps in the open tower.