Gregoric's Theorem

Rok Gregoric constructs a symmetric-monoidal equivalence of $\infty$-categories

$$Ind(Coh(\mathfrak{M}))\stackrel{\sim }{\to }Sp$$

where

• $\mathfrak{M}:={\mathcal{M}}_{or-fg}$ is the non-connective spectral DM stack of oriented spectral formal groups to Spectra.
• $Coh(X)\subseteq QCoh(X)$ is thick subcategory generated by the structure sheaf.

Details on the theorem

More precisely, what is shown is the following: consider the constant sheaves/global sections adjunction for $\mathfrak{M}$

$$\Delta :Sp\leftrightarrow QCoh(\mathfrak{M}):{\Gamma }_{\mathfrak{M}}$$

One shows that $\Delta$ becomes fully faithful when restricted along $S{p}^{fin}\to Ind(S{p}^{fin})\simeq Sp$. This is a consequence of a calculation of the global sections of the structure sheaf:

$${\Gamma }_{\mathfrak{M}}({\mathcal{O}}_{\mathfrak{M}})\simeq \mathbb{S}$$

which is related to how the ring spectrum $MU$ Amitsur-completes to $\mathbb{S}$. Ind-completing both sides of $Sp\stackrel{\sim }{\to }img({\Delta }^{fin})\simeq Coh({\mathcal{O}}_{\mathfrak{M}})$ yields the theorem.

Coherent Affinization

What the theorem is saying that the geometric objects $\mathfrak{M}$ and $Spec(\mathbb{S})$ are (1-categorically) Morita equivalent: that is, they have equivalent “(1-)categories of representations” (given here by the functor $Coh(-):P(Aff)\to Ca{t}_{\mathbb{S}}$.)

Furthermore the equivalence is induced by a map of the geometric objects, namely the map into the final object $\mathfrak{M}\to Spec(\mathbb{S})$. We can interpret this as follows:

• This is like saying, when using the sheaf theory $Coh$, the final map exhibits $Spec(\mathbb{S})$ as an affinization of $\mathfrak{M}$.
• That geometry over $\mathfrak{M}$ is a kind of geometry below $Spec(\mathbb{S})$ - it sees things that happen over ${\mathbb{F}}_1$.

Constructing ring spectra

Fix $x:Spec(R)\to {\mathcal{M}}_{or-fg}$, classifying some formal group $G$ over $R$, we can precompose $\Gamma :IndCoh({\mathcal{M}}_{or-fg})$ with pushing forward along $x$ to get a map

$$\Gamma \circ x_{!}:Mo{d}_R\to Sp$$

In particular, the structure sheaf $R\in CAlg(Mo{d}_R)$ gives rise to an $E_{\infty }$ ring spectrum

$$E(R,G):=\Gamma (x_{!}R)\in CAlg(Sp)$$

Question Can the map $Sp\to IndCoh({\mathcal{M}}_{or-fg})$ arise from a certain adjunction over or under Sp? In this sense, the moduli of formal groups is something that encodes descent over something “below $\mathbb{S}$"".

Question $THH(R)\in CRing$, together with its $U(1)$ symmetry, correspond to what construction in $IndCoh({\mathcal{M}}_{or-fg})$?