Local dimensions in derived spherical geometry

Let $X:CAlg\to S$ be a $E_{\infty }$ prestack, $x\in X(R)$ an $R$-point, and say $X$ admits a cotangent complex at $x$, this is an object

${\mathbb{L}}_x\in Mo{d}_R$ corepresenting the functor $X(R\oplus (-)){\times }_{X(R)}\{x\}:Mo{d}_R\to S$.

When $R$ is of the form $Hk$ for a field $k$ we have dimension counts

$$d_i:=di{m}_k({\pi }_i({\mathbb{L}}_x))$$

This sequence of (possibly infinite) numbers dimension at the point $x$. If $X$ is smooth $x$, we expect $d_i$ to only be non vanishing for $i=0$. More generally, positive degrees detect “non-smoothness/singularity”, negative detect infinitesimal symmetires/stackiness.

Now recall, Basterra-Mandell’s calculation

${\mathbb{L}}_{\mathbb{A}}\simeq \mathbb{S}[t]{\otimes }_{\mathbb{S}}H\mathbb{Z}$ We find that at any $x:Spec(R)\to \mathbb{A}$, ${\mathbb{L}}_{\mathbb{A},x}\simeq H\mathbb{Z}{\otimes }_{\mathbb{S}}R$ .

Generalizing, we find ${\mathbb{A}}^n$ pull back along any $R$-point to the module $H{\mathbb{Z}}^{\otimes n}{\otimes }_{\mathbb{S}}R$.

Setting: Setting - Spherical Derived Geometry