Spectral Superlines

Here we study the superlines ${\Omega }_0^n{\mathbb{A}}^1\simeq {\mathbb{A}}^{1|n}$ Here ${\mathbb{A}}^{1|0}:={\mathbb{A}}^1:=Spec(\mathbb{S}[\mathbb{N}])$ is the flat affine line in SAG.

Using the results thm - Koszul Duality exchanges invertibility and categorification, these objects have explicit forms. For example, ${\mathbb{A}}^{1|1}$ is the spherical group ring

\Sigma\mathbb N] \simeq \mathbb S[\mathbb T]$$ Then we see the higher superlines can be expressed as $$\mathbb A^{1| n} \simeq Spec(\mathbb S[B^{n-1}\mathbb T])$$ Sheaves are hence representations $$QCoh(\mathbb A^{1 |n +1}) \simeq Rep(B^n \mathbb T)$$ **Remark**: the appearance of $B^nT$ here is to be compared with $B^nC_p$ in definition of semi-additive height. Indeed, semi-additive height is a categorical support condition, and this is to be expressed in terms of singular support, as in [[Microlocal Smooth Geometry]].