Stacks of Prismatic Forms

Context: Picture - Koszul Duality and Prismatisation

Construction: A functor

$Form{s}_{\Delta }^k:CAlg(Sp)\to S$ $R\mapsto Ma{p}_R(R,{\Sigma }^{k}g{r}^k(Fi{l}_{\Delta }(THH(R{)}^{h\mathbb{T}}))$

These assemble into a “graded mixed complex”.

Example

$Form{s}_{\Delta }^0(R)\simeq R⟹Form{s}_{\Delta }^0\simeq {\mathbb{A}}_{smooth}^1$ So these higher forms generalize the smooth affine line.