Recollection - Traces and Prisms

Context: Picture - Koszul Duality and Prismatisation

For a certain class of discrete commutative rings (quasi-syntomic), the motivic filtration of BMS and HRW exhibits the following “deformations”:

• $TP(R):=THH(R{)}^{t\mathbb{T}}⇝\widehat{{\Delta }_R}\{i\}$
• $NC(R):=THH(R{)}^{h\mathbb{T}}⇝{\mathcal{N}}^{\ge i}\widehat{{\Delta }_R}\{i\}$
• $THH(R)⇝g{r}_{\mathcal{N}}^i\widehat{{\Delta }_R}\{i\}$

where $\widehat{{\Delta }_R}\{i\}$ is Nyagaard-completed absolute prismatic cohomology, Breuil-Kisin-twisted $i$ times. $\mathcal{N}$ is the Nyagaard filtration.

Furthermore, taking cyclotomic fixed-points recovers an object called the syntomic cohomology of $R$:

• $TC(R):=THH(R{)}^{cyc}⇝Z_p(i)(R)$

This suggests an interaction between genuine circle actions and geometric picture for absolute prismatic cohomology

• $\mathbb{T}$-fixed points $⇝$ Nyagaard-filtered prismatization $X^{\mathcal{N}}$
• $\mathbb{T}$-Tate construction $⇝$ (Nyagaard-complete) Prismatization $X^{\Delta }$
• $\mathbb{T}$-genuine (cyclotomic) fixed points $⇝$ Syntomification $X^{syn}$

Example

Take $R=k$, a perfect field. A calculation using Boksted periodicity and the homotopy-fixedpoint spectral sequence yields

$NC(k)=W(k)[t,u]/(tu-p)$ with $deg(t)=+2,deg(u)=-2$. On the other hand, the Nyagaard-filtered prismatization is

$Spec(k{)}^{\mathcal{N}}\simeq Spf(W(k)[\tau ,\mu ]/(\tau \mu -p))$ where $deg(T)=+1,deg(U)=-1$. We see that the formulas agree except the generator degrees are doubled - this has to do with the even filtration.