Hodge-Tate comparison

Background - prisms and Hodge degeneration

Input:

• $P:=(A,I)$ a prism
• $X$ a p-adic formal scheme over $A/I$.

Ingredients

• ${\Delta }_X^P\in CAlg(Shv(X_{et},Mo{d}_A))$, prismatic complex of $X$ relative to prism $P$. (Sheaf of $P$-prismatic jets)

The Hodge-Tate complex is obtained by pulling back to the “central fiber” of the prism, that is, along $Spec(A/I)\to Spec(A)$ ${\overline{\Delta }}_X^P:={\Delta }_X^P{\otimes }_{A}A/I\in CAlg(Shv(X_{et},Mo{d}_{A/I}))$ The Hodge-Tate cdga records pulling back ${\Delta }_X^P$ along the $I$-adic filtration on $Spec(A)$. Namely, we have

$\cdots \to {\Delta }_X^P{\otimes }_{A}A/I^{n+1}\to {\Delta }_X^P{\otimes }_{A}A/I^n\to \cdots \to {\Delta }_X^P{\otimes }_{A}A/I\simeq \overline{{\Delta }_X^P}$ View the completion of this filtered algebra as a algebra in $\infty$-categorical cochains, we denote the corresponding object by $HT(X)\in CAlg(Shv(X_{et},Mo{d}_{A/I}))$

The Hodge-Tate comparison map The functor projecting to degree-zero chains (on the filtration side, the functor $g{r}^0(-)$, is lax symmetric monoidal and admits a left adjoint given by the derived de Rham complex. This gives us a map of etale sheaves of $cdga$‘s on $X$. $\xi :DR(X)\to HT(X)$ The Hodge-Tate comparison theorem says $\xi$ is an equivalence.

The local case When $X=Spec(R)$ for a finitely generated polynomial ring $R$, we have an equivalence of sheaves of $cdga$‘s

$HT(X)\simeq Post(X)$ where the RHS is as below.

The Postinikov cdga The postonikov tower of this algebra object is a filtered algebra object ${\tau }_{\ge \bullet }({\overline{\Delta }}_X^P)\in CAlg(Fil(Shv(X_{et},Mo{d}_{A/I}))$ Under the filtrations-cochains correspondence $\phi$, this filtration, after completion, corresponds to an $\infty$-categorical cochain $Post(X):=\phi (\widehat{{\tau }_{\ge \bullet }({\overline{\Delta }}_X^P)})\in CAlg(Ch(Shv(X_{et},Mo{d}_{A/I}))$

Relationship to Brueil-Kisin twists

In our notation above, the $n$-th term of the algebra of cochains $HT(X)$ is

$H{T}_n(X):=coFib({\Delta }_X^P{\otimes }_{A}A/I^{n+1}\to {\Delta }_X^P{\otimes }_{A}A/I^n)$ Lemma: there’s an equivalence $H{T}_n(X)\simeq {\pi }_{-n}(\overline{{\Delta }_X^P})\{n\}$

Relationship to Sen operators