nLab prismatization

Idea

Prismatization is a stacky approach to prismatic cohomology developed independently by Drinfeld in #Drinfeld20 and Bhatt-Lurie in #BhattLurie22 and #BhattLurie22b.

Motivation: Linear algebra via stacks

The reference for this section is Section 2 of #BhattNotes.

Definitions

Definition

(Definition 3.1.4 of #BhattLurie22a, Definition 5.13 of #BhattNotes) Let $R$ be a p-nilpotent ring and $W(R)$ its ring of Witt vectors. A Cartier-Witt divisor on $R$ is an invertible $W(R)$-module $I$ together with a morphism $\alpha:I\to W(R)$ of $W(R)$-modules such that

• The image of the map $I\to W(R)\twoheadrightarrow R$ is a nilpotent ideal of $R$.
• The image of the map $I\to W(R)\xrightarrow{\delta} W(R)$ generates the unit ideal of $W(R)$.

Definition

(Definition 5.1.6 of #BhattNotes) Let $X$ be a bounded $p$-adic formal scheme. The prismatization $X^{\Delta}$ is the presheaf over $\mathrm{Spf}(\mathbb{Z}_{p})$ defined as follows. For a $p$-nilpotent ring $R$, $X^{\Delta}(R)$ is the groupoid of Cartier-Witt divisors on $R$ together with a map $\mathrm{Spec}(\mathrm{Cone}(W(R)/I))\to X$ of derived $\mathrm{Spf}(\mathbb{Z}_{p})$-schemes.

The Cartier-Witt stack

The prismatization $\mathrm{Spf}(\mathbb{Z}_{p})^{\Delta}$ of $\mathrm{Spf}(\mathbb{Z}_{p})$ is also called the Cartier-Witt stack, and denoted $\mathrm{WCart}$ or $\mathbb{Z}_{p}^{\Delta}$.

Filtered prismatization

The reference for this section is Section 5.3 of #BhattNotes.

Definition

(Definition 5.2.4 of #BhattNotes) Let $R$ be a $p$-nilpotent ring. Let $W$ be the ring scheme of Witt vectors over $\mathrm{Spf}(\mathbb{Z}_{p})$. Let $F:W\to F_{*}W$ be the Frobenius. An admissible $W$-module over $R$ is an affine $W$-module scheme $M$ which can be realized as an extension of a twisted form of $F_{*}M$ by a twisted form of the $\mathbb{G}_{a}^{#}$, where $\mathbb{G}_{a}^{#}$ is the PD-hull of the origin in $\mathbb{G}_{a}$ over $\mathbb{Z}$.

Definition

(Definition 5.3.1 of #BhattNotes) Let $R$ be a $p$-nilpotent ring. Let $W$ be the ring scheme of Witt vectors over $\mathrm{Spf}(\mathbb{Z}_{p})$. Let $F:W\to F_{*}W$ be the Frobenius. A filtered Cartier-Witt divisor over $R$ is an admissible $W$-module $M$ over $R$ and a map $M\to W$ such that the induced map $F_{\ast}M'\to F_{\ast}W$ of twisted forms of $F_{\ast}W$ comes from a Cartier-Witt divisor.

Definition

(Definition 5.3.10 of #BhattNotes)

Let $X$ be a bounded $p$-adic formal scheme. Define $\mathrm{Spf}(\mathbb{Z}_{p})^{\mathcal{N}}$ to be the stack which takes a $p$-nilpotent ring $R$ to the groupoid $\mathrm{Spf}(\mathbb{Z}_{p})^{\mathcal{N}}(R)$ of filtered Cartier-Witt divisors over $R$. The filtered prismatization of $X$ is the stack $X^{\mathcal{N}}$ over $\mathbb{Z}_{p}^{\mathcal{N}}$ obtained via transmutation from $\mathbb{G}_{a}^{\mathcal{N}}\to\mathbb{Z}_{p}^{\mathcal{N}}$, where $\mathbb{G}_{a}^{\mathcal{N}}$ is the stack over $\mathbb{Z}_{p}^{\mathcal{N}}$ taking a filtered Cartier-Witt divisor $d:M\to W$ to $R\Gamma(\mathrm{Spec}(R),W/M)$.

References

• Vladimir Drinfeld, Prismatization (2020) arXiv:2005.04746

• Bhargav Bhatt, Jacob Lurie, Absolute Prismatic Cohomology, preprint (2022) arXiv:2022.06120

• Bhargav Bhatt, Jacob Lurie, The Prismatization of $p$-adic Formal Schemes, preprint (2022) arXiv:2022.06124

• Bhargav Bhatt, Prismatic F-gauges, (Lecture notes)