A Tannakian framework for prismatic $F$-crystals

Technical Summary: A Tannakian Framework for Prismatic F-Crystals

Problem and Context
Integral $p$-adic Hodge theory seeks to understand the variation of $p$-adic geometric objects in families, particularly through the lens of crystalline $\mathbb{Z}_p$-local systems on a smooth formal scheme $X$ over the ring of integers $\mathcal{O}_K$ of a discretely valued field. Recent work by Bhatt and Scholze established the category $\text{Vect}^\varphi(X^\Delta)$ of prismatic $F$-crystals as a robust model for these local systems. However, applications to integral (local) Shimura varieties require a Tannakian formalism: a theory of $G$-objects (where $G$ is a smooth affine group scheme over $\mathbb{Z}_p$) within these categories.

A primary difficulty arises because the étale realization functor $T_{\acute{e}t}$, which relates prismatic $F$-crystals to crystalline local systems, is an equivalence of categories, but its quasi-inverse is not exact in general. Specifically, while $T_{\acute{e}t}$ maps the category of prismatic $F$-crystals to the subcategory of "prismatically good reduction" local systems, the inverse involves extending vector bundles from open subsets to their closures, a process that does not preserve exactness. This non-exactness threatens the construction of a Tannakian equivalence for $G$-objects, as the category of $G$-objects relies on the exactness of the underlying tensor category.

Methodology
The authors develop a Tannakian framework by analyzing the structure of $G$-objects in the category of prismatic $F$-crystals, denoted $G\text{-}\text{Vect}^\varphi(X^\Delta)$, and comparing them to $G$-objects in the category of crystalline local systems, $G\text{-}\text{Loc}^{\text{crys}}_{\mathbb{Z}_p}(X)$.

1. Tannakian Formalism for Prismatic Objects: The paper establishes that the category of prismatic $F$-crystals (and their analytic counterparts) possesses the necessary Tannakian properties. By utilizing the theory of torsors with Frobenius structure ($F$-structure), the authors show that $G$-objects in these categories correspond to $G$-torsors equipped with Frobenius isomorphisms.
2. Handling Non-Exactness: The core technical challenge is the non-exactness of the quasi-inverse to $T_{\acute{e}t}$. The authors address this by restricting attention to the case where $G$ is a reductive group scheme. They leverage a result by Colliot-Thélène and Sansuc, which states that for a reductive group $G$, a $G$-torsor on an open subscheme $U$ (where the complement has depth at least 2) extends to the whole scheme if the associated vector bundle extends. This allows them to bypass the failure of exactness for general groups by utilizing the specific geometric properties of reductive groups and the depth properties of the underlying rings (specifically, the Breuil–Kisin prism).
3. Shtuka Realization: The paper constructs a "shtuka realization functor" $T_{\text{sht}}$ that maps analytic prismatic $F$-crystals to $G$-shtukas. This construction relies on the relationship between prismatic cohomology and the theory of $v$-sheaves and diamonds developed by Scholze and others.

Key Contributions and Results

• Tannakian Equivalence for Reductive Groups: The central result (Theorem 2.29) establishes an equivalence of categories:
  $$T_{\acute{e}t}: G\text{-}\text{Vect}^\varphi(X^\Delta) \xrightarrow{\sim} G\text{-}\text{Loc}^{\Delta\text{-gr}}_{\mathbb{Z}_p}(X)$$
  where $G\text{-}\text{Loc}^{\Delta\text{-gr}}_{\mathbb{Z}_p}(X)$ is the category of $G$-local systems of "prismatically good reduction." Crucially, the authors prove that despite the quasi-inverse of $T_{\acute{e}t}$ not being exact on the level of vector bundles, the induced functor on $G$-objects is an equivalence when $G$ is reductive. This is achieved by showing that the obstruction to exactness vanishes for reductive groups due to the extension properties of their torsors.

• Analytic Prismatic $F$-Crystals: The paper refines the theory by working with analytic prismatic $F$-crystals ($\text{Vect}^{\text{an},\varphi}(X^\Delta)$), which are necessary to capture all crystalline $\mathbb{Z}_p$-local systems. It is shown that the étale realization functor induces a bi-exact equivalence between analytic prismatic $F$-crystals and crystalline local systems (Proposition 2.23).

• Shtuka Realization and Compatibility: The authors define a shtuka realization functor $T_{\text{sht}}: G\text{-}\text{Vect}^{\text{an},\varphi}(X^\Delta) \to G\text{-}\text{Sht}(X)$ (Theorem 3.21). They prove a comparison isomorphism $\varrho$ showing that this functor intertwines with the étale realization functor $T_{\acute{e}t}$ and the previously established functor $U_{\text{sht}}$ from de Rham local systems to shtukas (as defined in [PR24]).
  $$\varrho: (T_{\text{sht}})_\eta \xrightarrow{\sim} U_{\text{sht}} \circ T_{\acute{e}t}$$
  This result allows the extension of the shtuka realization functor to smooth $\mathcal{O}_K$-schemes (not just formal schemes) by gluing data from the generic fiber and the formal completion.

• Characterization of Good Reduction: The paper provides a criterion for a $G$-local system to be of prismatically good reduction: it suffices that the system associated with a single faithful representation of $G$ is of prismatically good reduction (Corollary 2.31).

Significance
The paper provides the foundational Tannakian theory required to rigorously apply prismatic cohomology to the study of integral Shimura varieties. By establishing the equivalence between $G$-objects in prismatic $F$-crystals and $G$-local systems of prismatically good reduction, the authors enable the use of prismatic methods to parameterize motivic objects over integral models of Shimura varieties.

The work is explicitly positioned as a companion to and a foundational component for [IKY23] and [IKY25]. In those works, the results of this paper are used to construct prismatic realization functors on integral Shimura varieties at hyperspecial level, leading to consequences such as an analogue of the Serre–Tate theorem. Furthermore, the compatibility with shtuka theory (Theorem 3) is pivotal for relating prismatic realizations to the Pappas–Rapoport conjecture regarding shtuka realizations for Shimura varieties of abelian type.

The authors note that while the main theorems require $G$ to be reductive, the framework is flexible enough to potentially accommodate parahoric group schemes, a case of significant interest for Shimura varieties, though this requires further investigation beyond the scope of the current results.