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Generalized Fontaine-Laffaille theory

Joint PU/IAS Arithmetic Geometry Seminar · September 14, 2026
Joint work with Shubhodip Mondal

Main Result

Cartesian Square of Perfect $F$-Gauges

$\mathfrak{X}$: $p$-adic (derived) formal scheme.

Theorem (Madapusi–Mondal [MM])

There is a Cartesian square:

\[ \require{amscd} \begin{CD} \Perf_{[0, p-2]}(\mathfrak{X}^\syn) @>>> \Perf_{[0, p-2]}(\mathfrak{X}^{\dR, +}) \\ @VVV @VVV \\ \Perf_{[0, p-2]}((\mathfrak{X} \otimes \F_p)^\syn) @>>> \Perf_{[0, p-2]}((\mathfrak{X} \otimes \F_p)^N) \end{CD} \]

(subscripts denote Hodge–Tate weights in $[0, p-2]$).

Remark 1

Here, $\mathfrak{X}^{\dR, +} = \mathfrak{X}^N \times_{(\Spf \Int_p)^N} (\A^1/\mathbb{G}_m \times \Spf \Int_p)$ is the Hodge filtered de Rham stack.

Remark 2

If $\mathfrak{X} = \Spf W(k)$, $k$: perfect field, then:

\[ (\mathfrak{X} \otimes \F_p)^N = \mathcal{R}(\Fil_p^\bullet W(k)), \qquad \mathfrak{X}^{\dR, +} = \A^1/\mathbb{G}_m \times \Spf W(k). \]

So:

\[ \require{amscd} \begin{CD} \left\{\begin{matrix} \text{Perfect } F\text{-gauges over } W(k) \\ \text{w. HT wts in } [0, p-2] \end{matrix}\right\} @>>> \left\{\begin{matrix} \text{Filtered perfect} \\ \text{complexes over } W(k) \end{matrix}\right\} \\ @VVV @VVV \\ \left\{\begin{matrix} \text{Perfect } F\text{-gauges over } k \\ \text{w. HT wts in } [0, p-2] \end{matrix}\right\} @>>> \left\{\begin{matrix} p\text{-adically filtered} \\ \text{perfect complexes} \\ \text{over } W(k) \end{matrix}\right\} \end{CD} \]

This recovers a theorem of Terentiuk–Vologodsky–Xu ([TVX]).

Remarks 3–7

(3) This recovers a theorem of Antieau–Mathew–Morrow–Nikolaus ([AMMN]) saying effectively that the syntomic cohomology of $\mathcal{O}\{m\}$ ($=\Int_p(m)(\mathfrak{X})$ for $0 \le m \le p-2$) can be computed using "differential forms".

(4) TVX prove their theorem for $\QCoh$, not just $\Perf$. This can be obtained from our methods as well.

(5) If $K/\Q_p$ is unramified, and $\mathfrak{X}/\mathcal{O}_K$ is $p$-completely smooth, then this is essentially due to Imai–Kato–Youcis ([IKY]) (for vector bundles).

Defn:

\[ \mathrm{FL}(\mathfrak{X}) = \Vect_{[0, p-2]}((\mathfrak{X} \otimes \F_p)^\syn) \times_{\Vect_{[0, p-2]}((\mathfrak{X} \otimes \F_p)^N)} \Vect_{[0, p-2]}(\mathfrak{X}^{\dR, +}). \]

Under the above hypotheses, Faltings ([Fal]) constructed a functor using local Frobenius liftings:

\[ \FL(\mathfrak{X}) \longrightarrow \mathrm{Loc}^{\crys}_{\Int_p}(\mathfrak{X}). \]

The comparison with syntomic vector bundles fits into the diagram:

\[ \require{amscd} \begin{CD} \FL(\mathfrak{X}) @>{\mathrm{Faltings}}>> \mathrm{Loc}^{\crys}_{\Int_p}(\mathfrak{X}) \\ @VVV @AA{\sim\;\mathrm{(Guo\text{–}Reinecke)}}A \\ \Vect(\mathfrak{X}^\syn) @>>> \left\{\begin{matrix} \text{Analytic prismatic} \\ F\text{-crystals over } \mathfrak{X} \end{matrix}\right\}. \end{CD} \]

The lower horizontal functor is fully faithful. Guo–Reinecke ([GR]) identify analytic prismatic $F$-crystals with crystalline $\Int_p$-local systems. Imai–Kato–Youcis ([IKY]) show that the left comparison is an equivalence onto $\Vect_{[0, p-2]}(\mathfrak{X}^\syn)$.

(6) Hokaj ([Hokaj]), a student of Tong Liu, proved a version for power series rings over unramified rings of integers.

(7) (Fontaine–Laffaille) The reason for the title of this talk is that Fontaine–Laffaille ([FL82]) proved:

\[ \mathrm{FL}(\Spf W(k)) \simeq \left\{\begin{matrix} \text{Galois-stable } \Int_p\text{-lattices in crystalline representations} \\ \text{of } \Gamma_{W(k)[1/p]} \text{ with Hodge–Tate weights in } [0, p-2] \end{matrix}\right\}. \]

This is now a consequence of TVX + Bhatt–Scholze ([BS23]).

Idea for Proof

Step 1: Construction of Factorization

Construction of factorization:

\[ \require{amscd} \begin{CD} \Perf_{[0, p-2]}(\mathfrak{X}^N) @>>> \Perf_{[0, p-2]}((\mathfrak{X} \otimes \F_p)^N) \\ @VVV @AAA \\ \Perf_{[0, p-2]}(\mathfrak{X}^{\dR, +}) @= \Perf_{[0, p-2]}(\mathfrak{X}^{\dR, +}) \end{CD} \]

This can be deduced from the fact (due to Bhatt–Lurie [BL]) that $\Fil_N^\bullet \mathbin{\Delta}_R \to \Fil_{\mathrm{Hdg}}^\bullet \widehat{\dR}_R$ induces isomorphisms:

\[ \mathbin{\Delta}_R / \Fil_N^i \mathbin{\Delta}_R \xrightarrow{\;\sim\;} \widehat{\dR}_R / \Fil_{\mathrm{Hdg}}^i \widehat{\dR}_R \qquad \text{for } 0 \le i \le p-1. \]

Step 1 gives us a canonical functor:

\[ (*) \quad \Perf_{[0, p-2]}(\mathfrak{X}^\syn) \longrightarrow \mathrm{PerfFL}(\mathfrak{X}) \quad (\text{perfect analogue of } \mathrm{FL}(\mathfrak{X})). \]

Step 2: Descent to Semiperfect $\F_p$-Algebras

Both sides of $(*)$ satisfy quasisyntomic & mod-$p$ Tot descent with respect to $\mathfrak{X}$. In particular, it suffices to show $(*)$ is an equivalence for $\mathfrak{X} = \Spec R$, where $R$ is a semiperfect $\F_p$-algebra.

Remark 8

It is crucial to work with animated commutative rings for this reduction. Even for the case $\mathfrak{X} = \Spf \Int_p$, we need to prove the result for the animated semiperfect $\F_p$-algebras $\F_p^{\otimes^{\mathbb{L}}m}$.

Observation

$R$: $\F_p$-algebra. $R \otimes \F_p \simeq R \oplus R[1]$, and we have a diagram:

\[ \require{amscd} \begin{CD} R @>>> R \otimes \F_p \\ @| @VVV \\ R @= R \end{CD} \]

Moreover, when $R$ is semiperfect, we have equivalences in the world of animated filtered commutative rings:

\[ \begin{aligned} \Fil_N^\bullet \mathbin{\Delta}_{R \otimes \F_p} &\simeq \Fil_N^\bullet \mathbin{\Delta}_R \otimes_{\Fil_N^\bullet \mathbin{\Delta}_{\F_p}} \Fil_N^\bullet \mathbin{\Delta}_{\F_p \otimes \F_p} \\ &\simeq \Fil_N^\bullet \mathbin{\Delta}_R \otimes_{\Fil_p^\bullet \Int_p} \left(\Fil_N^\bullet \mathbin{\Delta}_{R_\infty/p} \otimes_{\Fil_N^\bullet \mathbin{\Delta}_{R_\infty}} \Fil_p^\bullet \Int_p\right). \end{aligned} \]

Here $R_\infty = \Int_p[\zeta_{p^\infty}]_p^\wedge$, or any other perfectoid ring admitting $\F_p$ as a quotient. Moreover:

\[ \begin{aligned} \Fil_N^\bullet \mathbin{\Delta}_{R_\infty} &= \Fil_{(\xi)}^\bullet A_\mathrm{inf}(R_\infty), \\ \Fil_N^\bullet \mathbin{\Delta}_{R_\infty/p} &= \Fil_N^\bullet A_{\crys}(R_\infty). \end{aligned} \]
Remark 9
\[ \mathbin{\Delta}_{\F_p \otimes \F_p} \simeq \frac{\Int_p\langle x\rangle_p^\wedge}{(x-p)}, \]

where $\Int_p\langle x\rangle_p^\wedge$ is the free $p$-completed PD algebra in $x$. The description above enables a quick computation of $\Fil_N^\bullet \mathbin{\Delta}_{\F_p \otimes \F_p}$.

Step 3: Deformation Theory for Windows over Frames

Use deformation theory for perfect windows over frames (a derived enhancement of a definition of Lau [Lau]) to reduce to the following proposition:

Proposition

Set $\Fil_N^\bullet C = \cofib\big(\Fil_p^\bullet \Int_p \to \Fil_N^\bullet \mathbin{\Delta}_{\F_p \otimes \F_p}\big)$. Then:

  1. For $0 \le i \le p-1$, there is a canonical factoring:
    \[ \Fil_N^i \mathbin{\Delta}_{\F_p \otimes \F_p} \longrightarrow C \longrightarrow \Fil_N^i C. \]
    In particular, the divided Frobenius $\varphi_i: \Fil_N^i \mathbin{\Delta}_{\F_p \otimes \F_p} \to \mathbin{\Delta}_{\F_p \otimes \F_p}$ induces $\dot{\varphi}_i: C \to C$.
  2. For $0 \le i \le p-2$, $\dot{\varphi}_i$ is topologically nilpotent.
Remark 10

The proof is not difficult and reduces to the fact that $p^{[p]} = \frac{p^p}{p!}$ has $p$-adic valuation $p-1$.

Remark 11

If $p > 2$, then we obtain the Grothendieck–Messing square:

\[ \require{amscd} \begin{CD} \Vect_{[0, 1]}(R^\syn) @>>> \FilVect(R) \\ @VVV @VVV \\ \Vect_{[0, 1]}((R \otimes \F_p)^\syn) @>>> \FilVect(R \otimes \F_p) \times_{\Vect(R \otimes \F_p)} \Vect(R) \end{CD} \]

This is obtained from equivalences:

\[ \begin{aligned} \Vect_{[0, 1]}(R^{\dR, +}) &\simeq \Vect((R \otimes \F_p)^\Delta) \times_{\Vect(R)} \FilVect_{[0, 1]}(R), \\ \Vect_{[0, 1]}((R \otimes \F_p)^N) &\simeq \Vect((R \otimes \F_p)^\Delta) \times_{\Vect(R \otimes \F_p)} \FilVect_{[0, 1]}(R \otimes \F_p). \end{aligned} \]

By Anschütz–Le Bras ([ALB]) and Gardner–Madapusi–Mathew (see also [GM]), the left-hand categories in the square can be reinterpreted as categories of $p$-divisible groups over $R$ and $R \otimes \F_p$, respectively, and this is just the "usual" Grothendieck–Messing theory applied to the pro-nilpotent PD thickening $R \to R \otimes \F_p$.

Remark 12

The frame theoretic deformation techniques are also used in the proof of Grothendieck–Messing theory in Gardner–Madapusi ([GM]).

Bibliography

References

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