Prismatic \(p\)-adic Hodge Comparison

Job: construct the \(A_{\inf}\) cohomology of Bhatt–Morrow–Scholze and state the comparison isomorphisms to de Rham, crystalline, and étale cohomology via Hodge–Tate; record the Nygaard completion and the \(p\)-adic Poincaré lemma; give the dictionary prismatic \(\leftrightarrow\) Fontaine period rings.

Core

Setup: the perfectoid prism and \(A_{\inf}\)

Fix a perfectoid field \(K\) of char \(0\) with residue char \(p\) (e.g. \(K=\mathbb{C}_p\) or a finite \(K/\mathbb{Q}_p\)), ring of integers \(\mathcal{O}_K\), tilt \(K^\flat=\varprojlim_\varphi K/p\), and the Fontaine ring \(A_{\inf}=W(\mathcal{O}_K^\flat)\) with Fontaine's map \(\theta:A_{\inf}\to\mathcal{O}_K\), \(\ker\theta=(\xi)\). Then \((A_{\inf},(\xi))\) is a perfectoid prism (perfect: \(\varphi\) on \(A_{\inf}\) is an automorphism), and \(\tilde{\xi}:=\varphi(\xi)/p\in A_{\inf}\) is a unit (see Foundations).

For a smooth formal \(\mathcal{O}_K\)-scheme \(X\), \(A_{\inf}\) cohomology is the prismatic cohomology at this prism:

\[ A_{\inf} R\Gamma(X) := \triangle_{X/A_{\inf}}\in D(A_{\inf}), \]

carrying the Frobenius \(\varphi\) and a filtration. The BMS theorems identify its specializations with all classical \(p\)-adic cohomologies of \(X\).

The comparison theorems (Bhatt–Morrow–Scholze)

Write \(R\Gamma_{\mathrm{dR}}(X/\mathcal{O}_K)\), \(R\Gamma_{\mathrm{\acute{e}t}}(X_{\overline{K}},\mathbb{Z}_p)\), \(R\Gamma_{\mathrm{cris}}\) when applicable. Set \(B_{\mathrm{dR}}\) the de Rham period ring, \(B_{\mathrm{HT}}=B_{\mathrm{dR}}^{\mathrm{gr}}\) the Hodge–Tate ring, \(A_{\mathrm{cris}}\) the crystalline period ring.

Comparison Statement (smooth proper formal \(X/\mathcal{O}_K\), \(K\) perfectoid)
Hodge–Tate \(\triangle_{X/A_{\inf}}\otimes^{\mathbb{L}}_{A_{\inf}} A_{\inf}/\xi \;\cong\; \bigoplus_{i}\Omega^i_{X/\mathcal{O}_K}[-i]\) in \(D(\mathcal{O}_K)\) — recovers the Hodge filtration
de Rham \(\triangle_{X/A_{\inf}}\otimes^{\mathbb{L}}_{A_{\inf},\,\theta} \mathcal{O}_K \;\cong\; R\Gamma_{\mathrm{dR}}(X/\mathcal{O}_K)\)
Étale (after \(\otimes B_{\mathrm{dR}}\)) \(\triangle_{X/A_{\inf}}\otimes^{\mathbb{L}}_{A_{\inf}} B_{\mathrm{dR}} \;\cong\; R\Gamma_{\mathrm{\acute{e}t}}(X_{\overline{K}},\mathbb{Z}_p)\otimes_{\mathbb{Z}_p} B_{\mathrm{dR}}\)
Crystalline (good reduction case) at a crystalline/liftable prime, \(\triangle_{X/A_{\inf}}\otimes^{\mathbb{L}}_{A_{\inf},\,\theta\circ\varphi^{-1}} A_{\mathrm{cris}} \cong R\Gamma_{\mathrm{cris}}(X_0/W(k))\otimes A_{\mathrm{cris}}\)

Theorem (BMS, \(p\)-adic Poincaré lemma). For smooth \(X/\mathcal{O}_K\), the natural map

\[ \triangle_{X/A_{\inf}}\otimes^{\mathbb{L}}_{A_{\inf}} B_{\mathrm{dR}} \;\longrightarrow\; R\Gamma_{\mathrm{\acute{e}t}}(X_{\overline{K}},\mathbb{Z}_p)\otimes_{\mathbb{Z}_p} B_{\mathrm{dR}} \]

is an equivalence. This is the integral \(p\)-adic Hodge comparison — a single prismatic object, after inverting the period \(t\in B_{\mathrm{dR}}\) (where \(t=\theta(\log[\underline{\pi}])\) for a compatible system \(\underline{\pi}\) with \(\xi=[\underline{\pi}^{\flat}]\)), recovers the \(p\)-adic étale cohomology with \(B_{\mathrm{dR}}\)-coefficients.

Nygaard completion and the trace theorem

The integral (not just \(B_{\mathrm{dR}}\)-localized) comparison is captured by the Nygaard completion of prismatic cohomology. For the perfectoid prism,

\[ \widehat{\triangle}_{X/A_{\inf}}\{\varphi\} := \varprojlim_n \bigl(\triangle_{X/A_{\inf}}\xrightarrow{\varphi}\cdots\xrightarrow{\varphi}\triangle_{X/A_{\inf}}\otimes^{\mathbb{L}}_{A_{\inf}} A_{\inf}/\varphi(\xi)^n\bigr), \]

the derived \(\varphi(\xi)\)-adic Nygaard completion. The Frobenius \(\varphi\) becomes an isomorphism after inverting \(\varphi(\xi)\), hence after inverting \(p\).

Theorem (BMS, trace). For smooth proper \(X/\mathcal{O}_K\) with torsionfree \(H^*_{\mathrm{\acute{e}t}}\), the cyclotomic trace from algebraic \(K\)-theory to topological Hochschild homology identifies

\[ \pi_*\,\mathrm{TC}(X;\mathbb{Z}_p) \;\cong\; \pi_*\,\widehat{\triangle}_{X/A_{\inf}}\{\varphi\} \]

(after the standard \(p\)-adic completion / Nygaard adjustments). Combined with the Dundas–Goodwillie–McCarthy theorem (TC agrees with \(K\)-theory on nilpotent thickenings), this gives the BMS computation of \(p\)-adic algebraic \(K\)-theory of smooth \(\mathcal{O}_K\)-schemes in terms of Nygaard-completed prismatic cohomology — see algebraic \(K\)-theory.

Dictionary: prismatic \(\leftrightarrow\) Fontaine period rings

Prismatic object Fontaine / classical \(p\)-adic Hodge object Effect of specialization
\(A_{\inf}=W(\mathcal{O}_K^\flat)\) \(A_{\inf}\) (same) the universal \(p\)-adically complete period ring
\(\theta:A_{\inf}\to\mathcal{O}_K\), \(\ker\theta=(\xi)\) Fontaine's \(\theta\) \(A_{\inf}/\xi\cong\mathcal{O}_K\)
\(\tilde{\xi}=\varphi(\xi)/p\) (unit) generator of \(\ker\theta\circ\varphi^{-1}\) \(A_{\inf}/\tilde{\xi}\cong\mathcal{O}_K\) via \(\theta\circ\varphi^{-1}\)
\(\triangle_{X/A_{\inf}}\otimes^{\mathbb{L}} A_{\inf}/\xi\) Hodge–Tate cohomology \(H^*(X_{\overline{K}},\mathbb{C}_p)\otimes\Omega^i\) (conjugate HT) recovers \(B_{\mathrm{HT}}\)-comparison
\(\triangle_{X/A_{\inf}}\otimes^{\mathbb{L}}_{\theta} \mathcal{O}_K\) de Rham complex \(R\Gamma_{\mathrm{dR}}(X/\mathcal{O}_K)\) de Rham comparison
\(\triangle_{X/A_{\inf}}\otimes^{\mathbb{L}}_{\theta\circ\varphi^{-1}} A_{\mathrm{cris}}\) crystalline cohomology \(\otimes A_{\mathrm{cris}}\) crystalline comparison (good reduction)
\(\widehat{\triangle}_{X/A_{\inf}}\{\varphi\}\) \(\mathbb{Z}_p\)-adic étale cohomology (via TC / trace) integral comparison
\(\triangle_{X/A_{\inf}}\otimes^{\mathbb{L}} B_{\mathrm{dR}}\) \(R\Gamma_{\mathrm{\acute{e}t}}(X_{\overline{K}},\mathbb{Z}_p)\otimes B_{\mathrm{dR}}\) \(p\)-adic Poincaré lemma
\(\varphi\) on \(\triangle_{X/A_{\inf}}\) crystalline Frobenius on \(D_{\mathrm{cris}}\) the \(F\)-crystal structure
\(t=\theta(\log[\underline{\pi}])\in B_{\mathrm{dR}}\) period element \(\xi=[\underline{\pi}^{\flat}]\rightsquigarrow t\) via \(\theta\log\) relates \(\xi\)-adic to \(t\)-adic filtrations

This dictionary is the content of "integral \(p\)-adic Hodge theory is prismatic cohomology over the perfectoid prism": every Fontaine period ring and every classical comparison theorem is a specialization of the single prismatic object \(\triangle_{X/A_{\inf}}\) with its Frobenius and Nygaard filtration.

See it

An elliptic curve over \(\mathcal{O}_{\mathbb{C}_p}\) with good ordinary reduction. Let \(E/\mathbb{Z}_p\) be the canonical lift of an ordinary elliptic curve \(E_0/\mathbb{F}_p\) (so the Hasse invariant is a unit). Over \(K=\mathbb{C}_p\):

  • \(H^1_{\mathrm{dR}}(E/\mathcal{O}_K)\) is rank \(2\), with Hodge filtration \(0\to\omega_E\to H^1_{\mathrm{dR}}\to\omega_E^{-1}\to 0\).
  • The prismatic cohomology \(\triangle_{E/A_{\inf}}\) in degree \(1\) is a rank-\(2\) \(A_{\inf}\)-module with Frobenius \(\varphi\). For the ordinary lift, \(\varphi\) has slopes \(0\) and \(1\) on \(D_{\mathrm{cris}}(H^1_{\mathrm{\acute{e}t}})\) — the unit root and the non-unit root.
  • The Nygaard completion selects the unit-root part integrally: \(\widehat{\triangle}_{E/A_{\inf}}\{\varphi\}\big|_{H^1}\) is the \(A_{\inf}\)-lattice whose mod-\(\xi\) reduction is the Hodge filtration and whose Frobenius is the unit-root part. The BMS trace identifies \(\mathrm{TC}(E;\mathbb{Z}_p)\) at \(\pi_0\) (after \(\mathbb{A}^1\)-invariance) with this lattice, recovering the classical computation \(K_0(\mathcal{O}_E)\) via crystalline methods.
  • The \(B_{\mathrm{dR}}\)-comparison gives \(H^1_{\mathrm{\acute{e}t}}(E_{\overline{K}},\mathbb{Z}_p)\otimes B_{\mathrm{dR}}\cong H^1_{\mathrm{dR}}(E)\otimes B_{\mathrm{dR}}\) — the de Rham \(\leftrightarrow\) étale comparison for an elliptic curve — as a single consequence of the prismatic \(p\)-adic Poincaré lemma.

This is the elliptic-curve instance of the general BMS story: one prismatic computation (over \(A_{\inf}\)) with two specializations (\(\theta\) for de Rham, \(B_{\mathrm{dR}}\) for étale) yields every classical \(p\)-adic Hodge comparison for \(E\).

Check

Where does the \(p\)-adic Poincaré lemma use the perfectoid (perfect) hypothesis on the prism?

The identification \(\triangle_{X/A_{\inf}}\otimes B_{\mathrm{dR}}\cong R\Gamma_{\mathrm{\acute{e}t}}(X_{\overline{K}},\mathbb{Z}_p)\otimes B_{\mathrm{dR}}\) requires \(\varphi\) to be an automorphism on the base (so the Nygaard completion behaves) and the almost purity of \(\mathcal{O}_{\mathbb{C}_p}^\flat\) to invoke Faltings's \(p\)-adic Poincaré lemma on the perfectoid side; both fail for the crystalline or Breuil–Kisin prisms, which is why those give only crystalline / Breuil–Kisin (not étale) comparison directly. The Breuil–Kisin prism recovers étale comparison after a further base change to \(A_{\inf}\).

Sketch why the de Rham comparison \(\triangle_{X/A_{\inf}}\otimes^{\mathbb{L}}_\theta\mathcal{O}_K\cong R\Gamma_{\mathrm{dR}}\) is 'the Hodge–Tate comparison + \(p\)-adic Poincaré lemma' in disguise.

The Hodge–Tate comparison gives the graded pieces \(\mathrm{gr}^i\cong\Omega^i[-i]\) after mod \(\xi\). The \(\xi\)-adic completion of \(\triangle_{X/A_{\inf}}\) with its filtration (the prismatic filtration) is the prismatic avatar of the Hodge filtration on \(B_{\mathrm{dR}}\); specializing via \(\theta\) recovers the de Rham complex with its Hodge filtration, i.e. \(R\Gamma_{\mathrm{dR}}(X/\mathcal{O}_K)\) with filtration. The \(p\)-adic Poincaré lemma identifies the étale side after \(\otimes B_{\mathrm{dR}}\), and the de Rham side is the \(\theta\)-specialization, so the de Rham comparison is the filtered refinement — one statement, two specializations.

Pitfalls

  • Hodge–Tate vs de Rham normalization: the Hodge–Tate comparison uses the conjugate Hodge–Tate (Sen weight) normalization, \(\bigoplus_j \Omega^j[-j]\), not the analytic Hodge–Tate cohomology with \(\mathbb{C}_p(j)\) twists directly — the regrading introduces signs and Tate twists that the unwary reader mis-identifies with the naive \(H^i(X,\Omega^j)\) pairing.
  • Nygaard completion subtlety: \(\widehat{\triangle}\{\varphi\}\) is the derived \(\varphi(\xi)\)-adic Nygaard completion, not the naive inverse limit; for non-smooth or non-liftable \(X\) the Nygaard completion may fail to be bounded, and the BMS trace theorem requires the bounded prismatic hypothesis — without it, the cyclotomic trace does not identify \(K\)-theory integrally.
  • Étale comparison needs \(p\) invertible (after \(\otimes B_{\mathrm{dR}}\), \(p\) is invertible in \(B_{\mathrm{dR}}\)); the integral étale comparison (without inverting \(p\)) uses the Nygaard completion and TC, and is sharper but requires torsionfree hypotheses on \(H^*_{\mathrm{\acute{e}t}}\).
  • \(A_{\inf}\) is not \(A_{\mathrm{cris}}\) in general: \(A_{\mathrm{cris}}\) is the \(p\)-adic completion of a divided-power envelope of \(A_{\inf}\) over \(\mathcal{O}_K\); the crystalline comparison passes through \(A_{\mathrm{cris}}\), the étale through \(B_{\mathrm{dR}}\), and conflating the two period rings loses the distinction between crystalline and de Rham comparisons.
  • Non-liftable varieties: the BMS comparison theorems as stated require \(X\) smooth formal over \(\mathcal{O}_K\) (i.e. a lift). For a variety \(X_0/\mathbb{F}_p\) with no smooth lift, one works with derived/animated lifts; the Hodge–Tate comparison still holds after derived completion, but the Frobenius structure may be nilpotent / non-isogeny and the Nygaard trace theorem fails without the smooth-liftable hypothesis.

Refs

  • Bhatt–Morrow–Scholze, Integral \(p\)-adic Hodge Theory (Publ. Math. IHÉS 2019), §§1–10
  • Fontaine, Le corps des périodes \(p\)-adiques (Astérisque); \(B_{\mathrm{dR}},B_{\mathrm{HT}},A_{\mathrm{cris}}\)
  • Bhatt–Scholze, Prisms and Prismatic Cohomology, §17 (perfectoid prism)
  • Clausen–Mathew, Vinogradov sequences and prismatic cohomology (Nygaard refinements)

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Foundations · Prisms and Thickenings · Prismatic index · K-theory · AG cohomology · Arithmetic geometry · Derived AG