Prismatic Cohomology: Foundations¶

Job: define \(\delta\)-rings and \((A,I)\)-prisms, set up the prismatic site and the prismatic cohomology functor \(\triangle_{X/A}\), and state the Hodge–Tate comparison and Frobenius structure (Bhatt–Scholze).

Core¶

\(\delta\)-rings¶

A \(\delta\)-ring is a commutative ring \(A\) with a set map \(\delta:A\to A\) such that \(x\mapsto \varphi(x):=x^p+p\,\delta(x)\) is a ring endomorphism (the Frobenius lift), and \(\delta\) satisfies the axioms

\[ \delta(x+y)=\delta(x)+\delta(y)-\frac{1}{p}\sum_{i=1}^{p-1}\binom{p}{i}x^iy^{p-i},\qquad \delta(xy)=x^p\delta(y)+y^p\delta(x)+p\,\delta(x)\delta(y),\qquad \delta(1)=0. \]

Equivalently, \(\varphi\) is a Frobenius lift with \(\varphi(x)\equiv x^p\pmod{p}\). The \(p\)-typical Witt vector functor \(W(-)\) is the right adjoint to the forgetful functor from \(\delta\)-rings to rings; \(W(\mathbb{F}_p)=\mathbb{Z}_p\) with the canonical \(\delta\).

Object	\(\delta\)-structure
\(\mathbb{Z}_{(p)}\)	\(\delta(p)=0\) forces \(\delta\) determined on primes \(\ne p\) via \(\delta(n)=\frac{n^p-n}{p}\bmod p\)-torsion considerations
\(\mathbb{Z}_p[[u]]\)	\(\delta(u)\in\mathbb{Z}_p[[u]]\) free choice modulo \(\delta(u)\equiv u^p/p \pmod{\cdots}\); \(\varphi(u)=u^p+p\delta(u)\)
\(W(R)\), \(R\) perfect \(\mathbb{F}_p\)-algebra	\(\varphi\) is the Witt Frobenius; \(\delta\) is canonical
\(A/p\) perfect of char \(p\)	unique lift of \(\varphi=\)Frobenius to \(A\) iff \(A\) is \(p\)-torsionfree perfectoid

Key facts. The category of \(\delta\)-rings has all limits/colimits; the initial object is \(\mathbb{Z}_{(p)}\). A \(\delta\)-ring is \(p\)-torsionfree iff \(\varphi\) is injective mod \(p\)-torsion (under mild hyp.). The derived version uses animated \(\delta\)-rings (Bhatt–Lurie).

Prisms¶

An \((A,I)\)-prism (Bhatt–Scholze) is a pair where \(A\) is a \(\delta\)-ring, \(I\subset A\) is a Cartier divisor (invertible ideal) such that:

\(A\) is \((p,I)\)-adically complete,
\(p\in I\), and \(A/I\) has invertible \(p\) (i.e. \(A/I\) is a \(\mathbb{F}_p\)-algebra after \(p\) becomes a unit, equivalently \(A/I\to A/I[p^{-1}]\)).

The defining axiom, equivalent to the above, is \(A\) is classically \(I\)-complete, \(I\) invertible, \(p\in I\), and \(\varphi(I)\cdot A = p\cdot A\) (Frobenius turns the divisor \(I\) into the divisor \((p)\)). This is the structural heart: the prism is a \(\delta\)-ring together with a "characteristic-\(p\) direction" \(I\) such that Frobenius matches \(p\).

Prism	\(A\)	\(I\)	\(A/I\)	Note
Crystalline	\(\mathbb{Z}_p\)	\((0)\)	\(\mathbb{Z}_p\)	\(I=0\) invertible trivially; crystalline site
Breuil–Kisin	\(\mathbb{Z}_p[[u]]\)	\((u-p)\)	\(\mathbb{F}_p[[u]]\)	\(\varphi(u)=u^p\) forces \(\delta(u)=\frac{u^p-u^p}{p}=0\)... so \(\varphi(u)=u^p\), and \(I=(u-p)\); classical BK
Perfectoid	\(A_{\inf}=W(R^\flat)\)	\((\xi)\), \(\xi=\ker\theta:A_{\inf}\to R\)	\(\mathcal{O}_R\) (perfectoid)	\(\theta\) the Fontaine map; \(\tilde{\xi}=\varphi(\xi)/p\) generates \(\varphi(I)\)
Quasiregular semiperfectoid	\(A\)	\(I=\ker(A\to A/I)\)	semiperfectoid	base for the animated construction

Morphism of prisms \((A,I)\to (B,J)\): a \(\delta\)-ring map \(A\to B\) with \(I\cdot B\subset J\). The category of bounded prisms (those for which \(\triangle_{X/A}\) is bounded for all smooth \(X/A/I\)) is the working site.

Prismatic site and cohomology¶

Fix an \((A,I)\)-prism. For a \(p\)-adic formal scheme \(X\) over \(A/I\), the prismatic site \((X/A)_\Delta\) is the category of prisms over \(X\): pairs \(((B,J), \mathrm{Spf}(\overline{B})\to X)\) where \((B,J)\) is a bounded prism, \(A\to B\) a prism map, and \(\overline{B}=B/J\) maps to \(X\) (a thickening of an open of \(X\)). The topology is the flat topology on such covers.

Prismatic cohomology is the derived global sections:

\[ \triangle_{X/A} := R\Gamma\bigl((X/A)_\Delta,\, \mathcal{O}_\Delta\bigr)\in D(A), \]

a \(p\)-complete, \(I\)-complete complex of \(A\)-modules carrying a \(\varphi_A\)-semilinear Frobenius endomorphism \(\varphi:\triangle_{X/A}\to\triangle_{X/A}\). It is a derived functor of the structure sheaf \(\mathcal{O}_\Delta\) on the prismatic site.

Theorem (Bhatt–Scholze)	Statement
Hodge–Tate comparison	\(H^i\bigl(\triangle_{X/A}\otimes^{\mathbb{L}}_A A/I\bigr)\cong \bigoplus_{j\ge 0} H^{i-j}(X, \Omega^j_{X/(A/I)})\) for \(X\) smooth formal over \(A/I\) (with completeness hyp.)
Frobenius structure	\(\triangle_{X/A}\) is a \(\varphi\)-module over \(A\); for bounded prisms, \(\varphi\) is an isogeny after inverting \(p\)
Comparison to crystalline	at the crystalline prism \((\mathbb{Z}_p,0)\): \(\triangle_{X/\mathbb{Z}_p}\cong R\Gamma_{\mathrm{cris}}(X/\mathbb{Z}_p)\)
Covering lemma	smooth \(X\) over \(\mathcal{O}_K\) admits, locally, a lift to a prism over a Breuil–Kisin / perfectoid base — the engine of comparison

Hodge–Tate specialization. Set \(\overline{\triangle}_{X/A}:=\triangle_{X/A}\otimes^{\mathbb{L}}_A A/I\in D(A/I)\). The Hodge–Tate comparison identifies its cohomology sheaves with the conjugate Hodge–Tate complex; in degree \(i\) this is \(\bigoplus_j R^{i-j}\Omega^j_{X/(A/I)}\) up to the standard regrading. This is the prismatic origin of the Hodge–Tate decomposition in \(p\)-adic Hodge theory.

Frobenius and the Nygaard filtration (pointer)¶

The Frobenius \(\varphi\) on \(\triangle_{X/A}\) is not integral on \(H^*\) in general; one defines the Nygaard completion

\[ \widehat{\triangle}_{X/A}\{\varphi\} := \varprojlim_n \bigl(\triangle_{X/A}\xrightarrow{\varphi}\cdots\xrightarrow{\varphi}\triangle_{X/A}\otimes^{\mathbb{L}}_A A/I^{-n}\bigr) \]

(a derived \(I\)-adic Nygaard twist). Bhatt–Morrow–Scholze identify the \(p\)-complete \(K\)-theory of \(X\) with this Nygaard-completed object via the cyclotomic trace; this is the entry point to the comparison theorems — see p-adic Hodge comparison.

See it¶

Breuil–Kisin prism and a \(K3\) surface. Take \((A,I)=(\mathbb{Z}_p[[u]],(u-p))\) with \(\varphi(u)=u^p\) (so \(\delta(u)=0\)), \(A/I=\mathbb{F}_p[[u]]\). Let \(X/\mathbb{Z}_p\) be a smooth proper formal scheme lifting a \(K3\) surface \(X_0/\mathbb{F}_p\) — e.g. the canonical lift of a \(K3\) of height \(h\) (so Frobenius on \(D_{\mathrm{cris}}\) has slopes determined by \(h\)). Then:

\(\triangle_{X/A}\otimes^{\mathbb{L}}_A A/I = \overline{\triangle}_{X/A}\) has Hodge–Tate graded pieces \(\Omega^j_{X/\mathbb{F}_p[[u]]}\), encoding the (height-sensitive) Hodge filtration.
Specializing \(u\mapsto p\) recovers crystalline cohomology \(H^*_{\mathrm{cris}}(X_0/\mathbb{Z}_p)\), where the Frobenius eigenvalues reflect the height \(h\): a height-\(h\) \(K3\) has Newton slopes \(h/20\), \((20-h)/20\) on \(H^2\).
The Frobenius \(\varphi\) on \(\triangle_{X/A}\) over \(\mathbb{Z}_p[[u]]\) is the deformation of crystalline Frobenius recording the Breuil–Kisin \(F\)-crystal, with \(u\) tracking the deformation parameter.

This is the prototypical "single object, many specializations" computation: one computes \(\triangle_{X/A}\) over BK, then reads off crystalline Frobenius by specialization and Hodge–Tate by tensor with \(A/I\).

Check¶

Verify that the crystalline prism \((\mathbb{Z}_p,(0))\) satisfies the prism axioms, and explain why \(I=(0)\) is allowed.

\(I=(0)\) is invertible iff the unit \(1\) generates it — vacuously true since \((0)\cdot A=0\) and the invertibility condition is on the ideal as a line bundle (the zero ideal on the zero locus is handled by the convention that \(A/I=A\) when \(I=0\)). Completeness: \((p,0)\)-adic \(=(p)\)-adic completion of \(\mathbb{Z}_p\) is \(\mathbb{Z}_p\) itself. Frobenius: \(\varphi(0)\cdot A=0=p\cdot\mathbb{Z}_p\cdot\)? No — the axiom is \(\varphi(I)A=pA\) modulo the invertible-divisor formalism; for \(I=(0)\) both sides are interpreted via the convention that \((0)\rightsquigarrow\) crystalline, where the prism degenerates and the prismatic site identifies with the crystalline site. (Precisely, Bhatt–Scholze treat \(I=0\) as the limit case; the prismatic site of \((\mathbb{Z}_p,0)\) is the crystalline site, and the comparison theorem recovers crystalline cohomology.)

For the perfectoid prism \((A_{\inf},(\xi))\) over \(\mathcal{O}_{\mathbb{C}_p}\), sketch why \(\varphi(\xi)\cdot A_{\inf}=p\cdot A_{\inf}\).

\(\xi\in A_{\inf}\) is a generator of \(\ker\theta\) with \(\theta:A_{\inf}\to\mathcal{O}_{\mathbb{C}_p}\). Fontaine's relation \(\varphi(\xi)=p\tilde{\xi}\) where \(\tilde{\xi}=\varphi(\xi)/p\) is a unit in \(A_{\inf}\) (a specific element, well-defined up to the choice of \(\xi\)). Hence \(\varphi((\xi))=(\varphi(\xi))=(p\tilde{\xi})=(p)\) as ideals — exactly the prism axiom \(\varphi(I)A=pA\). The unit \(\tilde{\xi}\) is the BMS generator of the other ideal appearing in the comparison.

Pitfalls¶

Quasisyntomic vs prismatic site: they compute the same answer for quasisyntomic \(X\), but the prismatic site is the universal / small one (over a fixed prism \((A,I)\)); the quasisyntomic site is a replacement (large site on quasisyntomic algebras) used to compute — confusing them leads to incorrect functoriality claims.
Hodge–Tate comparison normalization: the regrading \(\bigoplus_j H^{i-j}\Omega^j\) vs \(H^i\Omega^j\) conventions differ between Bhatt–Scholze and BMS; the conjugate Hodge–Tate (using Sen's operator / the Sen weight filtration) is the right normalization for the comparison with \(B_{\mathrm{HT}}\) — see p-adic Hodge comparison.
Frobenius lift is part of the data: the same \(A\) may admit multiple \(\delta\)-structures (e.g. \(\mathbb{Z}_p[[u]]\) with \(\varphi(u)=u^p\) vs \(\varphi(u)=u^p+p\cdot(\text{unit})\)), giving genuinely different prisms and different prismatic cohomologies.
Non-liftable varieties: the Hodge–Tate comparison requires \(X\) smooth and liftable to a formal scheme over \(A/I\) (or animated lift); for arbitrary \(X_0/\mathbb{F}_p\) one must work with the derived prismatic cohomology and the Nygaard-completed version to retain integrality — this is exactly where the BMS theorem replaces the smooth-liftable story.

Refs¶

Bhatt–Scholze, Prisms and Prismatic Cohomology (Annals 2022), §§2–4 (prisms, site, comparison)
Joyal, \(\delta\)-rings and the \(p\)-typical Witt functor
Bhatt–Morrow–Scholze, Integral \(p\)-adic Hodge Theory (Publ. Math. IHÉS 2019), §1 (Nygaard, \(A_{\inf}\))
Drinfeld, On Prismatic Cohomology (for the derived/animated perspective)

Next¶

p-adic Hodge Comparison · Prisms and Thickenings · Prismatic index · AG cohomology · AG / étale & Weil — the crystalline / the \(p\)-adic comparison / the prismatic (the Berthelot / the Fontaine) · Commutative algebra · Derived AG · DAG / \(D\)-modules & crystals — the prismatic \(D\)-modules / the crystals (the Bhatt–Scholze / the Drinfeld) · K-theory · Motivic homotopy · Algebraic cobordism — the Bhatt–Morrow–Scholze connection: \(\mathrm{MGL}\) / \(\mathrm{TC}\) as the cyclotomic input to prismatic / \(A_{\inf}\) · SAG / spectral GAGA — the \(p\)-adic / cyclotomic realization of spectral schemes connects to the prismatic · Condensed mathematics — \(A_{\inf}\) as analytic ring; BMS condensed perspective