Prisms and Thickenings¶

Job: classify the prism typology (bounded, perfect, crystalline, Breuil–Kisin, perfectoid), introduce animated rings and quasiregular semiperfectoid frames as the derived input that makes the prismatic site work, and state Cartier–Witt divisors (Bhatt–Lurie) as the absolute version.

Core¶

The prism typology¶

A prism \((A,I)\) is determined by the geometry of \(A/I\) and the Frobenius lift \(\varphi\). The important subclasses:

Class	Definition / criterion	Geometric meaning
Bounded	for every smooth \(X/(A/I)\), \(\triangle_{X/A}\) is bounded (cohomology sheaves in \([0,2\dim X]\))	the working class — comparison theorems apply; the prismatic site is defined on bounded prisms
Perfect	\(\varphi:A\to A\) is an automorphism (so \(A/I\) is perfect of char \(p\))	perfectoid prism; \(A_{\inf}=W(R^\flat)\); étale comparison lives here
Crystalline	\(I=0\) (so \(A/I=A\) has \(p\) inverted trivially; the prism degenerates)	recovers crystalline cohomology over \(W(k)\)
Breuil–Kisin	\(A=\mathfrak S=\mathbb{Z}_p[[u]]\) with \(\varphi(u)=u^p\), \(I=(u-p)\)	the arithmetic prism over \(\mathcal{O}_K\); \(F\)-crystals / Kisin modules
Quasiregular semiperfectoid	\(A/I\) semiperfectoid and \(A\) is \(p\)-completely flat over \(A_{\inf}\) with \(I\) principal	base case for the animated construction of prismatic cohomology
Regular (lci)	\(A\) a \(p\)-completely flat lci lift of a regular ring \(A/I\) with \(\delta\)-structure	covers smooth \(\mathcal{O}_K\)-lifts and formal smooth bases

Crystalline prism as the special case. For \((A,I)=(W(k),(0))\) with \(W(k)\) the Witt vectors of a perfect \(k/\mathbb{F}_p\), the prism condition \(\varphi(I)\cdot A=p\cdot A\) becomes vacuous (interpreted as the limit), and the prismatic site of \(X_0/k\) over \((W(k),0)\) is identified with the crystalline site of \(X_0/W(k)\). Bhatt–Scholze prove:

\[ \triangle_{X_0/W(k)} \;\cong\; R\Gamma_{\mathrm{cris}}(X_0/W(k)) \]

as complexes with Frobenius. This recovers Berthelot's crystalline cohomology — including the Frobenius action — as the terminal point of the prism lattice. The prismatic framework generalizes crystalline: crystalline is the "trivial divisor" prism where the characteristic-\(p\) direction \(I\) degenerates to zero.

Breuil–Kisin prism. \(A=\mathfrak S=\mathbb{Z}_p[[u]]\), \(\varphi(u)=u^p\) (i.e. \(\delta(u)=0\)), \(I=(u-p)\), \(A/I=\mathbb{F}_p[[u]]\). Base change along \(u\mapsto p\) (i.e. \(A\to W(k)\)) recovers crystalline cohomology of the special fiber. Base change along \(u\mapsto[\underline{\pi}^{\flat}]\) into \(A_{\inf}\) (for a choice of compatible system of \(p\)-power roots \(\underline{\pi}\)) recovers \(A_{\inf}\) cohomology. Breuil–Kisin modules (Kisin) are the \(\varphi\)-modules over \(\mathfrak S\) arising from \(\triangle_{X/\mathfrak S}\) — the arithmetic \(F\)-crystals that classify crystalline local systems over \(\mathcal{O}_K\).

Animated rings and derived prisms¶

The construction of the prismatic site, and the comparison theorems in full generality (especially for non-liftable or singular \(X\)), require derived inputs. Following Bhatt–Lurie / Clausen:

An animated ring is a simplicial commutative ring that is \(\pi\)-finite (connective with finitely generated homotopy groups). Animated rings form an \(\infty\)-category \(\mathrm{AniRing}\) that resolves any (classical) ring by free polynomial resolutions — see DAG foundations.
An animated \(\delta\)-ring is an animated ring with a \(\delta\)-structure (Frobenius lift) in the derived sense; animated \(\delta\)-rings replace \(\delta\)-rings throughout the prism construction.
An animated prism \((A,I)\) is an animated \(\delta\)-ring \(A\) with an invertible ideal \(I\subset\pi_0 A\) (a Cartier divisor on the truncation) satisfying the prism axioms after derived completion.

The advantage: every \(p\)-torsionfree \(\delta\)-ring \(A\) with \(p\) invertible mod \(I\) admits an animated prism structure after derived completion, even when \(A\) is not classically \(I\)-complete. This is the framework in which the prismatic site is genuinely functorial in arbitrary (animated) thickenings.

Quasiregular semiperfectoid frames¶

The site-level construction of prismatic cohomology (Bhatt–Scholze, then Bhatt–Lurie's "absolute" version) proceeds by reducing to a base case. A quasiregular semiperfectoid ring \(S\) is a \(p\)-complete \(\mathcal{O}_C\)-algebra such that:

\(S\) is semiperfectoid (the Frobenius \(\varphi:S/p\to S/p\) is surjective, and \(S\) is \(p\)-completely flat over \(\mathcal{O}_C\)),
the cotangent complex \(\mathbb L_{S/\mathcal{O}_C}\) has Tor-amplitude in \([-1,0]\) (quasiregular — i.e. lci after \(p\)-completion).

The associated frame \((A_{\inf}(S), \ker\theta_S)\), with \(A_{\inf}(S)=W(S^\flat)\) and \(\theta_S:A_{\inf}(S)\to S\), is a quasiregular semiperfectoid prism — the base case from which every smooth \(X\) over a \(p\)-complete lci base is locally resolved by quasisyntomic maps. The quasisyntomic site on such \(S\) (the large site of quasisyntomic \(S\)-algebras) computes the same prismatic cohomology as the small prismatic site, by the Bhatt–Scholze comparison of sites. This is the technical engine making prismatic cohomology computable for arbitrary \(p\)-adic formal schemes.

Frobenius lifts and \(\delta\)-structures¶

The \(\delta\)-structure on \(A\) is equivalent to a Frobenius lift \(\varphi:A\to A\) with \(\varphi(x)\equiv x^p\pmod p\). Two themes:

Rigidity. For \(A\) \(p\)-torsionfree and \(A/p\) perfect of char \(p\), there is a unique lift of Frobenius — hence a unique \(\delta\)-structure. This is the perfectoid / semiperfectoid case: \(A_{\inf}\) and \(W(R^\flat)\) have canonical \(\delta\).
Freedom. For \(A=\mathbb{Z}_p[[u]]\) the choice \(\varphi(u)=u^p\) (\(\delta(u)=0\)) gives Breuil–Kisin, but \(\varphi(u)=u^p+p\cdot f(u)\) for other \(f\) gives genuinely different prisms — different \(\delta\)-structures on the same \(p\)-adic ring. The choice corresponds to a deformation of the Frobenius and changes the resulting \(F\)-crystal.

The derived \(\delta\)-structure (on animated rings) measures the obstruction to lifting Frobenius through square-zero extensions via the derived André–Quillen / cotangent theory — exactly the cotangent-complex control of DAG.

Cartier–Witt divisors (Bhatt–Lurie)¶

The absolute formulation of prisms (independent of a chosen base \(\mathbb{Z}_p\)) reinterprets a prism as a Cartier–Witt divisor:

A Witt vector \(a=(a_0,a_1,\ldots)\in W(A)\) corresponds, via the \(p\)-typical Witt construction, to a \(\delta\)-structure on a compatible lift; the Cartier–Witt divisor \(V(a)\) is the closed subscheme / ideal cut out by the Witt vector.
Concretely, a prism \((A,I)\) is equivalent to a Cartier–Witt divisor on an animated \(\delta\)-ring \(A\), i.e. a Witt vector \(d\in W(A)\) such that the ideal \(I=(d_0,d_1^p,\ldots)\) (the "Verschiebung" expansion) is a Cartier divisor and the prism axioms hold.
This makes the absolute prismatic site \((X)_\Delta^{\mathrm{abs}}\) a site of Cartier–Witt divisors over \(X\), base-change independent, and the absolute prismatic cohomology \(\triangle_X^{\mathrm{abs}}\) a functor of \(X\) alone (Bhatt–Lurie).

Object	Classical prism formulation	Cartier–Witt formulation
Prism \((A,I)\)	\(\delta\)-ring \(A\), invertible ideal \(I\), \(\varphi(I)A=pA\)	Cartier–Witt divisor \(d\in W(A)\) with \(I=(d_0,d_1^p,\ldots)\)
Crystalline prism	\((W(k),(0))\)	the zero Witt divisor \(d=0\)
Breuil–Kisin	\((\mathbb{Z}_p[[u]],(u-p))\)	the Witt vector \([u]-[p]\in W(\mathbb{Z}_p[[u]])\)
Perfectoid	\((A_{\inf},(\xi))\)	the Witt vector \([\xi^{\flat}]\) (tilt of \(\xi\))

The Cartier–Witt perspective makes prismatic cohomology base-independent and clarifies the relation to crystalline local systems (Bhatt–Lurie: a crystalline local system on a smooth \(p\)-adic formal scheme \(X\) is the same as a stratification on its absolute prismatic cohomology, i.e. descent data along the absolute prismatic site — a prismatic characterization of the Riemann–Hilbert correspondence in char \(0\) adapted to \(p\)-adic Hodge theory).

See it¶

The crystalline prism recovers crystalline cohomology. Take \(X_0/\mathbb{F}_p\) smooth proper and the crystalline prism \((A,I)=(\mathbb{Z}_p,(0))\). The prismatic site \((X_0/\mathbb{Z}_p)_\Delta\) is, by definition, the category of \(p\)-adically complete thickenings \(T\to X_0\) with a \(\delta\)-structure lifting over \(\mathbb{Z}_p\) — i.e. PD-thickenings (the divided-power envelope, since \(\delta\) on \(\mathbb{Z}_p\) encodes the canonical PD-structure). This is precisely Berthelot's crystalline site \((X_0/W(\mathbb{F}_p))_{\mathrm{cris}}\). Hence:

\[ \triangle_{X_0/\mathbb{Z}_p} \;=\; R\Gamma\bigl((X_0/\mathbb{Z}_p)_\Delta,\mathcal{O}_\Delta\bigr) \;\cong\; R\Gamma_{\mathrm{cris}}(X_0/\mathbb{Z}_p), \]

with Frobenius matching the crystalline Frobenius. For \(X_0=\mathbb{P}^n_{\mathbb{F}_p}\), this recovers \(H^*_{\mathrm{cris}}(\mathbb{P}^n/\mathbb{Z}_p)=\mathbb{Z}_p\{1,c_1,\ldots,c_n\}\) in degrees \(0,2,\ldots,2n\) with Frobenius acting by \(p^i\) on \(H^{2i}\) — a textbook computation now reorganized as "the crystalline prism is the trivial-divisor point of the prism lattice."

Check¶

For the Breuil–Kisin prism \((\mathbb{Z}_p[[u]],(u-p))\), exhibit the Witt vector \(d\in W(\mathbb{Z}_p[[u]])\) cutting out \(I=(u-p)\) and verify \(I=(d_0,d_1^p,\ldots)\) in the Cartier–Witt sense.

Take \(d=[u]-[p]\in W(\mathbb{Z}_p[[u]])\), where \([u]\) (resp. \([p]\)) is the Teichmüller lift. The \(0\)-th Witt component is \(d_0=u-p\in\mathbb{Z}_p[[u]]\), and \(d_i=0\) for \(i\ge 1\) (Teichmüller lifts have only the degree-\(0\) component). Hence \(I=(d_0)=(u-p)\) as required; the Cartier–Witt divisor \(V(d)\) has ideal \((u-p,u^p-p^2,u^{p^2}-p^{p},\ldots)\) but reduces to \((u-p)\) since the higher terms are powers of \((u-p)\) up to units. The prism axiom \(\varphi(I)A=pA\) becomes \(\varphi((u-p))=(u^p-p)=(u-p)(\cdots)\cdot p\cdot(\text{unit})\) — using \(\varphi(u)=u^p\) and \(u^p-p=(u-p)\cdot\frac{u^p-p}{u-p}\) with the quotient a unit in the \(I\)-adic completion.

Why does the quasiregular semiperfectoid condition (lci cotangent amplitude \([-1,0]\)) make the prismatic site computable, when a general prism does not?

The quasisyntomic site on a quasiregular semiperfectoid base \(S\) is covered by quasisyntomic maps from polynomial algebras (the lci / amplitude-\([-1,0]\) condition ensures the cotangent complex is controlled enough that derived prismatic cohomology is computed by a Čech / Koszul complex on these). General \(p\)-adic bases may have unbounded cotangent, and the prismatic site is defined but not directly computable; the quasisyntomic reduction is the technical heart of Bhatt–Scholze that bridges "prisms exist abstractly" to "we can compute prismatic cohomology of any smooth \(X\)."

Pitfalls¶

Bounded vs unbounded prisms: comparison theorems (Hodge–Tate, de Rham) require the bounded prism hypothesis. Unbounded prisms exist abstractly (e.g. for singular \(X\)), but their prismatic cohomology may be unbounded and the Frobenius may be nilpotent — the theorems degenerate.
Perfect vs perfectoid prism: a perfect prism has \(\varphi\) an automorphism on \(A\) (e.g. \(A_{\inf}\)); a perfectoid prism additionally has \(A/I\) perfectoid. Conflating these misses that perfectoid \(A/I\) is the geometric input for \(p\)-adic Hodge comparison, while perfect \(A\) is the algebraic input.
Animated \(\neq\) derived \(\neq\) spectral: animated rings are simplicial commutative (DAG), not \(\mathbb{E}_\infty\) (SAG) — the prismatic theory is built on animated/derived commutative rings, not spectral rings; see DAG vs SAG.
Cartier–Witt divisor is a Witt vector, not an ideal: the divisor \(V(d)\) is cut out by a Witt vector \(d\in W(A)\), and its ideal in \(A\) is the "Verschiebung expansion" \((d_0,d_1^p,\ldots)\) — taking only \(d_0\) loses the higher Witt components that carry the \(\delta\)-structure and Frobenius information.
Crystalline prism is the degenerate \(I=0\) case: the prismatic axioms must be interpreted via the limit / divided-power convention when \(I=0\); the crystalline site is not literally "prisms with zero ideal" in the naive sense but rather the prism whose \(\delta\)-structure encodes PD-structures — forgetting this leads to the wrong claim that "crystalline is a prism like any other."

Refs¶

Bhatt–Scholze, Prisms and Prismatic Cohomology, §§2 (prisms), 4 (site), 17 (perfectoid)
Bhatt–Lurie, Absolute Prismatic Cohomology (Cartier–Witt divisors; absolute site)
Kisin, Crystalline representations and \(F\)-crystals (Breuil–Kisin modules)
Bhatt, Lectures on Prismatic Cohomology (survey; quasisyntomic site)

Next¶

Foundations · p-adic Hodge Comparison · Prismatic index · Derived AG · Spectral AG · Commutative algebra · AG cohomology · K-theory