Prismatic Cohomology¶
Job: a single \(p\)-adic cohomology theory — the prismatic cohomology of Bhatt–Scholze and the \(A_{\inf}\) cohomology of Bhatt–Morrow–Scholze — that simultaneously interpolates crystalline, de Rham, and étale \(p\)-adic Hodge theory via a characteristic-\(p\) framework of prisms and \(\delta\)-rings.
Core¶
Prismatic cohomology replaces the ad-hoc patchwork of \(p\)-adic cohomology theories (crystalline for smooth proper \(W(k)\)-lifts, de Rham over \(\mathbb{Z}_p\), étale \(/B_{\mathrm{dR}}\) for \(\mathbb{Q}_p\)-varieties) by a relative construction over a base prism \((A,I)\). The output is a complex \(\triangle_{X/A}\) with a Frobenius structure and a Hodge–Tate specialization, from which all classical comparison isomorphisms are recovered by specialization of the prism — including crystalline cohomology at the crystalline prism and the \(A_{\inf}\) comparison at a perfectoid base.
| Layer | Content |
|---|---|
| Prisms | \(\delta\)-rings; \((A,I)\)-prisms; bounded / perfect / crystalline prisms |
| Functor | prismatic site \((X/A)_\Delta\); prismatic cohomology \(\triangle_{X/A}\) |
| Structure | Frobenius \(\varphi\); Hodge–Tate grading; Nygaard completion |
| Comparison | \(A_{\inf}\) cohomology (BMS); de Rham / crystalline / étale via Hodge–Tate |
| Frontiers | Cartier–Witt divisors; animated prisms; motivic / \(K\)-theoretic traces |
Reading map¶
| Page | Focus |
|---|---|
| Foundations | \(\delta\)-rings; prisms; prismatic site; \(\triangle_{X/A}\); Hodge–Tate; Frobenius; comparison theorem |
| p-adic Hodge Comparison | \(A_{\inf}\) cohomology (BMS); Nygaard completion; \(p\)-adic Poincaré lemma; dictionary \(\leftrightarrow\) Fontaine period rings |
| Prisms and Thickenings | prism typology (bounded / perfect / crystalline); animated rings; quasiregular semiperfectoid frames; Cartier–Witt divisors |
Dependency order¶
- Algebraic geometry: schemes, coherent cohomology, cohomology theories — crystalline cohomology assumed known.
- Commutative algebra: \(\mathfrak{m}\)-adic completion, regular sequences, derivations — \(\delta\)-rings are a deformation of a derivation; see CA foundations.
- Derived algebraic geometry: animated rings, derived completion, the cotangent complex — prismatic cohomology is genuinely derived; see DAG foundations.
- K-theory + motivic homotopy: cyclotomic trace, syntomic cohomology, the trace methods that prismatic cohomology explains; see algebraic \(K\)-theory.
- Foundations \(\to\) comparison \(\to\) prism typology; then frontier (Cartier–Witt, prismatic \(K\)-theory).
Interfaces¶
| Area | Contact |
|---|---|
| Algebraic geometry / cohomology | prismatic as the \(p\)-adic cohomology theory unifying crystalline / de Rham / étale |
| Commutative algebra | \(\delta\)-rings, \(p\)-adic completion, derived completions |
| Derived AG | animated rings; derived prismatic cohomology as a derived functor |
| K-theory | BMS identifies \(p\)-complete \(K\)-theory with Nygaard-completed prismatic cohomology (trace methods) |
| Motivic homotopy | motivic filtration on TC; prismatic / syntomic realization |
| Condensed mathematics | \(A_{\inf}^{\blacksquare}\) as analytic ring; BMS comparison in the condensed / analytic-ring language |
| Arithmetic geometry | \(p\)-adic periods of varieties over \(\mathcal{O}_K\); integral \(p\)-adic Hodge theory |
Notation (paper-standard)¶
- \((A,I)\) prism; \(A\) a \(\delta\)-ring, \(I\) a Cartier divisor defining \(\mathbb{Z}_p\to A\to A/I\)
- \(\triangle_{X/A}\in D(A)\) prismatic cohomology; \(\overline{\triangle}_{X/A}=\triangle_{X/A}\otimes^{\mathbb{L}}_A A/I\) Hodge–Tate
- \(\varphi_A:A\to A\) Frobenius lift; \(\varphi\)-module structure on \(\triangle_{X/A}\)
- \(\widehat{\triangle}_{X/A}\) \(p\)-complete (or \(I\)-complete) prismatic cohomology; \(\widehat{\triangle}_{X/A}\{\varphi\}\) Nygaard completion
- \(A_{\inf}=W(R^\flat)\) for perfectoid \(R\); \(\theta:A_{\inf}\to R\); \(\xi=\ker\theta\); \(\tilde{\xi}=\varphi(\xi)/p\) (perfectoid prism)
See it¶
The prism lattice interpolates \(p\)-adic theories. A single smooth proper formal scheme \(\mathfrak{X}/\mathbb{Z}_p\) produces one object \(\triangle_{\mathfrak{X}/A}\) over a chosen prism, and the prism selected dictates which classical theory one recovers:
| Prism \((A,I)\) | Recovered theory |
|---|---|
| \((\mathbb{Z}_p,0)\) crystalline | crystalline cohomology \(H^*_{\mathrm{cris}}(\mathfrak{X}/\mathbb{Z}_p)\) |
| Breuil–Kisin \((\mathbb{Z}_p[[u]],(u-p))\) | Breuil–Kisin cohomology; crystalline after base change along \(u\mapsto p\) |
| Perfectoid \(A_{\inf}\) over \(W(\mathcal{O}_{\mathbb{C}_p}^\flat)\) | \(A_{\inf}\) cohomology of BMS; étale comparison after inverting \(p\) |
| \(p\)-torsionfree lift of \(\mathcal{O}_K\) with \(\delta\)-structure | de Rham via Hodge–Tate specialization \(A/I\) |
This interpolation is the operational content of "prismatic is the universal \(p\)-adic cohomology theory."
Check¶
Why must a prism \((A,I)\) satisfy \(A\) be \(I\)-adically complete and \(p\in I\) invertible mod \(I\)?
The conditions encode "characteristic \(p\) quotient" (\(A/p\) lives over \(\mathbb{F}_p\) via \(\varphi\)) and the completeness needed for the Hodge–Tate and comparison theorems to descend from animated/local data; without \(I\)-adic completeness the prismatic site is not well-defined as a site of bounded prisms, and without \(A/I\) invertible-mod-\(p\) the Hodge–Tate comparison degenerates.
Which specialization of the prism recovers crystalline cohomology, and why is this a sanity check rather than a definition?
The crystalline prism \((A,I)=(\mathbb{Z}_p,0)\): \(A/I=\mathbb{Z}_p\), the prismatic site identifies with the crystalline site of \(X/W(\mathbb{F}_p)\), and \(\triangle_{X/A}\cong R\Gamma_{\mathrm{cris}}(X/\mathbb{Z}_p)\). This is a theorem (Bhatt–Scholze) — one does not define prismatic cohomology to equal crystalline; rather the prism axiomatizes the shape of all such theories and the crystalline case drops out as a terminal point.
Pitfalls¶
- Quasisyntomic vs prismatic site: the quasisyntomic site (large, on quasisyntomic \(A\)-algebras) computes the same cohomology for quasisyntomic \(X\) but is not the prismatic site — confusion arises because both host the "same" Frobenius on a \(p\)-complete derived object.
- Prismatic cohomology is relative to a prism \((A,I)\); the bare symbol \(\triangle_X\) is shorthand for a chosen base and the comparison theorems track base change carefully.
- Comparison theorems need hypotheses (boundedness of prismatic cohomology, smoothness / formal smoothness of \(X\) over \(A/I\)) — the cleanest statements are for smooth formal \(\mathcal{O}_K\)-schemes; non-liftable varieties require derived/animated refinements and Nygaard control.
- \(p\)-adic vs \(I\)-adic completion: for the Breuil–Kisin prism these coincide, but for \(A_{\inf}\) over a general perfectoid the \(p\)-completion differs from \(I\)-completion and the BMS comparison uses the Nygaard-completed \(p\)-adic object.
Refs¶
- Bhatt–Scholze, Prisms and Prismatic Cohomology (Annals 2022)
- Bhatt–Morrow–Scholze, Integral \(p\)-adic Hodge Theory (Publ. Math. IHÉS 2019)
- Bhatt–Lurie, Absolute Prismatic Cohomology / The Prismatic Characterization of Crystalline Local Systems (preprints)
- Drinfeld, On Prismatic Cohomology; for the Breuil–Kisin prism see Kisin, Crystalline representations and \(F\)-crystals
Next¶
Foundations · p-adic Hodge Comparison · Prisms and Thickenings · AG cohomology · Derived AG · Commutative algebra · K-theory · Motivic homotopy