Introduction
Prismatic cohomology gives a common source for several cohomology theories; its central achievement
is a system of integral comparison theorems: one cohomology theory recovers de Rham cohomology
of the total space, crystalline cohomology of the special fibre, and \(p\)-adic étale cohomology of
the geometric generic fibre. Working integrally means that we keep track of \(p\)-power torsion,
rather than passing to \(\mathbf{Q}_p\).
Let \(K/\mathbf{Q}_p\) be a finite field extension with ring of integers \(\mathcal{O}_K\),
residue field \(k\), and uniformiser \(\pi\). Consider a smooth projective morphism
\[X \longrightarrow S = \operatorname{Spec}(\mathcal{O}_K).\]
The generic fibre \(X_\eta = X_K\) is defined over a field of characteristic zero, while the special
fibre \(X_0 = X_k\) is defined over a field of characteristic \(p\). There are several natural ways
to study this family:
Comparing these cohomology groups is a central problem of \(p\)-adic Hodge theory. Prismatic cohomology
yields a very clean way to do exactly that, while also keeping track of mod-\(p\) information
(which is invisible in the more classical setting of \(p\)-adic Hodge theory, where one usually
inverts \(p\) in the coefficient rings). The prismatic site puts comparison theories into one framework
and yields new integral consequences, thereby constituting a reorganisation of the foundations of
integral \(p\)-adic Hodge theory.
Goal of this learning seminar
The aim of this learning seminar is to:
- Learn the basics of the foundations of prismatic cohomology.
- Study and understand the various comparison theorems.
- Learn to work with prismatic cohomology by working out some beautiful examples and applications.
A little bit more detail
Consider the smooth projective family \(X/\mathcal{O}_K\). One may ask:
Can the above different cohomology groups, together with their additional structures, be recovered
from a single integral object? This leads one to consider Breuil–Kisin prisms.
Let \(A = W(k)[[u]]\), let \(\theta:A\to\mathcal{O}_K\) send \(u\) to \(\pi\), and write
\(I = \ker\theta = (E(u))\), with \(E\) Eisenstein. Equip \(A\) with Witt-vector Frobenius on \(W(k)\)
and \(\varphi_A(u)=u^p\). The pair \((A,I)\) is the Breuil–Kisin prism.
For the \(p\)-adic completion \(\widehat{X}\), there are finitely generated \(A\)-modules
\[H^i_\Delta(\widehat{X}/A).\]
They have cup products and a semilinear Frobenius \(F\), so \(F(ax)=\varphi_A(a)F(x)\).
The induced map
\[H^i_\Delta(\widehat{X}/A)\otimes_{A,\varphi_A}A[1/E]
\xrightarrow{\sim}H^i_\Delta(\widehat{X}/A)[1/E]\]
is an isomorphism. The different classical theories are recovered by changing coefficients in this
cohomology theory. Write \(H^i_\Delta(\widehat{X}/A;B,f)\) for prismatic cohomology with coefficients
changed along \(f:A\to B\): change coefficients in a projective cochain complex computing the theory,
then take its \(i\)-th cohomology.
1. De Rham comparison
With \(\alpha=\theta\circ\varphi_A\), there are natural isomorphisms
\[H^i_\Delta(\widehat{X}/A;\mathcal{O}_K,\alpha)
\cong H^i_{\mathrm{dR}}(X/\mathcal{O}_K),\]
\[H^i_{\mathrm{dR}}(X/\mathcal{O}_K)\otimes_{\mathcal{O}_K}K
\cong H^i_{\mathrm{dR}}(X_K/K).\]
Thus this specialisation recovers differential-form cohomology of the family and its generic fibre.
2. Crystalline comparison
Let \(\beta:A\to W(k)\) send \(u\) to \(0\) and act by Witt-vector Frobenius on \(W(k)\). Then
\[H^i_\Delta(\widehat{X}/A;W(k),\beta)\cong H^i_{\mathrm{crys}}(X_0/W(k)),\]
compatibly with Frobenius. Changing coefficients further to \(k\), at the cochain level,
recovers \(H^i_{\mathrm{dR}}(X_0/k)\).
3. Hodge–Tate comparison
The untwisted specialisation along \(\theta\) is governed by the groups of differential forms.
Put \(L=I/I^2\), an invertible \(\mathcal{O}_K\)-module. There is a spectral sequence of cohomology groups
\[E_2^{a,b}=H^a(X,\Omega^b_{X/\mathcal{O}_K})\otimes_{\mathcal{O}_K}L^{\otimes(-b)}
\Longrightarrow H^{a+b}_\Delta(\widehat{X}/A;\mathcal{O}_K,\theta).\]
The terms with \(a+b=i\) contribute to degree \(i\). Integral degeneration or a direct-sum
decomposition is not being asserted.
4. Étale comparison
Set \(C=\widehat{\overline{K}}\), let \(\mathcal{O}_C^\flat\) be its tilt, and put
\(C^\flat=\operatorname{Frac}(\mathcal{O}_C^\flat)\). The rings
\(A_{\mathrm{inf}}=W(\mathcal{O}_C^\flat)\) and \(B=W(C^\flat)\) will be explained in
meeting 2. Base change of the family to \(\mathcal{O}_C\) gives
\[P_C^i:=H^i_\Delta(\widehat{X}_{\mathcal{O}_C}/A_{\mathrm{inf}};B)
\cong H^i_{\mathrm{\acute{e}t}}(X_{\overline{K}},\mathbf{Z}_p)\otimes_{\mathbf{Z}_p}B.\]
Here the coefficient map is the natural inclusion \(A_{\mathrm{inf}}\to B\).
The comparison respects Galois actions; Frobenius on the right acts only on \(B\).
Since \(B^{F=1}=\mathbf{Z}_p\), it also gives
\[H^i_{\mathrm{\acute{e}t}}(X_{\overline{K}},\mathbf{Z}_p)\cong(P_C^i)^{F=1},\]
where \(F=1\) now means ordinary fixed elements of a cohomology group.
Compatible roots of \(\pi\) give the prism map \(A\to A_{\mathrm{inf}}\)
linking this comparison to the original theory.
Some references: [BS], Theorem 1.8 and Example 1.9;
[BMS2], Theorem 1.2; [BMS1], Theorem 1.8
for the étale comparison; [K], Theorem 23.2.5 for the torsion inequality.
See also [FKW], §2.1.
Talk outlines
Part I: foundations
1. Motivation and the comparison problem
Thursday 22 October 2026 · 10:00–12:00
Introduce \(X/\mathcal{O}_K\), its two fibres and the three cohomology theories in the
introduction. Explain the Hodge, Frobenius and Galois structures,
and why retaining \(p\)-torsion matters. Describe the purpose of crystalline thickenings informally.
Preview how differential forms constrain mod-\(p\) restriction maps and how this leads to the two applications.
Goal of this talk: Describe degree-one cohomology of an elliptic curve with good reduction, and explain what the common prismatic complex must remember.
Scope: State the comparison package informally; no crystalline-site construction or proofs.
Reading: [B], I; [K], §1; [FKW] and [ANP], introductions; [N1]. [BdJ] is background.
2. \(\delta\)-rings, prisms and the \(A_{\mathrm{inf}}\) example
Friday 30 October 2026 · 10:00–12:00
Define a \(p\)-derivation and its Frobenius lift \(\varphi(x)=x^p+p\delta(x)\).
Recall Witt vectors and define a bounded prism, explaining the Cartier ideal, completeness
and \(p\in I+\varphi(I)A\). Work through three examples: the crystalline prism
\((W(k),(p))\), the Breuil–Kisin prism, and the perfect prism attached to \(\mathcal{O}_C\),
where \(C=\widehat{\overline{K}}\). For the last example, explain the tilt \(\mathcal{O}_C^\flat\),
the ring \(A_{\mathrm{inf}}=W(\mathcal{O}_C^\flat)\), and Fontaine’s map with its principal kernel.
State the perfect-prism/perfectoid correspondence.
Goal of this talk: Verify \((\mathbf{Z}_p[[u]],(u-p))\), distinguish the maps \(u\mapsto p,p^p,0\), and construct \(u\mapsto[\pi^\flat]\) into \(A_{\mathrm{inf}}\) using compatible roots. Check compatibility with Frobenius and reduction.
Scope: Devote roughly half the talk to definitions and the first two examples, and half to \(A_{\mathrm{inf}}\). Take the Witt-vector construction, general derived completeness and the perfectoid correspondence as inputs. Explain derived reduction through one two-term example; introduce \(W(C^\flat)\) only as needed for the comparison theorems above. Free \(\delta\)-rings and other period rings are optional reading.
Reading: [B], II–IV; [BS], §§2–3, definitions and examples; [BMS1], §3. [K], §§2–3, 5–8, and [N3] are companions.
3. The prismatic site and its cohomology
Friday 6 November 2026 · 10:00–12:00
For a smooth formal scheme \(Y/(A/I)\), define the objects and coverings of \((Y/A)_\Delta\),
and its structure and reduced structure sheaves. Explain how their cohomology gives the two complexes.
Describe the origin of Frobenius, functoriality and cup products. State completed base change and
proper finiteness, which make the comparisons usable.
Goal of this talk: Compute the relative point \(Y=\operatorname{Spf}(A/I)\), obtaining the complex \(A\), its Frobenius and its reduction \(A/I\).
Scope: Sketch the role of prismatic envelopes and Čech computations; take their existence and descent properties as inputs.
Reading: [B], V; [K], §§11–12; [BS], §4 and Theorem 1.8; [N5].
Part II: comparison theorems
4. Crystalline, Hodge–Tate and de Rham comparisons
Friday 13 November 2026 · 10:00–12:00
State the first three comparisons above, keeping the coefficient maps
\(\theta,\alpha,\beta\) distinct. Explain the Hodge–Tate cohomology sheaves and their global spectral
sequence. Use the Bockstein to illustrate how differential forms enter and how the de Rham differential
is recovered.
Goal of this talk: Work through the Hodge–Tate map in degrees 0 and 1 for one coordinate, including the twist \(I/I^2\).
Scope: Prove this local calculation; state the general comparison and gluing results. Full proofs of all three theorems are beyond this meeting.
Reading: [B], V–VI; [BS], Theorem 1.8 and §§5–6; [N5–N6]. [BdJ] and [K], §§14–15, are background.
5. Étale comparison and integral consequences
Friday 20 November 2026 · 10:00–12:00
State the finite-coefficient comparison over a perfect prism. Explain derived Frobenius fixed points
and their elementary characteristic-\(p\) model, the Artin–Schreier sequence. Indicate why perfection
and descent occur in the proof, taking the deeper results as inputs. Deduce the mod-\(p\) dimension
inequality from comparison and semicontinuity of a perfect complex.
Goal of this talk: Explain the group-level formula in the étale comparison above and its Galois action. Distinguish this special setting from the general derived fixed-point formula, where kernels alone need not suffice.
Scope: Prove the dimension inequality using the comparison theorem as an input.
Reading: [B], IX; [K], §§22–23; [BS], §9; [N2, N4].
Part III: applications
6. Essential dimension I: restriction of mod-\(p\) cohomology
Friday 27 November 2026 · 10:00–12:00
Use [FKW], §2.1, to translate a restriction map into a map of mod-\(p\)
prismatic complexes over \(k[[u]]\). Explain the small-degree freeness input and the role of Cartier
theory in obtaining a lower bound from differential forms. Keep the unramified good-reduction and
degree hypotheses visible.
Goal of this talk: Follow the image module through specialisation and localisation when the open is dense in the special fibre. Explain which comparison is used at each step.
Scope: Give this central argument; state the freeness and Cartier inputs, and outline the additional passage to arbitrary complex opens.
Reading: [FKW], §§2.1–2.2, especially the proof of the restriction bound; [N1–N2].
7. Essential dimension II: incompressibility of covers
Thursday 3 December 2026 · 10:00–12:00
Define essential dimension and \(p\)-essential dimension of a finite cover. Explain how characteristic
classes from the covering group and the restriction bound obstruct compression to a smaller-dimensional
variety. Show how trace handles auxiliary covers of degree prime to \(p\).
Goal of this talk: Explain why the mod-\(p\) homology cover of a complex abelian variety is \(p\)-incompressible for sufficiently large \(p\), using the cup products of degree-one classes.
Scope: Prove the deduction from the restriction theorem. Leave logarithmic extensions, Heisenberg groups and Shimura varieties for further study.
Reading: [FKW], introduction and §2.3, especially Corollary 2.3.12 and its inputs.
8. Wild Brauer classes I: the cohomological input
Friday 11 December 2026 · 10:00–12:00
Introduce the Brauer group, the Kummer sequence and evaluation at local points. Explain how restriction
maps and vanishing cycles enter [ANP]. Introduce the Nygaard filtration through
its relation to divisibility by powers of \(p\) under Frobenius, and state the syntomic/vanishing-cycle
comparison needed in the argument.
Goal of this talk: Unpack the Frobenius eigenspace and restriction map in [ANP], Theorem 1.2.2, and identify the role of each comparison theorem.
Scope: Explain the additional tools and their use; take their general construction and comparison proofs as inputs.
Reading: [ANP], §§2–3; [BMS2], §10; [BS], §§12, 15 as background.
9. Wild Brauer classes II: slopes and arithmetic
Friday 18 December 2026 · 10:00–12:00
Introduce Newton slopes and their relation to Hodge numbers, then explain the total-slope argument
of [ANP], §4. Combine it with meeting 8 to outline the
construction of a Brauer class with surjective evaluation after finite extension. Deduce the
weak-approximation consequence, stating all hypotheses.
Goal of this talk: Trace the implication from a nonzero global 2-form to nonconstant Brauer evaluation; use an abelian surface as a guiding geometric example.
Scope: Focus on the main argument and arithmetic deduction. Explicit products of elliptic curves and prismatic Dieudonné theory are optional extensions.
Reading: [ANP], §§4–5, and optionally §6; [ALB] for the last direction.