Last week, we had Peter Scholze give an interesting distinguished lecture series here at UCLA on “Prismatic Cohomology”, which is a new type of cohomology theory worked out by Scholze and Bhargav Bhatt. (Video of the talks will be available shortly; for now we have some notes taken by two notetakers in the audience on that web page.) My understanding of this (speaking as someone that is rather far removed from this area) is that it is progress towards the “motivic” dream of being able to define cohomology ${H^i(X/\overline{A}, A)}$ for varieties ${X}$ (or similar objects) defined over arbitrary commutative rings ${\overline{A}}$, and with coefficients in another arbitrary commutative ring ${A}$. Currently, we have various flavours of cohomology that only work for certain types of domain rings ${\overline{A}}$ and coefficient rings ${A}$:

• Singular cohomology, which roughly speaking works when the domain ring ${\overline{A}}$ is a characteristic zero field such as ${{\bf R}}$ or ${{\bf C}}$, but can allow for arbitrary coefficients ${A}$;
• de Rham cohomology, which roughly speaking works as long as the coefficient ring ${A}$ is the same as the domain ring ${\overline{A}}$ (or a homomorphic image thereof), as one can only talk about ${A}$-valued differential forms if the underlying space is also defined over ${A}$;
• ${\ell}$-adic cohomology, which is a remarkably powerful application of étale cohomology, but only works well when the coefficient ring ${A = {\bf Z}_\ell}$ is localised around a prime ${\ell}$ that is different from the characteristic ${p}$ of the domain ring ${\overline{A}}$; and
• Crystalline cohomology, in which the domain ring is a field ${k}$ of some finite characteristic ${p}$, but the coefficient ring ${A}$ can be a slight deformation of ${k}$, such as the ring of Witt vectors of ${k}$.

There are various relationships between the cohomology theories, for instance de Rham cohomology coincides with singular cohomology for smooth varieties in the limiting case ${A=\overline{A} = {\bf R}}$. The following picture Scholze drew in his first lecture captures these sorts of relationships nicely:

The new prismatic cohomology of Bhatt and Scholze unifies many of these cohomologies in the “neighbourhood” of the point ${(p,p)}$ in the above diagram, in which the domain ring ${\overline{A}}$ and the coefficient ring ${A}$ are both thought of as being “close to characteristic ${p}$” in some sense, so that the dilates ${pA, pA'}$ of these rings is either zero, or “small”. For instance, the ${p}$-adic ring ${{\bf Z}_p}$ is technically of characteristic ${0}$, but ${p {\bf Z}_p}$ is a “small” ideal of ${{\bf Z}_p}$ (it consists of those elements of ${{\bf Z}_p}$ of ${p}$-adic valuation at most ${1/p}$), so one can think of ${{\bf Z}_p}$ as being “close to characteristic ${p}$” in some sense. Scholze drew a “zoomed in” version of the previous diagram to informally describe the types of rings ${A,A'}$ for which prismatic cohomology is effective:

To define prismatic cohomology rings ${H^i_\Delta(X/\overline{A}, A)}$ one needs a “prism”: a ring homomorphism from ${A}$ to ${\overline{A}}$ equipped with a “Frobenius-like” endomorphism ${\phi: A \to A}$ on ${A}$ obeying some axioms. By tuning these homomorphisms one can recover existing cohomology theories like crystalline or de Rham cohomology as special cases of prismatic cohomology. These specialisations are analogous to how a prism splits white light into various individual colours, giving rise to the terminology “prismatic”, and depicted by this further diagram of Scholze:

(And yes, Peter confirmed that he and Bhargav were inspired by the Dark Side of the Moon album cover in selecting the terminology.)

There was an abstract definition of prismatic cohomology (as being the essentially unique cohomology arising from prisms that obeyed certain natural axioms), but there was also a more concrete way to view them in terms of coordinates, as a “${q}$-deformation” of de Rham cohomology. Whereas in de Rham cohomology one worked with derivative operators ${d}$ that for instance applied to monomials ${t^n}$ by the usual formula

$\displaystyle d(t^n) = n t^{n-1} dt,$

prismatic cohomology in coordinates can be computed using a “${q}$-derivative” operator ${d_q}$ that for instance applies to monomials ${t^n}$ by the formula

$\displaystyle d_q (t^n) = [n]_q t^{n-1} d_q t$

where

$\displaystyle [n]_q = \frac{q^n-1}{q-1} = 1 + q + \dots + q^{n-1}$

is the “${q}$-analogue” of ${n}$ (a polynomial in ${q}$ that equals ${n}$ in the limit ${q=1}$). (The ${q}$-analogues become more complicated for more general forms than these.) In this more concrete setting, the fact that prismatic cohomology is independent of the choice of coordinates apparently becomes quite a non-trivial theorem.