Introduction and overview

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Section 1 Introduction and overview

Reference.

[18], Lecture I.

We begin by describing some of the (global) context for the study of prisms and prismatic cohomology. We then take a more local view to explain what we are trying to do (with no proofs at this point). Keep in mind that it is not necessary to know about all of the topics I describe here in order to understand the rest of the course!

Subsection 1.1 (Co)homology of complex varieties

The dual notions of homology and cohomology first appeared in topology as ways to “linearize nonlinear geometry”; that is, to attach “linear” invariants (abelian groups, modules over commutative rings) to complicated geometric objects. This project proved to be quite successful, to the point that nowadays there are even significant real-life applications of these constructions; see for example [56].

In the theory of manifolds, there are two traditional approaches to homology and cohomology.

One is the combinatorial approach, in which one views a global space as being glued together from a small number of simple pieces (e.g., a triangulation of a surface). One can then extract the desired invariants by careful bookkeeping on the interactions between the pieces. The most robust version of this is singular homology/cohomology (also called Betti homology/cohomology).
The other is the cohomology of differential forms, which developed from the classical theorems in multivariable calculus about the relationship between integrals over a region and integrals over its boundary (and the physical laws from electromagnetism related to these), culminating in Stokes's theorem. The relationship between this and the singular theory was put on firm footing by the work of Georges de Rham, in whose honor the cohomology of differential forms is commonly referred to as de Rham cohomology.

Note that one cannot speak of differential forms without some additional structure on a manifold, at least a smooth ($C^\infty$) structure. For a complex manifold, one can do better: by Dolbeaut's theorem, one can compute de Rham cohomology using exclusively holomorphic forms (see [59], Chapter 3).

These two constructions are closely related via integration: for $C$ a homology class of dimension $k$ and $\omega$ a $k$-form on a complex manifold $X\text{,}$ there is a well-defined integral $\int_C \omega\text{.}$ Stokes's theorem then asserts that

\begin{equation*} \int_C d\omega = \int_{\delta(C)} \omega; \end{equation*}

we thus obtain a pairing

\begin{equation*} H_i(X, \CC) \times H^i_{\dR}(X) \to \CC \end{equation*}

which by the de Rham and Dolbeaut theorems is a perfect pairing; that is, the induced map

\begin{equation*} H^i_{\dR}(X) \cong H_i(X, \CC)^\dual \cong H^i(X, \CC) \end{equation*}

is an isomorphism.

While one can think of this isomorphism as asserting that singular and de Rham cohomology are “the same”, this is not the most useful conclusion to draw; it is better to interpret this as saying that “the whole is greater than the parts”.

The space $H^i(X, \CC)$ is really the base extension to $\CC$ of the $\QQ$-vector space $H^i(X, \QQ)\text{.}$ One can transport this rational subspace over to de Rham cohomology, but the result is rather mysterious! It can be described in terms of integrals of differential forms over rational homology classes (often called periods because the most basic example is the number $\pi$) but the arithmetic of these is quite subtle; there is a far-reaching conjecture about this due to Kontsevich and Zagier [87].
The singular cohomology depends only on the original manifold, whereas the de Rham cohomology depends on the extra data of a complex structure. For example, for Riemann surfaces of some genus $g \geq 2\text{,}$ the underlying manifolds are all homeomorphic, so all of the variation comes from the complex structure.
By Hodge's theorem, every real cohomology class admits a unique harmonic representative. This then leads to the Hodge decomposition on the de Rham side.
When $X$ is the base extension of an algebraic variety over a subfield $K$ of $\CC\text{,}$ $H^i_{\dR}(X)$ can also be computed using algebraic differential forms, by an argument of Grothendieck [60] using Serre's GAGA theorem [114], and therefore is really the base extension to $\CC$ of a certain $K$-vector space. One interesting consequence of this is that there is a strong relationship between the underlying topological spaces of the spaces obtained by taking different embeddings of $K$ into $\CC\text{;}$ this becomes more interesting when you realize that these spaces need not in general be homeomorphic! For instance, Serre found examples for which these spaces have distinct fundamental groups [115]; see also [32], [98], [102].

Note that the Hodge decomposition does not survive the descent to $K\text{,}$ but one of the filtrations derived from it does: this is the Hodge filtration.

To retain all of this data at once, Deligne defined the notion of a Hodge structure consisting of a $\CC$-vector space with a filtration plus a $\ZZ$-lattice. This captures much of the interesting data in the above picture; for example, from the Hodge structure of an abelian variety, one can recover the abelian variety by forming a complex torus (taking the quotient of the $\CC$-vector space by the $\ZZ$-lattice).

Subsection 1.2 The trouble with torsion

One thing that is missing from the previous discussion is the fact that singular homology can be defined over $\ZZ\text{,}$ and is not in general a subspace of the singular homology over $\QQ\text{;}$ that is, the singular homology can have nontrivial torsion. This is true even for algebraic varieties.

Example 1.2.1.

An Enriques surface (over an algebraically closed field) is a projective algebraic surface with irregularity 0 (in the sense of Riemann-Roch) for which the canonical bundle is nontrivial but its square is trivial (e.g., the quotient of a K3 surface by a fixed-point-free involution). The cycle class of the canonical bundle defines a nontrivial 2-torsion element of $H^2\text{.}$

The comparison isomorphism as formulated cannot really say anything meaningful about torsion in homology; one of the goals of prismatic cohomology is to provide a mechanism for interpreting this torsion via reductions to positive characteristic. Here is a sample statement.

Theorem 1.2.2.

Let $X$ be a smooth projective variety over $\QQ\text{.}$ Choose a prime number $p$ for which $X$ can be extended to a smooth proper scheme $\frakX$ over $\ZZ_{(p)}\text{,}$ and put $X_p = \frakX \times_{\ZZ_{(p)}} \FF_p\text{.}$ Then

\begin{equation*} \dim_{\FF_p} H^i(X^{\an}, \ZZ/p\ZZ) \leq \dim_{\FF_p} H^i_{\dR}(X_p). \end{equation*}

Proof.

See [22], Theorem 1.1.

One way to think of Theorem 1.2.2 is that while a nonzero rational homology class constitutes an obstruction to integrating differentials in characteristic 0, a nonzero $p$-torsion homology class constitutes an obstruction to integrating differentials in characteristic $p\text{.}$

Remark 1.2.3.

For various reasons, the inequality in Theorem 1.2.2 can be strict. One reason is that $X_p$ is not uniquely determined by $X\text{;}$ this has to do with birational geometry in mixed characteristic (e.g., one can perform flips in the special fiber). Another is that the left-hand side is not uniquely determined by $X_p\text{;}$ see [22], 2.1 for an example (a threefold which admits a nonsplit elliptic fibration over an Enriques surface, as compared with the split fibration).

Subsection 1.3 The $p$-adic situation

With the previous discussion in mind, let us now transition to the analogous discussion for algebraic varieties over not $\CC$ but a $p$-adic field (where $p$ denotes a fixed prime).

The discussion we conducted in over $\CC$ falls under the label of Hodge theory. There is a parallel discussion that happens for algebraic varieties over $p$-adic fields that is covered by the label of $p$-adic Hodge theory. In that context, there is no good analogue of singular (Betti) homology or cohomology, because the underlying topological spaces don't have the “right” homotopy type. (The homotopy type generally misses all of the “good reduction” information and only picks up “bad reduction” data. There is an extensive literature on this point; see [44] for an introduction.)

The best available replacement for singular homology/cohomology is étale homology/cohomology with $p$-adic coefficients, where crucially this is the same prime $p$ as the residue characteristic. This choice of characteristic can be thought of as an “ugly duckling”: underappreciated at first, but in fact a beautiful swan in the making.

One thing that étale cohomology with $p$-adic coefficients does not do gracefully is specialize to characteristic $p\text{;}$ it does not give a Weil cohomology in that setting (that is, you cannot use it to keep track of zeta functions and $L$-functions with complete accuracy). There are various ways to control this, which all amount to switching over to de Rham cohomology and making that work better in characteristic $p\text{;}$ a notable example is crystalline cohomology, which builds on Grothendieck's interpretation of de Rham cohomology via the infinitesimal site [61]). In any approach of this type, some effort is needed to overcome the fact that the Poincaré lemma doesn't hold in positive characteristic, the issue being that there are “too many constants”: in a characteristic-$p$ setting the formal derivative of any $p$-th power vanishes.

In this course we will consider an approach to $p$-adic cohomology via the mechanism of prisms (see Definition 5.3.1 for the definition). One benefit of this point of view is that almost everything we want to say appears already in a local setting, where we can talk very concretely about rings and complexes without having to keep track of too much fancy stuff (derived categories, simplicial objects, etc.). Another advantage is that it keeps track of “everything at once”; instead of constructing different cohomology theories and asserting comparison isomorphisms between them, we'll construct “one theory to rule them all”, in the manner of the universal coefficient theorem of algebraic topology. That is, we will have a single functor which we can postcompose with various simple algebraic functors to recover more classical constructions.

Subsection 1.4 The role of prisms

We give a representative statement of a prismatic cohomology isomorphism.

Definition 1.4.1.

Fix a prime $p$ and define the ring $A = \ZZ_p \llbracket u \rrbracket\text{;}$ this is a regular noetherian local ring of dimension 2 with residue field $\FF_p\text{.}$ Let $\phi\colon A \to A$ be the continuous homomorphism with $\phi(u) = u^p\text{;}$ this lifts the Frobenius endomorphism on $A/(p)\text{.}$ Let $I$ be the ideal $(u-p)$ of $A\text{;}$ let $\theta\colon A \to \ZZ_p$ be the identification of $A/I$ with $\ZZ_p$ taking $u$ to $p\text{.}$ The triple $(A, \phi, I)$ will later form a basic example of a prism; see Section 5 for the general definition.

The action of $\phi^*$ on $\Spec A$ fixes the axial points $(u)$ and $(p)$ and the closed point $(u,p)\text{,}$ but acts nontrivially on other points. For example, $(u-p)$ is carried to $(u-p^p)\text{.}$

Theorem 1.4.2.

For $R$ the $p$-adic completion of a smooth $\ZZ_p$-algebra, we can functorially define a complex $\Prism_{R/A}$ consisting of $(p, u)$-adically complete $A$-modules, as an object in the derived category $D(A)$ of $A$-modules; together with a morphism $\phi_{R/A}\colon \phi^* \Prism_{R/A} \to \Prism_{R/A}$ in $D(A)$ (so in particular $\phi^*$ is an operation in $D(A)$). Moreover, the pair $(\Prism_{R/A}, \phi_{R/A})$ will have the following additional properties.

The map $\phi_{R/A}$ becomes a quasi-isomorphism after inverting $u-p\text{.}$
There is a natural quasi-isomorphism

\begin{equation*} \phi^* \Prism_{R/A} \widehat{\otimes}^L_A \FF_p \cong \Omega^*_{R_{\FF_p}/\FF_p}. \end{equation*}

That is, at the closed point of $\Spec A\text{,}$ $\phi^* \Prism_{R/A}$ computes de Rham cohomology of $R_{\FF_p}\text{.}$ (There is also a version over $\ZZ_p$ using continuous differentials, which amounts to working at the point $(u-p^p)\text{.}$)
There is a quasi-isomorphism

\begin{equation*} (\Prism_{R/A} \widehat{\otimes}^L_A \overline{\FF_p((u))})^{\phi_{R/A} \otimes \phi=1} \cong R\Gamma_{\et}(\Spec R_{\CC_p}, \FF_p) \end{equation*}

where $\CC_p$ is a completed algebraic closure of $\QQ_p$ and $\phi$ denotes the absolute Frobenius on $\overline{\FF_p((u))}\text{;}$ this becomes natural if we relate the algebraic closures of $\QQ_p$ and $\FF_p((u))$ using the field of norms isomorphism (Theorem 8.3.4). That is, at the point $(p)$ of $\Spec A\text{,}$ $\Prism_{R/A}$ computes the $\FF_p$-étale cohomology of $R_{\CC_p}\text{.}$ (There is also a version with $\ZZ/(p^n)$-coefficients, which amounts to working in an infinitesimal thickening of $p=0\text{,}$ or if you prefer over $\Spec A[p/u]\text{.}$)
There is a natural identification

\begin{equation*} H^i(\Prism_{R/A} \widehat{\otimes}^L_{A,\theta} \ZZ_p) \cong \Omega^i_{R/\ZZ_p}. \end{equation*}

That is, at the point $(u-p)$ of $\Spec A\text{,}$ $\Prism_{R/A}$ computes the Hodge cohomology of $R\text{.}$ (A more robust version of this statement would include a twist; we'll come back to this later.)

Remark 1.4.3.

In case you know what this means, condition 1 in Theorem 1.4.2 is similar to a central restriction in the definition of a shtuka, or more precisely a shtuka with one leg. Recent developments in the Langlands correspondence over function fields, particularly the work of V. Lafforgue [91], makes heavy use of shtukas with multiple legs; while these have a geometric interpretation (see [47]), it is far from clear whether this can be integrated with the prismatic point of view.

Remark 1.4.4.

One important aspect of Theorem 1.4.2 is that we are not asserting a functorial construction of a complex of $A$-modules “on the nose”, but only in the derived category. This is in contrast with, say, de Rham cohomology, which is computed by a specific meaningful complex; it is more akin to the situation for étale cohomology in this respect.

However, in the local development one can mostly ignore derived aspects. They become unavoidable at the point when one wants to glue local structures together.

Remark 1.4.5.

The positioning of different cohomological invariants at different points in $\Spec A$ is illustrated in Figure 1.4.6. One can also observe in this picture the metaphor behind the term prism: the prism is an object that “refracts” the information from the original space into a “spectrum” of cohomological invariants.

Figure 1.4.6. The “values” of $\Prism_{R/A}$ at various points of $\Spec A = \ZZ_p\llbracket u \rrbracket$ as described by Theorem 1.4.2. The dashed arrow indicates where $\phi_{R/A}$ fails to be a quasi-isomorphism. Adapted from [18], Lecture I.

Notes on prismatic cohomology

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