Remark 11.0.1. Warning.
Our definition of the prismatic cohomology \(\Prism_{R/A}\) is preliminary; it will be overridden later by the construction of derived prismatic cohomology in Section 18.
Section 11 The prismatic site
Reference.
[18], lecture V.
In this section, we introduce the prismatic site of an affine scheme in the sense of [18]. This should perhaps be called the naive prismatic site because it does not give correct answers when one drops the affine hypothesis; see Remark 11.7.2.
Note that while we make some basic definitions in a rather expansive degree of generality, we will be unable to compute anything except under some smoothness hypotheses. We impose those starting in Section 12.
Subsection 11.1 Indiscrete Grothendieck topologies
Definition 11.1.1.
For a topological space \(X\text{,}\) let \(|X|\) be the (small) category consisting of the open subsets of \(X\text{,}\) where the set of morphisms from \(U\) to \(V\) is a singleton set if \(U \subseteq V\) and 0 otherwise.
A presheaf on \(X\) valued in some category \(\calC\) is nothing but a contravariant functor \(|X| \to \calC\text{.}\) A presheaf \(F\) is a sheaf if and only \(F\) preserves the colimit of the diagram \(\prod_{i,j \in I} V_i \cap V_j \rightrightarrows \prod_i V_i \to U\) for any covering of an open subset \(U\) by open subsets \(\{V_i\}_{i \in I}\text{.}\) (Since the functor is contravariant, that means we get a limit in \(\calC\text{.}\))
Remark 11.1.2.
Building upon this idea, one can define the notion of a Grothendieck topology on any category; the key point is to specify which families of morphisms to a given target are coverings of that target, and then the sheaf property on a presheaf is formulated in terms of diagrams as above. (A site means a category equipped with a Grothendieck topology.)
To deal with the naive prismatic site, we only need the case of an indiscrete (or chaotic) Grothendieck topology, in which no families of morphisms are coverings except isomorphisms, and there is consequently no distinction between presheaves and sheaves. What makes this interesting is that we do not assume that our category has a final object!
Definition 11.1.3.
Let \(\calC\) be a small category; we can then form the category \(\Pshv(\calC)\) of presheaves of abelian groups on \(\calC\text{.}\) The functor \(\Pshv(\calC) \to \Ab\) given by
\begin{equation*}
F \mapsto H^0(\calC, F) = \lim_{X \in \calC} F(X)
\end{equation*}
is left exact; we can then form its derived functor \(R\Gamma(C, \bullet)\text{.}\)
Example 11.1.4.
Let \(\calC\) be the category \(\{0 \to 1 \to 2 \to \cdots\}\text{;}\) that is, the objects are nonnegative integers, and the morphisms from \(i\) to \(j\) form a singleton set if \(i \leq j\) and the empty set otherwise. In this case, we have
\begin{equation*}
H^0(\calC, F) = \lim_n F(n), \qquad H^1(\calC, F) = R^1 \lim_n F(n)
\end{equation*}
While the indiscrete Grothendieck topology on a small category becomes trivial if the category admits a final object (i.e., an object to which every other object maps uniquely), we will be interested in a slightly less rigid situation where the topology becomes both interesting and computable.
Definition 11.1.5.
Recall that a final object in a category \(\calC\) is an object \(X \in \calC\) such that \(\Hom_{\calC}(Y, X)\) is a singleton set for every \(Y \in \calC\text{.}\) By contrast, a weakly final object in \(\calC\) is an object \(X \in \calC\) such that \(\Hom_{\calC}(Y, X) \neq \emptyset\) for every \(Y \in \calC\text{.}\) (That is, the natural map from the representable functor \(h_X\) to the forgetful functor \(\calC \to \Set\) is a bijection if \(X\) is final, but only surjective if \(X\) is weakly final.)
Example 11.1.6.
Let \(\calC\) be the category of algebraic field extensions of a fixed field \(F\text{,}\) viewed as a full subcategory of \(\Ring_F\text{.}\) Then every algebraic closure of \(F\) is a weakly final object of \(\calC\text{,}\) but not a final object unless \(F\) is itself separably closed.
Lemma 11.1.7. Čech-Alexander resolution.
Let \(\calC\) be a small category admitting finite nonempty products and containing a weakly final object \(X\text{.}\) Then for any \(F \in \Pshv(\calC)\text{,}\) \(R\Gamma(\calC, F)\) is computed by applying \(F\) to the Čech nerve of \(X\) (see Example 11.2.5); that is, it is given by the Čech-Alexander complex
\begin{equation}
0 \to F(X) \to F(X \times X) \to F(X \times X \times X) \to \cdots\tag{11.1}
\end{equation}
in which the differentials are given by alternating sums as per Definition 11.2.2.
Proof.
Using the fact that the map from \(h_X\) to the forgetful functor is a surjection (and that \(h_{X^n}\) is a sheaf for all \(n\) because we are using the indiscrete topology), this reduces to the general Čech spectral sequence for a (not necessarily indiscrete) Grothendieck topology. See [117], tag 07JM.
Subsection 11.2 A word on (co)simplicial objects
In preparation to use this language more extensively later, we introduce a bit of terminology that relates naturally to the previous discussion. Our conventions on simplicial sets and objects are taken to match [117], tag 0162.
Definition 11.2.1.
Let \(\Delta\) be the category of finite ordered sets. That is, the objects of \(\Delta\) are the sets \([n] = \{0,\dots,n\}\) for \(n = 0,1,\dots\) and a morphism \(f\colon [n] \to [m]\) is a nondecreasing map of sets (i.e., \(i \leq j\) implies \(f(i) \leq f(j)\)).
For \(n \geq 1\) and \(0 \leq j \leq n\text{,}\) let \(\delta_j^n\colon [n-1] \to [n]\) be the injective morphism in \(\Delta\) with \((\delta_j^n)^{-1}(\{j\}) = \emptyset\text{.}\) For \(n \geq 0\) and \(0 \leq j \leq n\text{,}\) let \(\sigma_j^n\colon [n+1] \to [n]\) be the surjective morphism in \(\Delta\) with \((\sigma_j^n)^{-1}(\{j\}) = \{j,j+1\}\text{.}\) Every morphism in \(\Delta\) can be factored into morphisms of these forms; see Exercise 11.8.2.
Definition 11.2.2.
A simplicial object of a category \(\calC\) is a covariant functor \(U\colon \Delta^{\op} \to \calC\text{.}\) A cosimplicial object of a category \(\calC\) is a covariant functor \(U\colon \Delta \to \calC\) (i.e., a simplicial object of \(\calC^{\op}\)). See Figure 11.2.3 and Figure 11.2.4 for graphical representations of simplicial and cosimplicial objects, respectively.
We will frequently consider (co)simplicial abelian groups, (co)simplicial (commutative) rings, and (co)simplicial modules over a (co)simplicial ring. Any cosimplicial abelian group \(U\) gives rise to a complex in which the differential are alternating sums of the maps \(\delta_j^n\text{:}\)
\begin{equation*}
d^n = \sum_{j=0}^{n+1} (-1)^j U(\delta_j^{n+1}).
\end{equation*}
Example 11.2.5.
Suppose that the category \(\calC\) admits finite nonempty products. Then for any \(X \in \calC\text{,}\) we can make a simplicial object \(U\) in \(\calC\) by taking \(U([n])\) to be the product of copies of \(X\) indexed by the elements of \([n]\text{.}\) This gives the Čech nerve of \(X\text{,}\) as in Lemma 11.1.7; for \(F\colon \calC \to \Ab\) a contravariant functor, the complex associated to the cosimplicial abelian group \(F(U)\) is the Čech-Alexander complex (Lemma 11.1.7).
Subsection 11.3 The prismatic site and “oppo-site”
Definition 11.3.1.
Let \((A, I)\) be a prism with slice \(\overline{A} = A/I\text{,}\) and let \(R\) be an \(\overline{A}\)-algebra. The prismatic oppo-site of \(R\) relative to \(A\text{,}\) denoted \((R/A)^{\op}_{\Prism}\text{,}\) will be the category in which an object consists of a morphism \((A,I) \to (B, IB)\) together with a morphism of \(\overline{A}\)-algebras \(R \to B/IB\text{.}\) (Recall that by Lemma 5.4.2, for any morphism \((A,I) \to (B, J)\) of prisms we must have \(J = IB\text{.}\)) We will typically notate such an object as \((R \to B/IB \leftarrow B)\) and depict such an object as a diagram as in Figure 11.3.2 (where \(\delta\) indicates a morphism in \(\Ring_\delta\)); a morphism of objects will consist of a morphism between the corresponding diagrams. Taking the opposite category yields the prismatic site \((R/A)_{\Prism}\text{.}\)
Remark 11.3.3.
Note that the category \((R/A)^{\op}_{\Prism}\) depends on the whole prism \((A,I)\) and not just on the underlying ring \(A\text{.}\) However, to keep the notation under control we leave \(I\) out, to be inferred from context (as in [18]).
Example 11.3.4.
For \(R = \overline{A}\text{,}\) \((R/A)^{\op}_{\Prism}\) is simply the category of prisms over \((A,I)\text{,}\) and thus has the initial object \((R \cong A/I \leftarrow A)\text{.}\) That is, the prismatic site in this case has a final object, and so cohomology on it is trivial.
Example 11.3.5.
Take \(R = \overline{A} \langle X \rangle\) to be the classical \(p\)-completion of \(\overline{A}[X]\text{.}\) In this case, the prismatic site does not have a final object; however, there are some useful test objects. For instance, let \(B\) be the \((p,I)\)-completion of \(A[X]\text{,}\) viewing the latter as a \(\delta\)-ring with \(\delta(X) = 0\text{;}\) the isomorphism \(B/IB \cong R\) gives us an object of \((R/A)_{\Prism}\text{.}\) (Compare [18], Lecture V, Example 2.7.)
One may generalize Example 11.3.5 as follows.
Proposition 11.3.6.
Let \((A,I)\) be a bounded prism and let \(R\) be a \(p\)-completely smooth \(A/I\)-algebra. Then there exists a prism \((B,J)\) over \((A,I)\) with \(B/IB \cong R\text{.}\)
Proof.
Using Remark 2.3.3, we may reduce the claim to Proposition 6.5.3.
Subsection 11.4 The case of a perfect prism
Definition 11.4.1.
Let \((A,I)\) be a perfect prism. Define the perfect prismatic site to be the subcategory of \((R/A)_{\Prism}\) consisting of objects of the form \((R \to B/IB \leftarrow B)\) in which \((B, IB)\) is a perfect prism. Recall by Theorem 7.3.5 that these objects are in one-to-one correspondence with lenses over \(R\text{.}\)
Example 11.4.2.
Let \((A, I)\) be a perfect prism. Take \(R = \overline{A} \langle X \rangle\) as in Example 11.3.5. Take \(S = \overline{A} \langle X^{p^{-\infty}} \rangle\text{;}\) we then have \(S = B/IB\) where \(B\) is the \((p, I)\)-completion of \(A[X^{p^{-\infty}}]\) for the \(\delta\)-structure under which \(X^{p^{-n}}\) is \(\delta\)-constant for all \(n\text{.}\) Note that \(R \to S\) is \(p\)-completely faithfully flat.
Remark 11.4.3.
If we further reduce the perfect prismatic site by considering only perfect prisms \((B, I)\) in which \(B/I\) is \(p\)-normal, we end up with the diamond of \(R\) in the sense of [110].
Subsection 11.5 Prismatic and Hodge-Tate cohomology
Definition 11.5.1.
With notation as in Definition 11.3.1, define the functors \(\calO_\Prism\) and \(\overline{\calO}_\Prism\) from \((R/A)^{\op}_{\Prism}\) taking \((R \to B/IB \leftarrow B)\) to \(B\) and \(B/IB\) respectively. We will think of these as the structure (pre)sheaf and the reduced structure (pre)sheaf.
The prismatic complex of \(R\) relative to \(A\) (or more precisely, relative to \((A/I)\)) is the object \(\Prism_{R/A} = R\Gamma((R/A)_{\Prism}, \calO_\Prism) \in D(A)\text{.}\) This is a derived \((p, I)\)-complete commutative algebra object in \(D(A)\text{;}\) the Frobenius action on \(\calO_{\Prism}\) induces a \(\phi\)-semilinear map \(\Prism_{R/A} \to \Prism_{R/A}\text{.}\)
The Hodge-Tate complex of \(R\) relative to \(A\) is the object \(\overline{\Prism}_{R/A} = R\Gamma((R/A)_\Delta, \overline{\calO}_\Delta) \in D(\overline{A})\text{.}\) By construction, we have \(\overline{\Prism}_{R/A} = \Prism_{R/A} \otimes_A^L \overline{A}\) (with no completion in the tensor product).
Remark 11.5.2.
To reiterate a point made in Section 1, the objects \(\Prism_{R/A}\) and \(\overline{\Prism}_{R/A}\) are by their nature intrinsic only in \(D(A)\) and \(D(\overline{A})\text{,}\) respectively; they do not come with distinguished representations as complexes.
Subsection 11.6 More on the prismatic site
We now verify the properties of the prismatic site needed in order to compute cohomology on it via the Čech resolution (Lemma 11.1.7).
Lemma 11.6.1.
Let \((A,I)\) be a prism. Then the forgetful functor from prisms over \((A,I)\) to \(\delta\)-pairs over \((A,I)\) admits a left adjoint (the prismatic envelope).
Proof.
We may check this locally on \(A\text{,}\) so by Lemma 5.2.5 we may assume that \(I\) is principal generated by a distinguished element \(d\text{.}\) Let \((A, I) \to (B, J)\) be a morphism of \(\delta\)-pairs. Let \(B'\) be the free \(\delta\)-ring over \(A\) in the generators \(x/d\) for \(x \in J\text{.}\) Let \(B_1\) be the derived \((p,d)\)-completion of \(B\) (viewed as a \(\delta\)-ring using Exercise 6.7.12). If \(B_1\) is \(d\)-torsion-free, then \((B_1, dB_1)\) has the desired universal property. Otherwise, we transfinitely iterate the operations of taking the maximal \(d\)-torsion-free quotient and taking the derived \((p,d)\)-completion; this terminates because a countably filtered colimit of derived \((p,d)\)-complete rings is again derived \((p,d)\)-complete (Remark 6.3.4), so we can stop taking the completions once we get to an uncountable ordinal.
Remark 11.6.2.
The proof of Lemma 11.6.1 gives very little insight into the structure of the resulting objects. See Lemma 14.4.2 for an example where we can make this construction explicit.
Remark 11.6.3.
For some purposes, it is more natural to modify the definition of a prism to replace the ideal \(I\) with a “virtual Cartier divisor”, to provide some missing stability under base change. In this context Lemma 11.6.1 becomes much more straightforward, as the issue with taking torsion-free quotients becomes irrelevant.
Lemma 11.6.4.
Let \((A, I)\) be a prism with slice \(\overline{A} = A/I\text{,}\) and let \(R\) be an \(\overline{A}\)-algebra. Then the category \((R/A)_{\Prism}\) admits finite nonempty products.
Proof.
It is equivalent to show that \((R/A)^{\op}_{\Prism}\) admits finite nonempty coproducts. Let \((R \to B/IB \leftarrow B)\) and \((R \to C/IC \leftarrow C)\) be two objects of \((R/A)^{\op}_{\Prism}\text{.}\) Form the \(\delta\)-ring \(D_0 = B \otimes_A C\) using Lemma 2.4.3. Let \(J\) be the kernel of the natural map
\begin{equation*}
D_0 \to B/IB \otimes_{A/IA} C/IC \to B/IB \otimes_R C/IC;
\end{equation*}
that is, \(J\) is generated by elements of the form \(x \otimes 1 - 1 \otimes y\) where \(x \in B, y \in C\) have the property that there is some \(z \in R\) mapping to \(x \in B/IB\) and to \(y \in C/IC\text{.}\) Apply Lemma 11.6.1 to the pair \((D_0, J)\) to obtain a prism \((D, ID)\text{;}\) the object \((R \to D/ID \leftarrow D) \in (R/A)^{op}_{\Prism}\) is the desired coproduct.
Proposition 11.6.5.
Let \((A, I)\) be a prism with slice \(\overline{A} = A/I\text{,}\) and let \(R\) be an \(\overline{A}\)-algebra. Then the category \((R/A)_{\Prism}\) admits a weakly final object.
Proof.
Let \(F_0\) be the free \(\delta\)-ring over \(A\) on the set \(R\text{,}\) so that there is a surjection of \(A\)-algebras \(F_0 \to R\text{;}\) let \(J\) be the kernel of this map. Applying Lemma 11.6.1 to the \(\delta\)-pair \((F_0, J)\) gives a prism \((F, IF)\) over \((A, I)\text{.}\)
We will check that \((F, IF)\) is a weakly initial object in \((R/A)^{\op}_{\Prism}\text{.}\) By the adjunction property from Lemma 11.6.1, it suffices to check that for any object \((R \to B/IB \leftarrow B)\) of \((R/A)_{\Prism}\text{,}\) there exists a morphism \(F_0 \to B\) of \(\delta\)-rings compatible with the map \(R \to B/IB\text{;}\) this holds because \(F_0\) is a free \(\delta\)-ring over \(A\text{.}\)
Remark 11.6.7.
To summarize, with notation as in Definition 11.3.1, we can compute the cohomology of either \(\calO_{\Prism}\) or \(\overline{\calO}_{\Prism}\) on \((R/A)_{\Prism}\) by choosing a weakly final object \((F, IF)\) and forming the cosimplicial \(A\)-algebra \(F^\bullet\) from Lemma 11.1.7; that is, \(F^n\) is the \((n+1)\)-fold completed tensor product of \(F\) over \(A\text{.}\)
Subsection 11.7 Additional remarks
Remark 11.7.1.
One awkward feature is that a morphism \(Y \to X\) does not give rise to a pullback functor \((X/A)_{\Prism} \to (Y/A)_{\Prism}\text{,}\) because there is no natural way to perform base change for prisms along a morphism at the level of slices. At the level of rings, this is saying that given an object \((R \to B/IB \leftarrow B)\) of \((R/A)_{\Prism}\) and a morphism \(R \to S\) of rings, there is no natural way to promote the map \(B/IB \to B/IB \widehat{\otimes}_R S\) to a morphism \(B \to *\text{.}\) This is in fact a rather common issue with Grothendieck topologies; it also arises for the infinitesimal and crystalline sites.
The standard fix for this is to replace the prismatic site with its associated category of sheaves of sets, the prismatic topos. In this language, one can show ([25], Remark 4.3) that the functor \(h_X\colon (B, IB) \mapsto \Hom_{\overline{A}}(\Spf(B/IB), X) \) is a sheaf on the site \((\overline{A}/A)_{\Prism}\) and the slice topos over this functor is naturally equivalent to the topos of \((R/A)_{\Prism}\text{.}\) (This also applies if we replace the indiscrete Grothendieck topology with the one in Remark 11.7.2.)
Remark 11.7.2.
As pointed out above, what we are calling the prismatic site here (following [18]) should really be called the naive prismatic site. The site defined in [25], Definition 4.1 has a different Grothendieck topology: a morphism \((B, IB) \to (C, IC)\) of prisms corresponds to a covering if and only if it is \(I\)-completely faithfully flat. This changes the resulting topos, but not the prismatic or Hodge-Tate cohomology; it also gives better results when replacing the ring \(R\) with a (usually smooth) \(p\)-adic formal scheme \(X\text{,}\) now with an object being given by a diagram as in Figure 11.7.3 (where \(\Spf\) is always taken with respect to the \(p\)-adic topology) to obtain the site \((X/A)_{\Prism}\text{.}\)
An alternate foundational treatment based on the prismatization functor on \(p\)-adic formal schemes and the absolute prismatic site (in which one does not fix a base prism, only the formal scheme \(X\text{;}\) for \(X = \Spf \ZZ_p\) this is just the category \(\Prism\) itself) can be found in work of Bhatt–Lurie (in preparation), Bhatt–Scholze (in preparation), and Drinfeld [42].
Exercises 11.8 Exercises
1.
Verify the claim of Example 11.1.4.
2.
Show that any morphism in \(\Delta\) can be factored as a composition of morphisms each of the form \(\delta_j^n\) or \(\sigma_j^n\) for some \(n, j\text{.}\)
