[18], Lecture III. The underlying reference is [25], section 2.

Using the framework of \(\delta\)-rings, we now set up the formalism of prisms, modulo a key technical detail: the difference between classical completion and derived completion with respect to an ideal. We postpone discussion of the latter until Section 6.

Notation.

For \(A \in \Ring\text{,}\) let \(\Mod_A\) denote the category of \(A\)-modules. For an ideal \(I\) of \(A\text{,}\) and an object \(M \in \Mod_A\text{,}\) write \(M[I]\) for the \(I\)-torsion submodule of \(M\) and \(M[I^\infty]\) for the union \(\bigcup_n M[I^n]\text{;}\) if \(I = (f)\) is principal, we also notate these as \(M[f]\) and \(M[f^\infty]\text{.}\)

Subsection5.1Distinguished elements and examples

We begin by singling out elements of a \(\delta\)-ring which behave as if they “vanish to order \(1\)”, as indicated by the \(p\)-derivation.

Definition5.1.1.

Let \(A\) be a \(\delta\)-ring. An element \(d \in A\) is distinguished if \((p, d, \delta(d))\) is the unit ideal of \(A\text{.}\) That is, the intersection of the zero loci of \(p, d, \delta(d)\) on \(\Spec A\) is empty.

If \(A\) is \((p, d)\)-local, then \(d \in A\) is distinguished if and only if \(\delta(d)\) is a unit in \(A\text{;}\) in fact this is the definition used in [18] and [25]. The discrepancy will not affect the definition of a prism because the latter already includes a completeness hypothesis (see Definition 5.3.1). One confusing aspect of our definition is that units in \(A\) are always distinguished.

In many arguments that follow, we can reduce to the \((p,d)\)-local case by localizing \(A\) at \((p,d)\text{.}\) By Remark 2.4.10, the result is still a \(\delta\)-ring.

Remark5.1.2.

Any morphism in \(\Ring_{\delta}\) carries distinguished elements to distinguished elements. The converse holds for the map \(\phi\text{;}\) see Exercise 5.5.1.

We describe a series of examples which will be related to various preexisting \(p\)-adic cohomology theories. We will promote these examples to prisms in Remark 5.3.4.

Example5.1.3.Crystalline cohomology.

Take \(A = \ZZ_p\) with \(d=p\text{.}\) Then \(\delta(d) = 1-p^{p-1} \equiv 1 \pmod{p}\text{,}\) so \(p\) is distinguished. By the same token, by Remark 2.2.3, \(p\) is distinguished in any \(\delta\)-ring.

Example5.1.4.\(q\)-de Rham cohomology and Wach modules.

Take \(A = \ZZ_p \llbracket q-1 \rrbracket\) with the \(\delta\)-structure for which \(\phi(q) = q^p\text{,}\) and define \(d\) to be the “\(q\)-analogue of \(p\)”:

\begin{equation*}
d = [p]_q = \frac{q^p-1}{q-1} = \sum_{i=0}^{p-1} q^i.
\end{equation*}

Under the map \(A \to \ZZ_p\) taking \(q\) to 1, \(d\) maps to \(p\) which is distinguished in the target; it follows that \(d\) is itself distinguished.

This example is closely related to Fontaine's theory of \((\varphi, \Gamma)\)-modules. The original construction of Fontaine [49] described an equivalence of categories between the continuous representations of the absolute Galois group of \(\QQ_p\) on finite \(\ZZ_p\)-modules and a certain category of finite modules over the \(p\)-adic completion of \(A[(q-1)^{-1}]\text{,}\) in which the continuous action of the monoid \(\ZZ_p \setminus \{0\}\) on \(A\) characterized by \(\gamma(q) = q^\gamma\) is extended to the module. The elements of the \(p\)-adic completion can be viewed as formal Laurent series in \(q-1\) with coefficients in \(\ZZ_p\text{;}\) it was later shown by Cherbonnier and Colmez [39] that the base ring can be shrunk down to the subring consisting of Laurent series whose negative tails converge on some region (see also [80]).

The ring \(A\) itself is the base ring of the theory of Wach modules [124], [13]; the \((\varphi, \Gamma)\)-module associated to a Galois representation descends to a Wach module if and only if the representation is crystalline in Fontaine's sense. Similar considerations apply if we enlarge \(A\) by replacing \(\ZZ_p\) with an unramified extension \(\frako_K\text{,}\) where now the Galois group in question is that of \(K\text{.}\)

Example5.1.5.Breuil-Kisin cohomology.

Let \(K/\QQ_p\) be a finite extension. Let \(\pi\) be a uniformizer of \(K\text{.}\) Let \(W \subseteq \frako_K\) be the maximal unramified subring (i.e., the ring \(W(k)\) where \(k\) is the residue field of \(\frako_K\)). Take \(A = W \llbracket u \rrbracket\) with the \(\delta\)-structure extending the canonical one on \(W\) for which \(\phi(u) = u^p\text{.}\) Take \(d\) to be a generator of the kernel of the map \(A \to \frako_K\) taking \(u\) to \(\pi\text{;}\) by projecting along the map \(u \mapsto 0\) as in Example 5.1.4, we see that \(d\) is distinguished.

The ring \(A\) is the base ring of the theory of Breuil-Kisin modules ([85]), which provides an alternative to Wach modules that can be used to classify crystalline representations of the Galois group of a ramified extension of \(\QQ_p\text{.}\) See [37] for more on the parallel between the two constructions.

Example5.1.6.\(\Ainf\)-cohomology.

Let \(A\) be the \((p, q-1)\)-adic completion of \(\ZZ_p[q^{p^{-\infty}}]\text{.}\) By Proposition 3.3.6, we have an isomorphism \(A \cong W(R)\) where \(R\) is the \((q-1)\)-adic completion of the coperfection of \(\FF_p[q-1]\text{.}\) In particular, \(A\) has a unique \(\delta\)-ring structure, for which \(\phi(q) = q^p\text{;}\) note that in this case \(\phi\) is an automorphism. By Example 5.1.4, \(d = [p]_q\) is a distinguished element, as is \(\phi^n(d)\) for any \(n \in \ZZ\text{.}\)

Let \(K\) be the \(p\)-adic completion of the \(p\)-cyclotomic extension \(\QQ_p(\mu_{p^\infty})\text{.}\) The ring \(R\) can then be identified with the perfection of \(\frako_K/(p)\) by fixing a choice of a coherent sequence \((\zeta_{p^n})\) of \(p\)-power roots of unity and identifying \(q\) with this sequence; this identifies \(R\) with the tilt of \(\frako_K\) (see Section 7 for further discussion). In this context, the ring \(A\) arises in Fontaine's notation as the value of the functor \(\Ainf\) evaluated at the valuation ring \(\frako_K\text{.}\)

Subsection5.2Properties of distinguished elements

We collect some lemmas about distinguished elements. See also Lemma 7.1.2 for a precise characterization of distinguished elements in \(W(R)\) when \(R\) is a perfect ring of characteristic \(p\text{.}\)

We first show that “distinguished elements are locally irreducible.”

Lemma5.2.1.

Let \(A\) be a \(\delta\)-ring and choose \(f,h \in A\text{.}\) Then \(fh\) is distinguished if and only if \(f\) and \(h\) are both distinguished and \((p,f,h)\) is the unit ideal of \(A\text{.}\)

If \(fh\) is distinguished, then \(\delta(fh)\) is a unit modulo \((p,fh)\) and hence also modulo \((p,f)\text{;}\) we deduce that both \(\delta(f)\) and \(h\) are invertible modulo \((p,f)\text{.}\) This means that \(f\) is distinguished and \((p,f,h)\) is the unit ideal; by symmetry, \(h\) is also distinguished.

Conversely, suppose that \(f\) and \(h\) are both distinguished and that \((p,f,h)\) is the unit ideal. To check that \((p, fh, \delta(fh))\) is the unit ideal, we may work in the localizations at \((p,f)\) and \((p,h)\text{;}\) without loss of generality, we may then assume that \(p,f \in \Rad(A)\text{.}\) In this case, \(\delta(f)\) and \(h\) are both units, and so (5.1) implies that \(\delta(fh)\) is a unit modulo \((p,f) = (p, fh)\text{;}\) hence \(fh\) is distinguished.

Remark5.2.2.

While Lemma 5.2.1 is written in a symmetric manner, in practice we will use it in the case where \(p, f \in \Rad(A)\text{.}\) We again reiterate that according to our conventions, any unit is a distinguished element.

We now see that the property of an element being distinguished depends only on the principal ideal generated by that element.

Lemma5.2.3.

Let \(A\) be a \(\delta\)-ring. Then an element \(f \in A\) is distinguished if and only if \(p \in (p^2, f, \phi(f))\text{.}\) (If \(A\) is \(p\)-local, this is equivalent to \(p \in (f, \phi(f))\text{.}\))

If \(f\) is distinguished, then \(ap + bf + c \delta(f) = 1\) for some \(a,b,c \in A\text{.}\) Since \(\phi(f) - f^p = p \delta(f)\text{,}\) we can write \(ap^2 + bfp + c\phi(f) - cf^p = p\text{,}\) yielding \(p \in (p^2, f, \phi(f))\text{.}\) Conversely, suppose that \(p \in (p^2, f, \phi(f))\) and (by way of contradiction) \((p, f, \delta(f))\) is not the unit ideal; using Remark 2.4.10, we may localize \(A\) to reduce to the case where \(p,f,\delta(f) \subseteq \Rad(A)\) (and \(A \neq 0\)). In this case, \(p \in (f, \phi(f))\text{,}\) so there exist \(a,b \in A\) such that \(p = af + b \phi(f)\text{;}\) that is,

\begin{equation*}
p(1 - b \delta(f)) = af + b f^p = f(a + bf^{p-1}).
\end{equation*}

Since \(p\) is distinguished (Example 5.1.3), so is \(f\) by two applications of Lemma 5.2.1 (one in each direction); this yields the desired contradiction.

Corollary5.2.4.

For \(A\) a \(\delta\)-ring, the property of \(d \in A\) being distinguished depends only on the image of \(d\) in \(A/p^2\text{.}\)

It will be convenient later to globalize the notion of an ideal generated by a distinguished element. Fortunately, the resulting condition still has a convenient characterization.

Lemma5.2.5.

Let \(A\) be a \(p\)-local \(\delta\)-ring. Let \(I\) be a locally principal ideal of \(A\) contained in \(\Rad(A)\text{.}\) Then the following conditions are equivalent.

We have \(p \in I + \phi(I)A\text{.}\)

There exists a faithfully flat map \(A \to A'\) of \(p\)-local \(\delta\)-rings which is an ind-Zariski localization, such that \(IA' = (f)\) for some distinguished element \(f\) of \(A'\) contained in \(\Rad(A')\text{.}\)

Moreover, if these conditions hold, then \(p \in I^p + \phi(I)A\text{.}\)

The equivalence of (1) and (2) is a consequence of Remark 2.4.10 (which allows us to construct \(A'\) such that \(IA'\) is principal) and Lemma 5.2.3. Compare [18], Lecture III, Corollary 1.9 or [25], Lemma 3.1.

To check that (1) and (2) imply \(p \in I^p + \phi(I)A\text{,}\) we may reduce to the case where \(I = (f)\) for some distinguished element \(f\) of \(A\text{.}\) In this case, the equation \(\phi(f) = f^p + p\delta(f)\) shows that \(p \in (f^p, \phi(f))\) because \(\delta(f)\) is a unit.

Subsection5.3Prisms

A prism will consist of a \(\delta\)-ring \(A\) and an ideal \(I\) such that the closed subschemes of \(\Spec A\) defined by \(I\) and \(\phi^{-1}(I)\) intersect “as transversely as possible” along the closed subscheme defined by \(p\text{.}\)

Definition5.3.1.

A \(\delta\)-pair consists of a pair \((A, I)\) in which \(A\) is a \(\delta\)-ring and \(I\) is an ideal.

A prism is a \(\delta\)-pair \((A, I)\) satisfying the following conditions.

The ideal \(I\) defines a Cartier divisor on \(\Spec A\) (i.e., \(I\) is an invertible \(A\)-module, or equivalently \(I\) is locally principal generated by a non-zerodivisor). In most of our examples, \(I\) will be principal; see Exercise 6.7.14 for a restriction that applies otherwise.

The ring \(A\) is derived \((p, I)\)-complete (as a module over itself). We will define this condition a bit later (see Definition 6.2.1); for the moment, note that it implies \((p,I) \subseteq \Rad(A)\) (see Corollary 6.3.2) and hence also \(\phi(I) \subseteq \Rad(A)\text{.}\) See also Remark 5.3.2.

We have \(p \in I + \phi(I) A\text{.}\) By Lemma 5.2.3, this holds if \(I\) is generated by a distinguished element.

A prism \((A, I)\) is orientable if the ideal \(I\) is principal. A prism \((A,I)\) is oriented if it is orientable and we have fixed the choice of a generator \(d\text{,}\) which by Lemma 5.2.3 is a distinguished element (and a non-zerodivisor).

A prism \((A,I)\) is bounded if \(A/I\) has bounded \(p^\infty\)-torsion; that is, there is a positive integer \(n\) such that \((A/I)[p^n] = (A/I)[p^\infty]\text{.}\)

Remark5.3.2.

Definition 5.3.1 includes a condition on derived completeness that we have not yet defined. We insert a few remarks in order to maintain the narrative flow.

If \(A\) is classically \((p,I)\)-complete, then \(A\) is derived \((p,I)\)-complete. The converse holds if \(A\) is \((p,I)\)-adically separated; this will be true in particular if \((A,I)\) is a bounded prism (see Lemma 6.4.2).

For these reasons, on first reading it is safe to pretend that Definition 5.3.1 requires \(A\) to be classically \((p,I)\)-complete rather than derived \((p,I)\)-complete. However, when proving theorems it will be problematic to take completions due to the bad behavior of this functor in some situations (Remark 6.1.2). The notion of derived completeness will help mitigate this, as will the odd definition of flatness for morphisms of prisms (Definition 5.4.3).

Example5.3.3.

A \(\delta\)-pair \((A, I)\) with \(I = (p)\) is a prism if and only if \(A\) is \(p\)-torsion-free and classically \(p\)-complete. We say that such a prism is crystalline.

Remark5.3.4.

By Lemma 5.2.3 (and the fact that the rings in question are all integral), all of the examples of distinguished elements enumerated in Subsection 5.1 give rise to prisms (taking \(I = (d)\)). These examples are all bounded.

Example 5.1.3 is an example of a crystalline prism. Example 5.1.6 is an example of a perfect prism; we will describe these in terms of perfectoid rings in Section 7.

Example5.3.5.The universal oriented prism.

Let \(A_0 = \ZZ_{(p)}\{d\}\) be the free \(\delta\)-ring in a single variable \(d\) over \(\ZZ_{(p)}\text{.}\) Let \(S\) be the multiplicative subset of \(A_0\) generated by \(\phi^n(\delta(d))\) for all \(n \geq 0\text{.}\) By Lemma 2.4.8, the localization \(A_1 = S^{-1} A_0\) is also a \(\delta\)-ring. Let \(A\) be the derived \((p,d)\)-completion of \(A_1\text{;}\) since \(A_1\) is \(p\)-torsion-free, \(A\) is classically \(p\)-complete. By construction, \(d\) is a distinguished element of \(A\) and \((A, dA)\) is a bounded prism. Moreover, \(p,d\) is a regular sequence in \(A\) and \(\phi\colon \overline{A} \to \overline{A}\) is \(p\)-completely flat (see Definition 6.5.1).

Lemma5.3.6.

Let \((A, I)\) be a prism. Then the ideal \(\phi(I) A\) is principal, and any generator of it is a distinguished element.

It will be enough to produce a single generator of \(\phi(I) A\text{,}\) as then Lemma 5.2.3 (which applies because \(pA + \phi(I) \subseteq \Rad(A)\)) will imply that any other generator is also distinguished.

By definition, we have \(p = a + b\) with \(a \in I^p, b \in \phi(I) A\text{;}\) we will show that \(b\) generates \(\phi(I) A\) and is distinguished. Choose a faithfully flat map \(A \to A'\) of \(\delta\)-rings as per Lemma 5.2.5; it will suffice to show that \(b\) generates \(\phi(I)A'\) and is distinguished in \(A'\text{.}\) By construction, \(IA'\) is generated by a distinguished element \(d \in A'\text{.}\) Write \(a = xd^p, b = y\phi(d)\) for some \(x,y \in A'\text{.}\) Since \(\phi(d)\) is also distinguished, it will suffice to show that \(y\) is a unit in \(A'\text{.}\) Since \(pA + I \subseteq \Rad(A)\text{,}\) it will further suffice to show that \(pA' + IA' + yA' = A'\text{.}\)

Suppose the contrary; using Remark 2.4.10, we may choose a further localization \(A' \to A''\) of \(\delta\)-rings such that \(pA'' + IA'' + yA'' \subseteq \Rad(A'')\text{.}\) The equation \(p = a+b = xd^p + y\phi(d)\) yields

Since \(1-y\delta(d)\) is a unit in \(A''\) and \(p\) is distinguished, we may apply Lemma 5.2.1 twice to deduce that \(d\) is distinguished in \(A''\) and \(d^{p-1}(x+y)\) is a unit; this is impossible because \(d \in \Rad(A'')\text{.}\) (Compare [18], Lecture III, Lemma 3.5 or [25], Lemma 3.6.)

Remark5.3.7.

Let \((A, I)\) be a prism. Since \(I\) is an invertible \(A\)-module, \(I \otimes_A A/I = I/I^2\) is an invertible \(A/I\)-module, as is \(I^n/I^{n+1}\) for any nonnegative integer \(n\text{.}\) These will appear in the discussion of Hodge-Tate cohomology.

Subsection5.4The category of prisms

Definition5.4.1.

The category of \(\delta\)-pairs is defined so that a morphism \((A,I) \to (B,J)\) is a morphism \(f\colon A \to B\) of \(\delta\)-rings such that \(f(I) \subseteq J\text{.}\) The category of prisms, denoted \(\Prm\text{,}\) is defined as the full subcategory of the category of \(\delta\)-pairs consisting of prisms.

Lemma5.4.2.Rigidity of prisms.

Let \((A, I) \to (B,J)\) be a morphism in \(\Prm\text{.}\) Then the natural map \(I \otimes_A B \to J\) is an isomorphism of \(B\)-modules. In particular, \(J = IB\text{.}\)

Since the map in question is between invertible \(B\)-modules, it is enough to check that it is surjective. Using Lemma 5.2.5, we may reduce to the case where \(I = (f)\) and \(J = (g)\) are both principal ideals generated by distinguished elements. Then \(f\) is a multiple of \(g\) in \(B\text{,}\) so we may apply Lemma 5.2.1 to conclude. (Compare [18], Lecture III, Lemma 3.7 or [25], Lemma 3.5.)

Definition5.4.3.

A map \((A, I) \to (B, J)\) in \(\Prm\) is (faithfully) flat if \(B\) is \(I\)-completely (faithfully) flat in the sense of Definition 6.5.1. This holds in particular if \(A \to B\) is (faithfully) flat.

Exercises5.5Exercises

1.

Let \(A\) be a \(\delta\)-ring. Prove that an element \(d \in A\) is distinguished if and only if \(\phi(d)\) is distinguished.