In this lecture, we study perfect prisms (i.e., prisms with a bijective Frobenius map) in detail. These end up being closely related to perfectoid rings, which appear frequently in \(p\)-adic Hodge theory; however, we will not use too much of the existing theory of perfectoid rings, and in fact we will end up recovering some of it via a different approach.

Notation.

For \(I\) an ideal in a commutative ring, write \(\sqrt{I}\) for its radical.

Subsection7.1Distinguished elements in perfect \(\delta\)-rings

Recall that the condition of an element of a \(\delta\)-ring being distinguished is meant to capture the idea that “the \(p\)-adic order of vanishing equals 1”. For perfect \(\delta\)-rings, we can further develop this metaphor to assert that “the linear coefficient in \(p\) is a unit”.

The following will be used later in the discussion of perfect prisms (see Lemma 7.1.2).

Lemma7.1.1.

Let \(A\) be a \(p\)-local, \(p\)-torsion-free, \(p\)-adically separated \(\delta\)-ring in which \(A/p\) is reduced (e.g., \(W(R)\) where \(R\) is a perfect \(\FF_p\)-algebra). Suppose that \(d \in \Rad(A)\) is a distinguished element.

In the ring \(A\text{,}\)\(d\) is a non-zerodivisor.

To prove (1), suppose by way of contradiction that \(df = 0\) for some nonzero \(f \in A\text{.}\) Since \(A\) is \(p\)-torsion-free and \(p\)-adically separated, we may divide \(f\) by a suitable power of \(p\) to reduce to the case where \(f \notin pA\text{.}\) Now

Since \(A\) is \((p,d)\)-local, \(\delta(d)\) is a unit in \(A\text{,}\) so \(f^p \phi(f) = 0\text{.}\) Reducing modulo \(p\text{,}\) we obtain \(f^{2p} \equiv 0 \pmod{p}\text{.}\) Since \(A/p\) is reduced, this implies \(f \equiv 0 \pmod{p}\text{,}\) contradicting our earlier choice of \(f\) and thus proving the claim.

To prove (2), it is enough to show that \((A/d)[p^2] = (A/d)[p]\text{.}\) That is, given \(f,g \in A\) with \(p^2 f = gd\text{,}\) we must have \(pf \in dA\text{.}\) Since \(gd \in p^2 A\text{,}\) we have \(\delta(gd) \in pA\) and hence \(\phi(g) \delta(gd) \in pA\text{.}\) Rewriting this as \(\delta(d) g^p \phi(g) + \delta(g) \phi(gd)\text{,}\) we see that \(\delta(d) g^p \phi(g) \in pA\text{.}\) Since \(A\) is \((p,d)\)-local, \(\delta(d)\) is a unit in \(A\text{,}\) so \(g^p \phi(g) \in pA\) and so \(g^{2p} \in pA\text{.}\) Because \(A/p\) is reduced, this implies \(g \in pA\text{;}\) since \(A\) is \(p\)-torsion-free, this implies that \(pf \in dA\) as desired. (Compare [25], Lemma 2.34.)

Lemma7.1.2.

Let \(R\) be a perfect \(\FF_p\)-algebra. Then \(d = \sum_{n=0}^\infty [x_n] p^n \in W(R)\) is distinguished if and only if \((x_0, x_1)\) is the trivial ideal of \(R\text{.}\) In particular, if \(d \in \Rad(W(R))\) (which means \(x_0 \in \Rad(R)\)), then \(d\) is distinguished if and only if \(x_1\) is a unit.

Use (3.1) to write \(d^p \equiv [x_0]^p \pmod{p^2}\) and

\begin{equation*}
p\delta(d) = \phi(d)-d^p \equiv p [x_1]^p \pmod{p^2}
\end{equation*}

to deduce that the ideals \((p, d, \delta(d))\) and \((p, [x_0], [x_1])\) coincide.

Remark7.1.3.

The criterion for distinguished elements in Lemma 7.1.2 coincides with Fontaine's notion of a primitive element of degree \(1\). While this terminology was introduced in [50], it echoes similar constructions found elsewhere (e.g., [79]).

Subsection7.2Perfect prisms

Definition7.2.1.

A prism \((A,I)\) is perfect if \(A\) is a perfect \(\delta\)-ring.

Theorem7.2.2.

Let \((A, I)\) be a perfect prism.

The ideal \(I\) is principal, and any generator \(d\) of \(I\) is a distinguished element and a non-zerodivisor.

The ring \(A\) is \(p\)-torsion-free and classically \((p, I)\)-complete.

We have a canonical isomorphism \(A \cong W(A/(p))\) of \(\delta\)-rings.

We have \((A/I)[p^\infty] = (A/I)[p]\) and \((A/p)[I^\infty] = (A/p)[I]\text{.}\) In particular, \((A,I)\) is a bounded prism.

By Lemma 5.3.6, the ideal \(I\) is principal and any generator \(d\) of \(I\) is a distinguished element. By Lemma 2.2.8, \(A\) is \(p\)-torsion-free.

The ring \(A/(p)\) is perfect (by functoriality) and derived \(I\)-complete (by Proposition 6.3.1, it being the cokernel of \(A \stackrel{\times p}{\to} A\)). By Lemma 6.4.3, \(A/(p)\) is also classically \(I\)-complete. By induction on \(n\) using the exact sequence

and the isomorphism \(A/p \cong p^{n-1}A/p^n A\) of \(A\)-modules (a consequence of \(A\) being \(p\)-torsion-free), we deduce that each quotient \(A/(p^n)\) is classically \(I\)-complete.

Since \(A\) is \(p\)-torsion-free and derived \(p\)-complete, it is also classically \(p\)-complete by Lemma 6.4.2. By the previous paragraph, it is also classically \((p, I)\)-complete.

By Proposition 3.3.6, \(A \cong W(A/p)\text{.}\) By Lemma 7.1.2, any generator \(d\) of \(I\) is a non-zerodivisor. By Lemma 6.4.3, \((A/p)[I^\infty] = (A/p)[I]\text{.}\) By Lemma 7.1.2, \((A/I)[p^\infty] = (A/I)[p]\text{.}\)

Proposition7.2.3.

The inclusion of the category of perfect prisms into \(\Prm\) admits a left adjoint. Given a prism \((A, I)\text{,}\) the left adjoint is obtained by taking the classical \((p,I)\)-completion of the coperfection of \(A\) (which we call the coperfection of \((A,I)\)).

Let \(A'\) be the coperfection of \(A\text{;}\) by Lemma 2.2.8, \(A'\) is \(p\)-torsion-free. Let \(A''\) be the classical \(p\)-completion of \(A'\text{;}\) by Lemma 6.4.2, \(A''\) is also the derived \(p\)-completion. By Exercise 2.5.8, \(A''\) can be canonically promoted to a \(\delta\)-ring over \(A\text{.}\) Now Proposition 3.3.6 implies \(A'' \cong W(A''/p)\text{.}\)

For each positive integer \(n\text{,}\) we may now argue as in the proof of Theorem 7.2.2 that the derived \(I\)-completion of \(A/p^n\) coincides with the classical completion. Consequently, if we take \(A'''\) to be the classical \((p, I)\)-completion of \(A''\) (or equivalently of \(A'\)), then \(A'''\) also equals the derived \((p,I)\)-completion of either \(A'\) or \(A''\text{.}\) By Exercise 2.5.8, \(A'''\) can be canonically promoted to a \(\delta\)-ring over \(A''\text{.}\) Again, Proposition 3.3.6 implies \(A''' \cong W(A'''/p)\text{.}\)

At this point, \((A''', IA''')\) is a prism (the conditions on the ideal \(IA'''\) are implied by the corresponding conditions on \(I\)) and \(A'''\) is universal for maps of \(A\) to derived \((p,I)\)-complete \(\delta\)-rings. Thus the proof is complete. (Compare [18], Lecture IV, Lemma 1.3 or [25], Lemma 3.9.)

Subsection7.3Tilting and slicing

Definition7.3.1.

For any prism \((A,I)\) (perfect or not), define the slice (or face) of \((A,I)\) as the ring \(\overline{A} = A/I\text{.}\) Define the tilt of \((A,I)\) (or of \(\overline{A}\)), denoted \(\overline{A}^\flat\text{,}\) as the perfection of \(\overline{A}/p\text{.}\)

Suppose that \((A,I)\) is bounded, so that \(\overline{A}\) is classically \(p\)-complete. Using Lemma 3.3.5, we may lift the projection map \(\overline{A}^\flat \to \overline{A}/p\) uniquely to a map \(\theta_A\colon W(\overline{A}^\flat) \to \overline{A}\text{.}\)

Remark7.3.2.

The term slice is not standard terminology. Another reasonable name would be the special fiber, in the sense that the prism is some sort of “thickening” of the slice.

Proposition7.3.3.

Let \((A,I)\) be a perfect prism with slice \(\overline{A}\) and tilt \(\overline{A}^\flat\text{.}\) We then have a commutative diagram as in Figure 7.3.4 in which the horizontal arrows are all surjective, the vertical arrows are all reductions modulo \(p\text{,}\) and the diagonal arrows are all isomorphisms. Moreover, this diagram is natural in \((A, I)\text{.}\)

Everything will follow once we construct a natural isomorphism \(A \cong W(\overline{A}^\flat)\text{.}\) By Theorem 7.2.2, it will suffice to construct a natural isomorphism \(A/p \cong \overline{A}^\flat\text{.}\)

By Theorem 7.2.2, \(I\) admits a generator \(d\) which is a distinguished element. By definition, we have \(\overline{A}/p = A/(p,d)\text{.}\) For each positive integer \(n\text{,}\) the \(n\)-fold Frobenius \(A/(p,d) \to A/(p,d)\) identifies with the canonical map \(A/(p,d^{p^n}) \to A/(p,d)\) compatibly with \(n\text{,}\) so the limit \(\lim_\phi \overline{A}/p\) gets identified with \(\lim_\phi A/(p, d^{p^n})\text{.}\) The latter is naturally isomorphic to \(A/(p)\) because the latter is clasically \(d\)-complete (Lemma 6.4.3).

Theorem7.3.5.

The slice functor \((A,I) \mapsto \overline{A}\) restricts to a fully faithful functor from perfect prisms to \(\Ring\text{.}\)

It will suffice to explain how to recover \(A\) and \(I\) functorially from \(\overline{A}\text{.}\) Since \(\overline{A}\) is in the essential image of the functor, \(\phi\colon \overline{A}/p \to \overline{A}/p\) is surjective and so \(\overline{A}^\flat \to \overline{A}/p\) is surjective. Consequently, \(\theta_A\colon W(\overline{A}^\flat) \to \overline{A}\) is also surjective. We can now reconstruct the diagram of Figure 7.3.4 to recover \(A = W(\overline{A}^\flat)\) and \(I = \ker(A \to \overline{A})\text{.}\)

We will study the essential image of this functor in more detail in Section 8.

Exercises7.4Exercises

1.

Show that the category of perfect \(\FF_p\)-algebras is closed under arbitrary limits and colimits in \(\Ring\text{.}\)

2.

Let \(R\) be a \(p\)-adically complete ring and set \(R^\flat = \lim_\phi R/p\text{.}\) Prove that the natural map

\begin{equation*}
\lim_{x \mapsto x^p} R \to \lim_{\phi} R/p
\end{equation*}

is a multiplicative bijection. This gives the set on the left a ring structure; can you describe the addition law explicitly?

3.

Let \(R\) be a perfect \(\FF_p\)-algebra. Choose \(f \in R\) and define the ideal \(I = \sqrt{(f)}\) of \(R\text{.}\) Prove that \(R/I \in \Mod_R\) has Tor-dimension at most 1.