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Notes on prismatic cohomology

Kiran S. Kedlaya

Section 19 Coperfections in mixed characteristic

Reference.

[18], lecture VIII; [25], section 7.
Given a perfect prism \((A, I)\text{,}\) we apply prismatic cohomology to construct a “canonical coperfection” of a \(p\)-complete \(A/I\)-algebra. In general this will not be a true ring but rather something derived; only in certain (important) special cases will we end up with a genuine ring. Nonetheless, this construction is quite useful in mixed-characteristic commutative algebra; for instance, it recovers the André flatness lemma, whose earliest proofs [5], [16] depended heavily on the theory of perfectoid spaces. We incur no such dependence here; we make the argument entirely in the world of rings and schemes, with no recourse to nonarchimedean analytic geometry.
From now on, we write \(\Prism_{R/A}\) and \(\overline{\Prism}_{R/A}\) to mean derived prismatic and Hodge-Tate cohomology (which were previously denoted \(L\Prism_{R/A}\) and \(L\overline{\Prism}_{R/A}\)), as we will have no further use for the underived versions.

Subsection 19.1 Coperfections in characteristic \(p\) revisited

To motivate the mixed-characteristic construction, we start by reconstructing the coperfection of an \(\FF_p\)-algebra in a somewhat exotic-looking fashion.

Definition 19.1.1.

Recall that for \(R\) an \(\FF_p\)-algebra, we have defined the coperfection of \(R\) as the image \(R_{\perf}\) of \(R\) under the left adjoint of the forgetful functor from perfect \(\FF_p\)-algebras to arbitrary \(\FF_p\)-algebras. Concretely,
\begin{equation*} R_{\perf} = \colim(R \stackrel{\phi}{\to} R \stackrel{\phi}{\to} R \to \cdots). \end{equation*}
Now suppose that \(R\) is an algebra over a perfect field \(k\) of characteristic \(p\text{.}\) Let \(R^{(1)} \to R\) be the relative Frobenius map (Definition 14.1.2); then the induced map \((R^{(1)})_{\perf} \to R_{\perf}\) is an isomorphism. See also Exercise 19.6.2.

Remark 19.1.2.

A fundamental theorem of Kunz (see [117], tag 0EC0) asserts that a noetherian \(\FF_p\)-algebra is regular if and only if its Frobenius map is flat. (As a reminder, if \(R\) is a finite type \(k\)-algebra for some perfect field \(k\) of characteristic \(p\text{,}\) then \(R\) is regular if and only if it is a smooth \(k\)-algebra; see [117], tag 00TQ.) Since flatness is preserved by colimits, we see that if \(R\) is a noetherian regular \(\FF_p\)-algebra, then \(R \to R_{\perf}\) is flat; the converse is also true (see Exercise 19.6.1). For an analogue in mixed characteristic, see Remark 25.5.3.
The following can be seen as another instance of the same phenomenon that gives rise to the vanishing of the cotangent complex for a morphism of perfect rings in characteristic \(p\) (as in the proof of Lemma 3.3.5).
Since everything is defined using the formalism of nonabelian derived functors (Definition 17.2.1), it suffices to treat the case where \(R\) is a polynomial ring over \(k\) in finitely many variables. In this case, it will suffice to check that for each \(i \gt 0\text{,}\)
\begin{equation*} \colim(\Omega^i_{R/k} \stackrel{\phi_R}{\to} \Omega^i_{R/k} \stackrel{\phi_R}{\to} \cdots) \end{equation*}
vanishes. This follows from the fact that Frobenius kills differential forms: in degree 1 we have
\begin{equation*} \phi_R(x\,dy) = x^p d(y^p) = px^p y^{p-1} dy = 0 \end{equation*}
and similarly in higher degrees.
To get closer to the mixed-characteristic case, let us reformulate in terms of (derived) prismatic cohomology.
Again, we formally reduce to the case where \(R\) is a polynomial ring in finitely many variables. To deduce this from Proposition 19.1.3, we need to check that the map
\begin{equation*} \gr_i^{\HT}(\phi_R)\colon \gr_i^{\HT}(\overline{\Prism}_{R/A}) \to \gr_i^{\HT}(\overline{\Prism}_{R/A}) \end{equation*}
induced by the Frobenius on \(\overline{\Prism}_{R/A}\) conicides with the map \(\Omega^i_{R/k} \to \Omega^i_{R/k}\) induced by the Frobenius on \(R\) via the identification \(\gr_i^{\HT}(\overline{\Prism}_{R/A}) \cong \Omega^i_{R/k}\) of Proposition 18.1.4 (note that now we have reverted from derived to ordinary prismatic cohomology). By functoriality, it suffices to treat the case \(R = k[x]\text{;}\) this amounts to a direct calculation in the style of Lemma 12.3.4, which we leave to the reader. (Compare [18], Lecture XIII, Proposition 1.6.)
We make one more change to prepare for the passage to mixed characteristic: we replace the Frobenius action on Hodge-Tate cohomology, which has no analogue in mixed characteristic, with the prismatic Frobenius.

Subsection 19.2 The mixed characteristic case

Definition 19.2.1.

Let \((A,I)\) be a perfect prism with slice \(\overline{A}\text{.}\) For \(R \in \Ring_{\overline{A}}\) derived \(p\)-complete, define the prismatic coperfection
\begin{equation*} \Prism_{R/A,\perf} = \colim(\Prism_{R/A} \stackrel{\phi_R}{\to} \Prism_{R/A} \stackrel{\phi_R}{\to} \cdots)^{\wedge}_{(p,I)} \in D_{\comp}(A) \end{equation*}
using the \(A\)-linear structure on the initial term. This corresponds to the perfection in [18], [25].
Define the lens coperfection as
\begin{equation*} R_{\lens} = \Prism_{R/A,\perf} \otimes_A^L \overline{A} \in D_{\comp}(R) \end{equation*}
(the derived completion being \(p\)-adic) using the \(R\)-linear structure coming from \(R \to \overline{\Prism}_{R/A} \to \Prism_{R/A,\perf} \otimes_A^L \overline{A}\) (the latter map coming from the identification of \(\Prism_{R/A}\) with the first term of the colimit defining \(\Prism_{R/A,\perf}\)). This corresponds to the perfectoidization in [18], [25].
By construction, \(\Prism_{R/A,\perf}\) and \(R_{\lens}\) are commutative algebra objects in \(D_{\comp}(A)\) and \(D_{\comp}(R)\text{,}\) respectively. The Frobenius on \(\Prism_{R/A}\) induces an automorphism of \(\Prism_{R/A,\perf}\) denoted \(\phi_R\text{.}\)

Remark 19.2.2.

The notation \(R_{\lens}\) suggests that the lens coperfection of \(R\) depends only on \(R\) and not on its description as an \(\overline{A}\)-algebra. This will be confirmed by Lemma 19.2.3.
By derived Nakayama (Remark 6.6.6), the first statement implies the third. To check the first and second statements, we may reduce to comparing graded pieces of the Hodge-Tate filtration. Using Proposition 18.1.4 to translate the statement in terms of cotangent complexes plus (17.1), we reduce to checking that the derived \(p\)-completion of \(L_{\overline{B}/\overline{A}}\) vanishes. This holds because both rings are lenses; see Exercise 17.5.4.
Let us consider some examples.

Example 19.2.4. Coperfection for a crystalline prism.

Suppose that \((A,I)\) is crystalline, that is, \(I = (p)\) and \(A = W(\overline{A})\text{.}\) By Corollary 19.1.5, we have
\begin{equation*} R_{\lens} \cong R_{\perf}, \qquad \Prism_{R/A,\perf} \cong W(R_{\perf}) \end{equation*}
with everything concentrated in degree \(0\text{.}\)

Example 19.2.5. Coperfection for a lens.

Let \((A,I)\) be a perfect prism and suppose that \(R \in \Ring_{\overline{A}}\) is itself a lens. By Lemma 19.2.6, \(\Prism_{R/A} \cong W(R^\flat)\) concentrated in degree \(0\text{.}\) Since Frobenius is already an automorphism on \(W(R^\flat)\text{,}\) it follows that \(\Prism_{R/A,\perf} \cong W(R^\flat)\) and \(R_{\lens} \cong R\text{,}\) both concentrated in degree \(0\text{.}\)
Write \(R = \overline{B}\) for some perfect prism \((B,J)\text{;}\) by Theorem 7.3.5, the map \(\overline{A} \to \overline{B}\) promotes uniquely to a morphism of prisms \((A,I) \to (B,J)\text{.}\) Now \((R \to B/J \leftarrow B)\) is an object of \((R/A)_{\Prism}\text{,}\) so we have a natural map \(\Prism_{R/A} \to B = W(R^\flat)\text{.}\) To check that this is an isomorphism, by derived Nakayama (Remark 6.6.6) it suffices to do this after applying \(\bullet \otimes_A^L \overline{A}\text{;}\) that is, we must check that \(\overline{\Prism}_{R/A} \cong R\text{.}\) This follows from Lemma 19.2.3.
We are now ready to consider a simple example where the lens coperfection is not concentrated in degree \(0\text{,}\) although the verification of this will come later (see Section 26). This should not necessarily be viewed as a bad thing, as the higher cohomology will carry some important geometric information.

Example 19.2.7. The \(q\)-torus.

Let \((A,I)\) be the coperfection of \((\ZZ_p\llbracket q-1 \rrbracket, ([p]_q))\text{,}\) so that \(A\) is the classical \((p, [p]_q)\)-completion of \(\ZZ_p[q^{p^{-\infty}}]\text{.}\) Take \(R = \overline{A}[x^{\pm}]^\wedge_{(p)}\text{.}\)
We will see later (see Section 26) that in this example \(H^1(\Prism_{R/A,\perf})\) and \(H^1(R_{\lens})\) are both nonzero. This will follow by our later computation of \(\Prism_{R/A}\) using a \(q\)-de Rham complex (compare Example 12.4.3). We will eventually see that \(\Prism_{R/A,\perf}\) is given by the \((p, [p]_q)\)-completion of
\begin{equation*} A[x^{\pm p^{p^{-\infty}}}] \stackrel{\gamma-\id}{\to} J A[x^{\pm p^{p^{-\infty}}}] \end{equation*}
where
\begin{equation*} J = \left(\bigcup_n (q^{p^{-n}}-1) \right)^\wedge_{(p,[p]_q)} = \ker(A \to \ZZ_p, q^{p^{-n}} \mapsto 1) \end{equation*}
and \(\gamma\) is characterized by
\begin{equation*} \gamma(x^i) = q^i x^i \qquad (i \in \ZZ[p^{-1}]). \end{equation*}
In particular, \((q-1) \cdot 1\) in degree \(1\) is not a coboundary even modulo \([p]_q\text{.}\)

Subsection 19.3 More properties of coperfection

We treat here only the case where \(R\) has bounded \(p\)-power torsion. See [25] for a broader result that includes the general case of this assertion.
Let \((A,I) \to (B,IB)\) be a faithfully flat map of perfect prisms. Put \(S = R \widehat{\otimes}^L_{\overline{A}} \overline{B}\text{;}\) then \(S\) is \(p\)-completely flat over \(R\) and thus concentrated in degree 0 (because \(R\) has bounded \(p\)-power torsion). We need to show that \(R_{\lens} \widehat{\otimes}^L_{\overline{A}} \overline{B} \cong S_{\lens}\text{;}\) by compatibility with filtered colimits, this reduces to showing that \(\overline{\Prism}_{R/A} \otimes^L_{\overline{A}} \overline{B} \cong \overline{\Prism}_{S/B}\text{.}\) This follows by comparing the Hodge-Tate filtrations on both sides using Proposition 18.1.4, then using the analogous compatibility for the cotangent complex and its exterior powers (Proposition 17.1.2).
The following can be viewed as a refinement of Exercise 17.5.1.
For a given \(i \gt 0\text{,}\) set \(S_i = \Sym_{\FF_p} \FF_p[i]\text{.}\) By construction, \(H^{-i}(S_i)\) is nonzero; moreover, any class of \(H^i(R_\bullet)\) is in the image of \(H^{-i}(S_i)\) along some map \(S_i \to R_\bullet\text{.}\) Hence it suffices to check that Frobenius kills \(H^{-i}(S_i)\text{.}\)
For \(i=1\text{,}\) we may write
\begin{equation*} S_1 = \FF_p \otimes^L_{\FF_p[x]} \FF_p, \end{equation*}
from which we read off that \(H^{-1}(S_1) \cong (x)/(x^2)\text{,}\) which is evidently killed by Frobenius.
For \(i \gt 1\text{,}\) we may write
\begin{equation*} S_{i+1} = \FF_p \otimes_{S_i}^L \FF_p \end{equation*}
to obtain an identification
\begin{equation*} H^{-i-1}(S_{i+1}) \cong H^{-i-1}(\FF_p \otimes_{S_i}^L \FF_p) \cong H^{-i}(S_i) \end{equation*}
that is compatible with Frobenius. By induction on \(i\text{,}\) we deduce the desired result. (Compare [24], Proposition 11.6.)

Remark 19.3.3.

It is noted in [24], Remark 11.8 that Lemma 19.3.2 admits a generalization which makes no reference to Frobenius or characteristic \(p\text{:}\) for any simplicial commutative ring \(R_\bullet\text{,}\) the multiplication map \(R_\bullet \times R_\bullet \to R_\bullet\) induces the zero map on \(H^{-i}(R_\bullet)\) for all \(i \gt 0\text{.}\)
From its construction, \(\Prism_{R/A,\perf}/p\) carries a natural Frobenius endomorphism; by Lemma 19.3.2, its negative cohomology groups must vanish. By applying derived Nakayama (Exercise 6.7.5) to the canonical truncation \(\tau^{\leq -1}(\Prism_{R/A,\perf})\text{,}\) we deduce the claim. (Compare [18], Lecture VIII, Remark 2.5(1) or [25], Lemma 8.4.)
Let \((A,I)\) be a perfect prism and let \(R = \overline{A}/J\) be a derived \(p\)-complete quotent. Since \(\overline{A} \to R\) is surjective, \(\Omega^1_{R/\overline{A}} = 0\) and so \(L_{R/\overline{A}}[-1] \in D^{\leq 0}_{\comp}(R)\text{.}\) This in turn implies that \(\wedge^i L_{R/\overline{A}}[-i] \in D^{\leq 0}_{\comp}(R)\) for all \(i\text{,}\) and similarly after derived \(p\)-completion. By the Hodge-Tate filtration (Proposition 18.1.4), we deduce that \(\overline{\Prism}_{R/A} \in D^{\leq 0}_{\comp}(R)\) and hence \(\Prism_{R/A} \in D^{\leq 0}_{\comp}(A)\text{.}\) Now apply Lemma 19.3.5 to deduce that \(R_{\lens}\) is concentrated in degree 0, where it is a lens.

Remark 19.3.7.

As indicated in [18], Lecture VIII, Remark 2.5, Lemma 19.3.4 and Lemma 19.3.5 are concrete consequences of the statement that the action of \(\phi_R\) gives \(\Prism_{R/A,\perf}\) the structure of a “derived perfect \(\delta\)-ring”. We will not try to unpack this statement further here.

Subsection 19.4 André flatness

We next use prismatic coperfections to construct faithfully flat morphisms of prisms; this recovers an important assertion of mixed-characteristic commutative algebra.

Definition 19.4.1.

A ring \(R\) is absolutely integrally closed if every monic polynomial over \(R\) has a root. We often abbreviate this to AIC.
By [117], tag 0DCS, the localization of \(R\) at any prime ideal is strictly henselian; this implies the claim at once.
Define the ring
\begin{equation*} R = \overline{A}[x^{p^{-\infty}}]^{\wedge}_{(p)}/(P); \end{equation*}
by construction, \(R\) is a regular semilens and \(\overline{A} \to R\) is \(p\)-completely faithfully flat. By Remark 18.2.3, \(\Prism_{R/A}\) is concentrated in degree \(0\text{,}\) where it is a \((p,I)\)-completely flat \(A\)-algebra. By the Hodge-Tate comparison (Proposition 18.1.4), \(R \to \overline{\Prism}_{R/A}\) is \(p\)-completely faithfully flat.
By Corollary 19.3.6, \(\Prism_{R/A,\perf}\) is concentrated in degree \(0\text{,}\) where it is a perfect \((p,I)\)-complete \(\delta\)-ring which we call \(B\text{.}\) By the previous paragraph, \(R \to \overline{\Prism}_{R/A,\perf}\) is \(p\)-completely faithfully flat, so \((A,I) \to (B,IB)\) is faithfully flat. By construction, \(\overline{B}\) is an \(R\)-algebra, so it contains a root of \(P\text{.}\) (Compare [25], Proposition 7.11.)
This follows directly from Lemma 19.4.3 via transfinite induction. (Compare [25], Theorem 7.12.)
By Corollary 19.3.6, \(R\to R_{\lens}\) is the universal map from \(R\) to a lens, which we wish to show is surjective. Since we may check this after a \(p\)-completely faithfully flat base extension, using Theorem 19.4.4 we may reduce to the case where the multiplicative map \(\sharp\colon \overline{A}^\flat \to \overline{A}\) is surjective. Let \(J\) be the kernel of \(\overline{A} \to R\text{.}\) We can then choose elements \(x_i \in \overline{A}^\flat\) for \(i\) running over some index set \(I\) such that the elements \(x_i^\sharp \in \overline{A}\) form a set of generators of \(J\text{,}\) and check directly that the quotient \(R'\) of \(\overline{A}\) by the \(p\)-completion of the ideal generated by \(x_i^{p^{-j}}\) for all \(i \in I, j \geq 0\) is a lens. The natural map \(R \to R'\) satisfies the same universal property as \(R \to R_{\lens}\text{,}\) so \(R_{\lens} \cong R'\) is indeed a quotient of \(R\text{.}\) (Compare [18], Corollary 3.2.)

Remark 19.4.7.

In the theory of perfectoid spaces, the surjectivity assertion in Corollary 19.4.6 corresponds to the fact there is no difference between Zariski closed subsets and strongly Zariski closed subsets of a perfectoid space. These concepts had previously been distinguished in [108], Remark II.2.4.

Subsection 19.5 Examples of lens coperfection

Remark 19.5.1.

In each of the following examples, we exhibit the lens coperfection of a semilens \(S\) not from its definition (Definition 19.2.1), but from the adjunction property (Corollary 19.3.6).
We have the following analogue of Example 3.4.3.

Example 19.5.2.

Let \(R\) be a lens and let \(S\) be the regular semilens \(R[x^{p^{-\infty}}]^\wedge_{(p)}/(x)\text{.}\) Then \(S_{\lens} \cong R\) with the kernel of \(S \to S_{\lens}\) being the closure of the ideal \((x^{p^{-\infty}})\text{.}\) We check this from the adjunction property (Corollary 19.3.6): if \(S \to T\) is a morphism with \(T\) a lens, then \(T\) is reduced (Corollary 8.4.7) and so \(x^{p^{-n}} \in \ker(S \to T)\) for all \(n\text{.}\)
In this case, it is easy to see that the kernel of \(S \to S_{\lens}\) is strictly larger than the radical of the ideal \(x\text{.}\) For example, if \(R\) is \(p\)-torsion-free, then the element
\begin{equation*} \sum_{n=1}^\infty p^n x^{p^{-n}} \end{equation*}
belongs to the kernel but no power of it is divisible by \(x\text{.}\)

Example 19.5.3.

Let \(R\) be a completed algebraic integral closure of \(\ZZ_p\) and let \(S\) be the regular semilens \(R[x^{p^{-\infty}}]^{\wedge}_{(p)}/(x-1)\text{.}\) Fix a coherent sequence \((\zeta_{p^n})\) of \(p\)-power roots of unity in \(R\text{.}\) Then \(S_{\lens}\) can be described as the ring of continuous functions \(\ZZ_p \to R\text{,}\) viewed as a subring of the product \(\prod_{c \in \ZZ_p} R\text{,}\) via the map taking \(x^{p^{-n}}\) to \((\zeta_{p^n}^c)_{c \in \ZZ_p}\text{.}\) As in Example 19.5.2, this can be checked using the adjunction property of lens coperfection (Corollary 19.3.6).
The kernel of the map \(S \to S_{\lens}\) has been analyzed in [53]: it is the radical of the ideal \((x-1)\text{,}\) but is strictly larger than \((x-1)\) itself. However, it is difficult to exhibit “explicit” elements witnessing the difference between the two ideals.
Here is a variation of the previous example.

Example 19.5.4.

Let \(R\) be a \(p\)-torsion-free lens and let \(S\) be the nonregular semilens \(R[x^{p^{-\infty}}, y^{p^{-\infty}}]^{\wedge}_{(p)}/(x^{p^{-n}} - y^{p^{-n}}\colon n =0,1,\dots)\text{.}\) In this case, \(S_{\lens} \cong R[x^{p^{-\infty}}]^{\wedge}_{(p)}\) via the map \(y^{p^{-n}} \mapsto x^{p^{-n}}\text{.}\) By contrast, if we take the quotient of \(R[x^{p^{-\infty}}, y^{p^{-\infty}}]^{\wedge}_{(p)}\) by the ideal \((x-y)\text{,}\) we end up with something more similar to Example 19.5.3 (particularly if \(R\) contains a coherent \(p\)-power sequence of roots of unity).
Note that in the previous examples, the complications all arise from the kernel of the map to the lens coperfection. If we exclude this by requiring the semilens to be \(p\)-torsion-free, then one can express the lens coperfection in more classical language. (One can also make some statements in the more general case, for which we defer to [70] for details.)

Definition 19.5.5.

For \(R\) a \(p\)-torsion-free ring, the \(p\)-root closure (or \(p\)-normalization) of \(R\) is the minimal subring \(S\) of \(R[p^{-1}]\) containing \(R\) and closed under taking \(p\)-th roots. That is, if \(x \in R[p^{-1}]\) and \(x^p \in S\text{,}\) then also \(x \in S\text{.}\)
It is clear that every \(x\) of this form belongs to the \(p\)-root closure. It thus suffices to check that the resulting set is a ring, as then it is clear that it contains \(R\) and is closed under taking \(p\)-th roots. We leave the verification to the reader; alternatively, see [105] where the concept of the \(p\)-root closure was first considered in detail.
See [70], Main Theorem C.

Exercises 19.6 Exercises

1.

Show that for \(R \in \Ring_{\FF_p}\text{,}\) the Frobenius map \(\phi_R\colon R \to R\) is flat if and only if the canonical map from \(R\) to its coperfection \(R_{\perf}\) is flat.
Hint.
Let \(R_1\) be a copy of \(R\) viewed as an \(R\)-algebra via \(\phi\text{.}\) The map \(R_1 \to R_{\perf}\) induces a surjection on spectra; hence if \(I\) is a finitely generated ideal of \(R\) and
\begin{equation*} 0 \to K \to I \otimes_R R_1 \to IR_{1} \to R_{1}/IR_1 \to 0 \end{equation*}
is exact with \(K \neq 0\text{,}\) then \(I \otimes R_{\perf} \to I R_{\perf}\) is not injective either. This checks a standard criterion for flatness ([117], tag 00M5).

2.

Let \(R \to S\) be a morphism of \(\FF_p\)-algebras such that the corresponding map \(\Spec S \to \Spec R\) is a universal homeomorphism. Show that the induced map \(R_{\perf} \to S_{\perf}\) of coperfections is an isomorphism.
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