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Notes on prismatic cohomology

Kiran S. Kedlaya

Section 2 \(\delta\)-rings

References.

We have followed [18], Lecture II, fairly closely. Some of the exercises were taken from [30], others from [25], section 2.
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In this section, we introduce the fundamental notion of a \(\delta\)-ring. This definition was introduced by Joyal [74], [75] with a view towards applications in \(K\)-theory; it is closely related to the older notion of a \(\lambda\)-ring, which we will discuss briefly in the next section (Subsection 4.1). However, this development did not gain much attention until the same idea was rediscovered by Buium [34] under the guise of arithmetic differentiation. (Buium had the original goal of adapting Manin’s proof of the finiteness of the set of torsion points on a hyperbolic algebraic curve embedded into its Jacobian from the function field case to the number field case; see [35] for a slightly later treatment presenting arithmetic differentiation on its own terms.)
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Definition 2.0.1.

For the remainder of the course (except as specified), fix a prime number \(p\text{.}\) Define the following standard categories:
  • \(\Set\text{:}\) sets.
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  • \(\Ab\text{:}\) abelian groups.
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  • \(\Ring\text{:}\) commutative unital rings.
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  • \(\Mod_A\text{:}\) modules over \(A\) (where \(A \in \Ring\)).
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Let \(\Rad(A)\) denote the Jacobson radical of \(A \in \Ring\text{.}\) For \(I\) an ideal of \(A\text{,}\) we say that \(A\) is \(I\)-local if \(I \subseteq \Rad(A)\text{;}\) if \(I = (f)\text{,}\) we also say that \(A\) is \(f\)-local.
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Subsection 2.1 \(p\)-derivations and Frobenius lifts

We begin by formalizing the fundamental idea of “differentiation with respect to a prime number”.
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Definition 2.1.1.

Following Joyal, we define a \(\delta\)-ring to be a pair \((A, \delta)\) in which \(A \in \Ring\) and \(\delta\colon A \to A\) is a map of sets satisfying the following conditions for all \(x,y \in A\text{.}\)
\begin{gather} \delta(1) = 0;\tag{2.1}\\ \delta(xy) = x^p \delta(y) + y^p \delta(x) + p \delta(x) \delta(y);\tag{2.2}\\ \delta(x+y) = \delta(x) + \delta(y) - \sum_{i=1}^{p-1} \frac{(p-1)!}{i!(p-i)!} x^i y^{p-i}\text{.}\tag{2.3} \end{gather}
The last condition implies that
\begin{equation*} p (\delta(x+y) - \delta(x) - \delta(y)) = x^p + y^p - (x+y)^p \end{equation*}
and conversely if \(A\) is \(p\)-torsion-free. (In some sources, a map \(\delta\) satisfying these conditions is called a \(p\)-derivation.)
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We will habitually abuse notation and terminology and say that “\(A\) is a \(\delta\)-ring” when it is meant to be clear from context what the map \(\delta\) is supposed to be. We will also apply adjectives to a \(\delta\)-ring (e.g., “\(p\)-torsion-free”) when they are meant to apply to the underlying ring.
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Remark 2.1.2.

Note that (2.3) also implies that \(\delta(0) = 0\text{,}\) so we don’t need to include this condition separately. By contrast, (2.2) does not by itself imply that \(\delta(1) = 0\text{;}\) see Exercise 2.5.1.
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The form of the previous definition is partly explained by the following construction.
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Proof.

It was already pointed out in Definition 2.1.1 that \(\delta\) satisfies (2.3) if and only if \(\phi\) is additive, and conversely if \(A\) is \(p\)-torsion-free. Meanwhile, (2.1) implies that \(\phi(1) = 1\text{,}\) and conversely if \(A\) is \(p\)-torsion-free; while (2.2) implies that \(\phi(xy) = \phi(x) \phi(y)\text{,}\) and conversely if \(A\) is \(p\)-torsion-free.
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Remark 2.1.4.

It is possible for a \(p\)-torsion-free ring to admit no \(\delta\)-ring structures, and hence no Frobenius lifts. A simple example (taken from [25], Lemma 2.35) is \(A = \ZZ_{(p)}[x, x^p/p]\text{:}\) if \(\delta\colon A \to A\) were a \(p\)-derivation with associated Frobenius lift \(\phi\text{,}\) we would have
\begin{align*} \frac{1}{p} (x^p/p)^p &= \frac{1}{p} (\phi(x^p/p) - p \delta(x^p/p))\\ &= \frac{\phi(x)^p}{p^2} - \delta(x^p/p)\\ &= \frac{(x^p + p \delta(x))^p}{p^2} - \delta(x^p/p)\\ &= p^{p-2} (x^p/p + \delta(x))^p - \delta(x^p/p) \in A\text{,} \end{align*}
a contradiction. See Exercise 2.5.10 for a consequence of this calculation.
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Definition 2.1.5.

By (2.1) and (2.2), in any \(\delta\)-ring \(A\) the elements \(x \in A\) for which \(\delta(x) = 0\) form a monoid under multiplication. By analogy with the case of an ordinary derivation, we call these the \(\delta\)-constant elements of \(A\text{.}\) These elements also satisfy \(\phi(x) = x^p\text{,}\) and conversely if \(A\) is \(p\)-torsion free. (In [25] these elements are said to be of rank \(1\); this terminology will make more sense in the context of big Witt wectors, as in Remark 4.2.1.)
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Remark 2.1.6.

One might ask to what extent the notion of a \(p\)-derivation is a “natural” modification of the definition of a usual derivation. One answer to this question can be found in [35]: one can define the notion of a jet operator \(\delta\) on a local domain \(A\) of characteristic 0 and show that any such map is either an ordinary derivation, a \(\pi\)-difference operator for some \(\pi \in A\) (i.e., \(x \mapsto x + \pi \delta(x)\) is a ring homomorphism), or a \(\pi\)-derivation for some \(\pi \in A\) (i.e., \(x \mapsto x^q + \pi \delta(x)\) is a ring homomorphism for some prime power \(q\)). This can be thought of as a loose analogue of Ostrowski’s classification of valuations.
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Subsection 2.2 Examples of \(\delta\)-rings

Using Lemma 2.1.3, it is not difficult to generate examples of \(\delta\)-rings. Here are a few illustrative cases.
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Example 2.2.1.

If \(p\) is invertible in the ring \(A\text{,}\) then every endomorphism of \(A\) is a Frobenius lift, and thus gives rise to a \(p\)-derivation.
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Example 2.2.2.

By Lemma 2.1.3, there is a unique way to equip \(\ZZ\) with the structure of a \(\delta\)-ring, namely via the map
\begin{equation} \delta(x) = \frac{x-x^p}{p}\text{.}\tag{2.4} \end{equation}
The \(\delta\)-constant elements are \(\{0,1\}\) if \(p = 2\) and \(\{0,1,-1\}\) if \(p \gt 2\text{.}\)
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By the same token, \(\ZZ_p\) has trivial automorphism group (even if we ignore its topology!) and so admits a unique \(\delta\)-ring structure. The \(\delta\)-constant elements are \(\{0\} \cup \mu_{p-1}\text{.}\) This example is the first case of the general construction of rings of Witt vectors; see Subsection 3.1.
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Remark 2.2.3.

Using (2.3), we can start from the equality \(\delta(0) = \delta(1) = 0\) and reconstruct the values of \(\delta\) on arbitrary integers. Consequently, for any \(\delta\)-ring \(A\text{,}\) the action of \(\delta\) on integers is given by (2.4) even if \(A\) is not \(p\)-torsion-free. In particular, for any positive integer \(n\text{,}\)
\begin{equation*} \delta(p^n) = p^{n-1} (1 - p^{n(p-1)}) \end{equation*}
and the second factor is not divisible by \(p\) unless \(p\) is invertible in \(A\) (see Example 2.2.1); that is, \(\delta\) “lowers the \(p\)-adic order of vanishing by 1”. By the same token, for any \(x \in A\text{,}\)
\begin{equation*} \delta(p^n x) = p^{np} \delta(x) + x^p \delta(p^n) + p \delta(x) \delta(p^n) \equiv p^{n-1} x^p \pmod{p^n}\text{.} \end{equation*}
See Exercise 2.5.5 for a related observation.
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Example 2.2.4.

Building on Example 2.2.2, take \(A = \ZZ[\mu_n\colon \gcd(n,p) = 1]\text{.}\) The automorphism \(\phi\colon A \to A\) taking \(\zeta_n\) to \(\zeta_n^p\) for every positive integer \(n\) coprime to \(p\) is a Frobenius lift; for \(\delta\) the corresponding \(p\)-derivation, the \(\delta\)-constant elements are \(\{0\} \cup \bigcup_{\gcd(n,p)=1} \mu_n\text{.}\)
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Example 2.2.5.

Take \(A = \ZZ[x]\text{.}\) For any \(y \in A\text{,}\) there is a unique Frobenius lift \(\phi\) of \(A\) for which \(\phi(x) = x^p + py\text{;}\) consequently, there is a unique \(p\)-derivation \(\delta\) on \(A\) with \(\delta(x) = y\text{.}\) It is tempting to interpret this as the statement that “the set of \(p\)-derivations on \(A\) is a free \(A\)-module of rank \(1\)”, but in fact there is no natural module structure on the set of \(p\)-derivations on a general ring.
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You may have noticed that none of these examples has \(p\)-torsion. That is not entirely an accident.
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Proof.

We prove the claim by induction on \(n\text{,}\) with the base case \(n=0\) being vacuously true.
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Suppose that \(n>0\text{.}\) Then \(A\) is a \(\ZZ_{(p)}\)-algebra. Consequently, by (2.4) and Remark 2.2.3,
\begin{equation*} 0 = \delta(0) = \delta(p^n) = p^{n-1} (1 - p^{np-n}) \end{equation*}
and the second factor is a unit in \(A\text{.}\) Hence \(p^{n-1} = 0\) in \(A\) also, and the induction hypothesis applies.
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Remark 2.2.7.

Notwithstanding Lemma 2.2.6, there do exist examples of \(\delta\)-rings in which the underlying ring is not \(p\)-torsion-free; it is difficult to write these down concretely (in part because it is not enough to specify the associated Frobenius lift), but they will be generated naturally by Definition 3.1.1. See for instance Exercise 3.6.1.
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Fortunately, quite often we can ignore these examples when checking basic properties of \(\delta\)-rings by appealing to the existence of free \(\delta\)-rings; see again Definition 2.4.5. See also Exercise 2.5.4 for a variant of Lemma 2.2.6 that applies to this situation.
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Proof.

Since \(x\) maps to zero in \(A[p^{-1}]\text{,}\) so then does \(\phi(x)\text{.}\) It is thus sufficient to check the claim after localizing at \((p)\text{,}\) that is, we may assume that \(A\) is a \(\ZZ_{(p)}\)-algebra. Now apply (2.2) to write
\begin{equation*} 0 = \delta(0) = \delta(px) = x^p \delta(p) + p^p \delta(x) + p \delta(x) \delta(p) = p^p \delta(x) + \phi(x) \delta(p)\text{.} \end{equation*}
By Remark 2.2.3, \(\delta(p)\) is a unit in \(A\text{;}\) it will thus suffice to check that \(p^p \delta(x) = 0\text{.}\) This follows by writing
\begin{equation*} p^p \delta(x) = p^{p-1} (\phi(x) - x^p) = p^{p-2} (\phi(px) - px^p) = 0\text{.} \end{equation*}
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Subsection 2.3 Truncated Witt vectors

Just as derivations can be naturally interpreted as giving first-order deformations of a ring, one can interpret \(p\)-derivations in the following manner.
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Definition 2.3.1.

For \(A\) a ring, let \(W_2(A)\) be the set \(A \times A\) equipped with binary operations \(+, \times\) defined as follows:
\begin{align*} (x_0, x_1) + (y_0, y_1) &= (x_0+y_0, x_1+y_1 - \sum_{i=1}^{p-1} \frac{(p-1)!}{i!(p-i)!} x_0^i y_0^{p-i})\\ (x_0, x_1) \times (y_0, y_1) &= (x_0 y_0, x_0^p y_1 + y_0^p x_1 + p x_1 y_1)\text{.} \end{align*}
Note the relationship between these formulas to the definition of a \(\delta\)-ring (Definition 2.1.1); see Remark 2.3.3.
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Proof.

One may see directly from the definitions that:
  • addition is commutative and the element \(0 = (0, 0)\) is an identity element;
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  • every element has an additive inverse;
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  • multiplication is commutative and the element \(1 = (1, 0)\) is an identity element;
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  • the operation \(A \mapsto W_2(A)\) defines a functor from the category of commutative rings to the category of sets equipped with two binary operations.
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We thus need to check that addition and multiplication are associative and that multiplication distributes over addition. These are all conditions asserting the validity of certain polynomial identities in two arbitrary elements \(x,y \in A\text{;}\) thanks to the functoriality, these can be checked after lifting from \(A\) to some ring that surjects onto it. In particular, we may take \(A\) to be a polynomial ring over \(\ZZ\text{,}\) which in particular is \(p\)-torsion-free.
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In this setting, the map
\begin{equation*} W_2(A) \to A \times A, \qquad (x_0, x_1) \mapsto (x_0, x_0^p + px_1) \end{equation*}
is a monomorphism (in the category of sets equipped with two binary operations) for the usual ring operations on \(A \times A\text{.}\) Consequently, we may deduce the desired properties by transferring the knowledge from \(A \times A\text{.}\) (This map is related to the ghost map on Witt vectors; see Subsection 3.2.)
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Remark 2.3.3.

There are two natural (in \(A\)) ring homomorphisms \(\epsilon_1, \epsilon_2\colon W_2(A) \to A\) given by
\begin{equation*} \epsilon_1((x_0, x_1)) = x_0, \qquad \epsilon_2((x_0, x_1)) = x_0^p + px_1\text{.} \end{equation*}
In this notation, a \(\delta\)-ring structure on \(A\) corresponds to a ring homomorphism \(w\colon A \to W_2(A)\) such that \(\epsilon_1 \circ w = \id_A\text{.}\) In this way, the ring \(W_2(A)\) plays a role comparable to that of the ring of dual numbers \(k[\epsilon]/(\epsilon^2)\) over a field \(k\text{.}\)
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On a related note, for \(A \in \Ring_\delta\text{,}\) \(B \in \Ring\text{,}\) and \(f\colon A \to B\) a morphism in \(\Ring\text{,}\) the formula
\begin{equation*} a \mapsto (f(a), f(\delta(a))) \end{equation*}
defines a homomorphism \(A \to W_2(B)\) in \(\Ring\text{:}\) namely, this is the composition of \(w\colon A \to W_2(A)\) with the functorial map \(W_2(A) \to W_2(B)\text{.}\) This map will reappear later via the adjunction property of Witt vectors (Definition 3.1.1).
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Remark 2.3.4.

For \(A\) a \(p\)-torsion-free ring, \(W_2(A)\) can also be described as the fiber product of the reduction map \(A \to A/(p)\) with the composition \(A \to A/(p) \stackrel{\phi}{\to} A/(p)\) where \(\phi\) denotes the Frobenius on \(A/(p)\text{;}\) the two projection maps \(W_2(A) \to A\) are the ones from Remark 2.3.3. That is, \(\Spec W_2(A)\) consists of two copies of \(\Spec A\) glued along \(\Spec A/(p)\) via Frobenius.
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In [18], Lecture II, Remark 3.4, Bhatt also suggests a version of this statement without the \(p\)-torsion-free condition: for a general ring \(A\text{,}\) a \(\delta\)-structure on \(A\) corresponds to an endomorphism \(\phi\colon A \to A\) which is a “derived Frobenius lift”. That is, for \(\overline{A} = A \otimes^L_{\ZZ} \ZZ/(p)\text{,}\) there exists (and is specified!) a homotopy between the composition \(A \stackrel{\phi}{\to} A \to \overline{A}\) and the composition \(A \to \overline{A} \stackrel{\mathrm{Frob}}{\to} \overline{A}\text{.}\) This description follows from the previous discussion by interpreting \(W_2(A)\) as the fiber product of \(A \to \overline{A}\) and \(A \to \overline{A} \stackrel{\mathrm{Frob}}{\to} \overline{A}\text{.}\)
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Subsection 2.4 The category of \(\delta\)-rings

Definition 2.4.1.

A morphism of \(\delta\)-rings \((A, \delta) \to (A', \delta')\) is a homomorphism \(f\colon A \to A'\) of rings such that \(f \circ \delta = \delta' \circ f\) (again as maps of sets only). It is evident that with this definition, \(\delta\)-rings form a category, denoted \(\Ring_\delta\text{.}\)
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Proof.

All we need to check is that if \(x \equiv y \pmod{I}\text{,}\) then \(\delta(x) \equiv \delta(y) \pmod{I}\text{.}\) This is apparent from (2.3), which implies that \(\delta(x) \equiv \delta(y) + \delta(x-y) \pmod{x-y}\text{.}\)
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Proof.

For limits, this is pretty straightforward. For colimits, it is perhaps easiest to use Remark 2.3.3: if \(A\) is the colimit of a diagram \(\{A_i\}\text{,}\) then we get maps \(\colim A_i \to \colim W_2(A_i) \to W_2(\colim A_i)\) whose composition splits the projection map, and then we recover a \(\delta\)-structure on \(\colim A_i\text{.}\)
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Remark 2.4.4.

The analogue of Lemma 2.4.3 fails for the category of rings with a Frobenius lift; see Exercise 2.5.9. This is one reason why to prefer the category of \(\delta\)-rings as a basic object of study.
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Definition 2.4.5.

By Lemma 2.4.3 plus Freyd’s adjoint functor theorem (see [94], section V.8 or [117], tag 0AHM) and a set-theoretic consideration (see Remark 2.4.12), the forgetful functor \(\Ring_\delta \to \Ring\) admits both a left adjoint and a right adjoint. (More precisely, existence of limits gives the left adjoint, existence of colimits gives the right adjoint.) The right adjoint gives rise to Witt vectors; see Section 3.
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The left adjoint can be described concretely in the case of the polynomial ring \(\ZZ[S]\) (for \(S\) an arbitrary set, not necessarily finite); it produces the free \(\delta\)-ring on \(S\) which we denote by \(\ZZ\{S\}\text{.}\) Concretely, the underlying ring of \(\ZZ\{S\}\) is given by \(\ZZ[S_0, S_1, \dots]\) where each \(S_i\) is a copy of \(S\text{.}\) The map \(\delta\) acts on these elements as follows: for \(s \in S\) corresponding to \(s_i \in S_i\text{,}\) we have \(\delta(s_i) = s_{i+1}\text{.}\) (To evaluate the left adjoint on an arbitrary ring, we can write it as a quotient of \(\ZZ[S]\) for some \(S\text{,}\) say by using the adjunction between commutative rings and sets, and then take a quotient of the resulting free \(\delta\)-ring using Lemma 2.4.2.)
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An important corollary of this observation is that every \(\delta\)-ring can be written as a quotient of some \(\)\(p\)-torsion-free \(\delta\)-ring (e.g., if \(R \in \Ring_\delta\) then it is a quotient of \(Z\{R\}\)). This will allow us to reduce many computations to the \(p\)-torsion-free case.
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Remark 2.4.6.

Note that in Definition 2.4.5, applying the left adjoint to a finitely generated polynomial ring over \(\ZZ\) produces a \(\delta\)-ring whose underlying ring is not noetherian. This suggests that we cannot entertain any hope of staying within the noetherian realm as we go along. This is similar to what happens in difference algebra, the study of rings equipped with an endomorphism.
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Proof.

We are claiming that \(\delta \circ \phi = \phi \circ \delta\text{;}\) this is a polynomial identity on a single element \(x \in A\) and its image under \(\delta\text{,}\) so by lifting to a free \(\delta\)-ring using Definition 2.4.5, we may reduce to the case where \(A\) is \(p\)-torsion-free. In this case, the claim reduces to checking that \(\phi\) is an endomorphism of the difference ring \((A, \phi)\text{,}\) which is obviously true.
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Proof.

Suppose first that \(A\) is \(p\)-torsion-free; then so is \(S^{-1} A\text{.}\) Since \(\phi(S) \subseteq S\text{,}\) the map \(\phi\colon A \to A\) extends uniquely to a morphism \(\phi\colon S^{-1} A \to S^{-1} A\) which is again a Frobenius lift. We thus recover a unique \(\delta\)-structure on \(S^{-1} A\text{.}\)
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In the general case, form a surjection \(F \to A\) of \(\delta\)-rings with \(F\) being \(p\)-torsion-free. Let \(T\) be the preimage of \(S\) in \(F\text{;}\) it is again a multiplicative subset such that \(\phi(T) \subseteq T\text{,}\) so we may uniquely promote \(T^{-1} F\) to a \(\delta\)-ring over \(T\text{.}\) Since Figure 2.4.9 is a pushout diagram in \(\Ring\text{,}\) the pushout in \(\Ring_\delta\) (which is a colimit, and hence covered by Lemma 2.4.3) gives us the unique \(\delta\)-ring structure on \(S^{-1} A\text{.}\) (One can also argue more explicitly using Lemma 2.4.2.)
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Figure 2.4.9.
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Remark 2.4.10.

An important special case of Lemma 2.4.8 is localization at a closed subscheme on which \(p\) vanishes, taking \(S\) to be the complement of the radical ideal defining this subscheme. In this case, the hypothesis \(\phi(S) \subseteq S\) is automatically satisfied.
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Remark 2.4.11.

We do not know whether the following natural generalization of Lemma 2.4.8 holds: if \(A\) is a \(\delta\)-ring and \(A \to B\) is an étale morphism of rings, then the ways to promote this to a morphism of \(\delta\)-rings correspond precisely to the ways to extend the action of \(\phi\) on \(A\) to \(B\text{.}\) (This is true if \(A\) is \(p\)-torsion-free, but the reduction to this case is a bit subtle.)
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Remark 2.4.12.

We fill in the missing set-theoretic consideration from Definition 2.4.5. In order to apply the adjoint functor theorem to a functor \(F\colon \calC \to \calC'\) to get the left adjoint, one must know that for every \(y \in \calC'\text{,}\) there is a set of elements \(x_i \in \calC\) such that for any \(x \in \calC\text{,}\) any morphism \(f\colon y \to F(x)\) factors as \(F(g) \circ f_i\) for some \(i\text{,}\) some \(f_i\colon y \to F(x_i)\text{,}\) and some \(g\colon x_i \to x\text{.}\) This is needed to ensure that when we construct the image of an object under the adjoint functor, we are not trying to take a limit indexed by a class which is too large to be a set.
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Similarly, to get the right adjoint, one must know that for every \(y \in \calC'\text{,}\) there is a set of elements \(x_i \in \calC\) such that for any \(x \in \calC\text{,}\) any morphism \(f\colon F(x) \to y\) factors as \(f_i \circ F(g)\) for some \(i\text{,}\) some \(g\colon x \to x_i\text{,}\) and some \(f_i\colon F(x_i) \to F(y)\text{.}\)
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Typically conditions like these are established by taking all of the pairs \((x_i, f_i)\) which satisfy some cardinality bound. For the case of \(\Ring_\delta \to \Ring\text{,}\) see Exercise 2.5.12 and Exercise 2.5.14.
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Exercises 2.5 Exercises

1.

Show that in Definition 2.1.1, condition (2.1) cannot be omitted.
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Hint.
Check that the map \(\delta(x) = -x^p/p\) satisfies the other conditions when it is well-defined. (Note that it corresponds to \(\phi\) being the zero map, which is additive and multiplicative but not a homomorphism of unital rings.)
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2.

Suppose that \(p=2\text{.}\) For \(x,y\) in a \(\delta\)-ring, compute \(\delta(\delta(xy))\) and \(\delta(\delta(x+y))\text{.}\)
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3.

Assume that \(p > 2\text{.}\) Show that in Example 2.2.4, if we replace \(A\) with \(A[\zeta_p]\) and extend the map so that it fixes \(\zeta_p\text{,}\) we get an endomorphism which is not a Frobenius lift. (There is a more permissive definition of Frobenius lifts that would allow this, but it does not integrate neatly with the theory of \(\delta\)-rings.)
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4.

Prove the following refinement of Lemma 2.2.6: in any \(\delta\)-ring, every \(p\)-power-torsion element is nilpotent. In particular, any reduced \(\delta\)-ring is \(p\)-torsion-free.
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Hint.
Adapt the proof of Lemma 2.2.8 to show that if \(p^n x = 0\text{,}\) then \(p^{n-1} \phi(x) = 0\text{.}\) See also [25], Lemma 2.28.
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5.

(Emerton) Prove that for any \(\delta\)-ring \(A\) and any \(x \in A\text{,}\)
\begin{equation*} \phi(x) = p^{p-1} x^p + \delta(px)\text{.} \end{equation*}
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Hint.
Reduce to the \(p\)-torsion-free case, then use the fact that \(\phi\) is a ring homomorphism.
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6.

Let \(A\) be a \(p\)-adically separated \(\delta\)-ring. Show that any element of \(A\) which admits \(p^n\)-th roots in \(A\) for all positive integers \(n\) must be \(\delta\)-constant.
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Hint.
It is enough to show that for each positive integer \(n\text{,}\) for any \(y \in A\) we have \(\delta(y^{p^n}) \in p^n A\text{.}\) This can be checked by reducing to the \(p\)-torsion-free case and computing in terms of \(\phi\text{.}\)
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7.

Let \(A\) be a \(\delta\)-ring and let \(x \in A\) be an element. Prove that there exists a faithfully flat map \(A \to B\) of \(\delta\)-rings such that the image of \(x\) in \(B\) belongs to the image of \(\phi\text{.}\) That is, “\(\phi\) is fpqc-locally surjective.”
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Hint.
See [25], Corollary 2.12.
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8.

Let \(A\) be a \(\delta\)-ring. Let \(I\) be a finitely generated ideal of \(A\) containing \(p\) (we do not assume any compatibility with \(\delta\)). Then the \(I\)-adic completion of \(A\) admits a unique \(\delta\)-structure compatible with \(A\text{.}\)
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Hint.
See [25], Lemma 2.17.
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9.

Let \(\Ring_\phi\) be the category of rings equipped with a Frobenius lift.
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  1. Show that \(\Ring_\phi\) admits arbitrary colimits and products, and that these commute with the forgetful functor to rings.
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  2. Show that \(\Ring_\phi\) also admits equalizers (and hence arbitrary limits), but these do not commute with the forgetful functor to rings. This is a reason why to prefer the category \(\Ring_\delta\) over \(\Ring_\phi\text{.}\)
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10.

Let \(A\) be a \(p\)-torsion-free \(\delta\)-ring over \(\ZZ_{(p)}\text{.}\) Define the divided power operations \(\gamma_n\colon A \to A[p^{-1}]\) by
\begin{equation*} \gamma_n(x) = \frac{x^n}{n!}\text{.} \end{equation*}
Show that if \(x \in A\) satisfies \(\gamma_p(x) \in A\text{,}\) then \(\gamma_n(x) \in A\) for all \(n \geq 0\text{.}\) In particular, this conclusion does not depend on the \(\delta\)-ring structure, so it can be used to exhibit an obstruction to the existence of Frobenius lifts on some rings, as in Remark 2.1.4.
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Hint.
First adapt the calculation from Remark 2.1.4 to show that \(\gamma_{p^2}(x) \in A\text{.}\) Then use this as the basis for an induction on \(n\) (uniformly over all \(A\) and \(x\)), by comparing \(\gamma_{kp}(x)\) to \(\gamma_k(\gamma_p(x))\text{.}\) See also [25], Lemma 2.35.
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11.

Let \(A = \ZZ_{(p)}\{x\}\) and let \(D\) be the divided power envelope of \(A\) with respect to the ideal \((x)\text{,}\) that is, the smallest subring of \(A[p^{-1}]\) with the property that the divided power operations (Exercise 2.5.10) carry every element of the ideal \(xA\) into \(D\text{.}\) (They also carry every element of the ideal \(xD\) into \(D\text{.}\)) Prove that \(D = \ZZ_{(p)}\{x, \frac{\phi(x)}{p} \}\text{.}\)
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Hint.
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12.

Show that for the forgetful functor \(\Ring_\delta \to \Ring\text{,}\) the set-theoretic condition for the left adjoint in Remark 2.4.12 is satisfied by taking all objects \(x_i\) with \(|x_i| \leq \max\{|x|, \aleph_0\}\text{.}\)
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Hint.
Let \(f\colon A \to B\) be a morphism in \(\Ring\) with \(B \in \Ring_\delta\text{.}\) Then the \(\delta\)-subring of \(B\) generated by \(f(A)\) has cardinality at most \(\max\{|A|, \aleph_0\}\text{.}\)
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13.

Let \(A\) be a \(\delta\)-ring. Let \(I\) be an ideal of \(A\text{.}\) Show that the set of \(x \in A\) such that both \(x\) and \(\delta(x)\) belong to \(I\) is also an ideal of \(A\text{.}\)
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14.

Show that for the forgetful functor \(\Ring_\delta \to \Ring\text{,}\) the set-theoretic condition for the right adjoint in Remark 2.4.12 is satisfied by taking all objects \(x_i\) with \(|x_i| \leq 2^{\max\{|y|, \aleph_0\}}\text{.}\)
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Hint.
Let \(f\colon A \to B\) be a morphism in \(\Ring\) with \(A \in \Ring_\delta\text{.}\) Let \(I\) be the set of \(x \in A\) for which \(\delta^m(x) \in \ker(f)\) for all \(m \geq 0\text{.}\) Then \(I\) is a \(\delta\)-stable ideal of \(A\) (by repeated application of Exercise 2.5.13) and the map
\begin{equation*} A/I \to B \times B \times \cdots, \qquad x \mapsto (f(x), f(\delta(x)), f(\delta^2(x)), \dots) \end{equation*}
is injective.
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15.

Show that the functor \(W_2\) on \(\Ring\) commutes with filtered colimits, but not with coequalizers.
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