Construction 3.1. Let $(A, I)$ be a preprism. We denote $\varphi \colon A\to A$ a Frobenius lift as usual. We have a tower of $\delta $ -rings

$$ \begin{align*} A_0 \xrightarrow{\varphi} A_1 \xrightarrow{\varphi} A_2 \xrightarrow{\varphi} \dots \xrightarrow{\varphi} A_i \xrightarrow{\varphi} \dots, \end{align*} $$

where $A_i$ is the same as A and $\varphi $ is the Frobenius lift on A. The map of rings $\varphi \colon A_i\to A_{i+1}$ induces the following maps of rings:

where $\varphi ^i(I)A_i$ is the ideal of $A_i$ generated by the image $\varphi ^i(I) \subseteq A_i$ of $I \subseteq A_i$ . Note that $\overline {\varphi }_{(p, I)}^{(i)}$ is the pth power map $A_i/(p, I^{[p^i]})A_i \xrightarrow {a \mapsto a^p} A_{i+1}/(p, I^{[p^{i+1}]})A_{i+1}$ . Those maps give three towers of rings:

$$ \begin{align*} (\{A_i\}, \{\varphi\}) & = A_0 \xrightarrow{\varphi} A_1 \xrightarrow{\varphi} A_2 \xrightarrow{\varphi} \dots, \\ (\{A_i/\varphi^i(I)A_i\}, \{\overline{\varphi}^{(i)}_I\}) & = A_0/\varphi(I)A_0 \xrightarrow{\overline{\varphi}_I^{(0)}} A_1/\varphi(I)A_1 \xrightarrow{\overline{\varphi}_I^{(1)}} A_2/\varphi^2(I)A_2 \xrightarrow{\overline{\varphi}_I^{(2)}} \dots, \\ (\{A_i/(p, I^{[p^i]})\}, \{\overline{\varphi}_{(p, I)}^{(i)}\}) & = A_0/(p, I)A_0 \xrightarrow{\overline{\varphi}_{(p,I)}^{(0)}} A_1/(p, \varphi(I))A_1 \xrightarrow{\overline{\varphi}_{(p,I)}^{(1)}} A_2/(p, \varphi^2(I))A_2 \xrightarrow{\overline{\varphi}_{(p,I)}^{(2)}} \dots. \end{align*} $$

The first tower becomes a tower of (pre)prism $(\{A_i\}, \{\varphi \}, I)$ .

Proof. Fix an orientation d of I. To prove the animated prism $(I \xrightarrow {\varphi ^i} A_i)$ is a discrete prism, it suffices to show that $\varphi ^i(d)$ is a non-zero-divisor in $A_i = A$ by [Reference Bhatt and Lurie5, Lemma 2.13], which follows from the transversal property of $(A, I)$ . The transversal property of $(A_i, \varphi ^i(I)A_i)$ also follows since $p, \varphi ^i(d)$ becomes a regular sequence in A for each $i \geq 0$ .

Construction 3.3. Let $(A, I)$ be a preprism. For each $i \geq 0$ , by [Reference Bhatt and Scholze7, Remark 2.7] (or [Reference Bhatt and Lurie5, Proposition A.20(1)]), we can define a $\delta $ -A-algebra $A^{1/p^i}$ as the (finite) colimit in the category of $\delta $ -rings

which is the colimit consisting of $(i+1)$ -copies of A with the Frobenius lift $\varphi $ . The underlying ring of $A^{1/p^i}$ is the colimit of A, that is, $A^{1/p^i} \cong \operatorname *{\mathrm {colim}} \{A \xrightarrow {\varphi } A \xrightarrow {\varphi } \dots \xrightarrow {\varphi } A\}$ as rings. In particular, if A is derived $(p, I)$ -complete, $A^{1/p^i}$ is a derived $(p, I)$ -complete $\delta $ -ring by Lemma 3.4 below.

We denote the canonical map $A \to A^{1/p^i}$ of $\delta $ -rings from the jth term of the colimit as $c_j^i$ for each $0 \leq j \leq i$ , namely, we have a map of $\delta $ -rings

$$ \begin{align*} c_j^i \colon A \to A^{1/p^i}. \end{align*} $$

In the following, $A^{1/p^i}$ is a $\delta $ -A-algebra via $c_0^i\colon A\to A^{1/p^i}$ . Maps $c_0^{i+1}, \dots , c_i^{i+1}$ of $\delta $ -rings uniquely induce a map of $\delta $ -A-algebras

$$ \begin{align*} t_i \colon A^{1/p^i} \to A^{1/p^{i+1}}. \end{align*} $$

Set the ideal

then $t_i$ induces a map of $\delta $ -A-algebras

Consequently, we have a tower of preprisms $(\{A^{1/p^i}\}, \{t_i\}, I)$ and two towers of rings $(\{A^{1/p^i}/I_i\}, \{t_{i,I}\})$ and $(\{A^{1/p^i}/(p, I_i)\}, \{t_{i,(p,I)}\})$ . If $(A, I)$ is a prism, $(\{A^{1/p^i}\}, \{t_i\}, I)$ is a tower of prisms.

By using commutativity of filtered colimits and tensor products, we have the following isomorphisms of A-algebras:

$$ \begin{align*} A^{1/p^i}/I_i & \cong \operatorname*{\mathrm{colim}} \{A/I \xrightarrow{\overline{\varphi}_I^{(0)}} A/\varphi(I)A \xrightarrow{\overline{\varphi}_I^{(1)}} \dots \xrightarrow{\overline{\varphi}_I^{(i-1)}} A/\varphi^i(I)A\}, \\ A^{1/p^i}/(p, I_i) & \cong \operatorname*{\mathrm{colim}} \{A/(p, I)A \xrightarrow{\overline{\varphi}_{(p,I)}^{(0)}} A/(p, \varphi(I))A \xrightarrow{\overline{\varphi}_{(p,I)}^{(1)}} \dots \xrightarrow{\overline{\varphi}_{(p,I)}^{(i-1)}} A/(p, \varphi^i(I))A\} \\ & \cong (A/pA)^{1/p^i}/I(A/pA)^{1/p^i}. \end{align*} $$

As in $c_j^i\colon A\to A^{1/p^i}$ , there exist the canonical maps of rings

$$ \begin{align*} c_{j, I}^i &\colon A/\varphi^j(I)A \to A^{1/p^i}/I_i, \\c_{j, (p,I)}^i &\colon A/(p, I^{[p^j]})A \to A^{1/p^i}/(p, I_i) \end{align*} $$

for each $i \geq 0$ and $0 \leq j \leq i$ .

Proof. Those isomorphisms follow from the definition of colimits. Since A (resp. $A_i/\varphi ^i(I)A_i$ ) is derived $\varphi ^i(p, I)$ -complete (resp. derived p-complete), the completeness also follows.

Proof. The first isomorphism is because the maps $c_i^i$ are compatible with $t_i$ by those constructions. The second and third isomorphisms follow from the same argument.

Proof. If $\varphi (x) = 0$ for $x \in A$ , then $\overline {x}^p = 0$ in the reduced ring $A/pA$ and thus there exists $x_1 \in A$ such that $x = px_1$ . Since A is p-torsion-free, the equation $0 = \varphi (x) = p \varphi (x_1)$ implies $\varphi (x_1) = 0$ in A. Repeating this argument, x is contained in $\cap _{n \geq 0} p^nA = 0$ . This shows the injectivity of $\varphi $ .

Proof. Since $p, d$ is a regular sequence on A, so is $p, \varphi ^{i+1}(d)$ and especially $A/\varphi ^{i+1}(d)A$ is p-torsion-free. Take an element $a \in A$ such that $\varphi (a) \in \varphi ^{i+1}(d)A$ . Taking modulo p, this implies that $\overline {a}^p = \overline {d}^{p^{i+1}}A/pA$ . The p-root closed assumption says that $\overline {a} \in \overline {d}^{p^i}A/pA$ and there exist elements $a_1, b_1 \in A$ such that $a = pa_1 + \varphi ^i(d)b_1 \in (p, \varphi ^i(d))A$ . Applying $\varphi (-)$ to this equation, we have $\varphi ^{i+1}(d)A \ni \varphi (a) = p\varphi (a_1) + \varphi ^{i+1}(d)\varphi (b_1)$ and thus $\varphi (a_1) \in \varphi ^{i+1}(d)A$ , since $A/\varphi ^{i+1}(d)A$ is p-torsion-free. The same argument shows that there exist $a_2, b_2 \in A$ such that $a_1 = pa_2 + \varphi ^i(d)b_2 \in (p, \varphi ^i(d))A$ and thus $a \in (p^2, \varphi ^i(d))A$ . Repeating this process, we have $a \in (p^j, \varphi ^i(d))A$ for all $j \geq 0$ and thus $\overline {a} \in \bigcap _{j \geq 0} p^jA/\varphi ^i(d)A = 0$ by the p-adic separatedness of $A/\varphi ^i(d)A$ . This shows the injectivity.

Construction 3.11. Let $(A, I)$ be a preprism. Note that any ring $A_i$ is A itself. Considering the following commutative diagram of rings:

(3.2)

Taking the colimits for the horizontal directions and taking care of those A-algebra structures, the Frobenius lift $\varphi $ on the $\delta $ -ring $A^{1/p^{i+1}}$ factors through $t_i$ in the category of A-algebras as follows:

We call the isomorphism $\varphi _i$ of A-algebras the ith Frobenius lift projection of the tower of prisms $(\{A^{1/p^i}\}, I)$ .

Taking the quotient in the category of rings (not of A-algebras), the Frobenius lift $\varphi $ of $A^{1/p^{i+1}}$ induces the following commutative diagrams of rings:

Here, by the diagram (3.2), note that the left lower isomorphism $\varphi _i/(p, I)$ is deduced from $I_i = c^0_i(I)A^{1/p^i}$ and $\varphi _i(I_{i+1})A^{1/p^i} = \varphi _i(c^0_{i+1}(I)A^{1/p^{i+1}})A^{1/p^i} = c^0_i(\varphi (I))A^{1/p^i} = \varphi (I_i)A^{1/p^i}$ in $A^{1/p^i}$ by the following commutative diagram:

The lower surjective map

(3.3) $$ \begin{align} \varphi_{i, (p, I)} \colon A^{1/p^{i+1}}/(p, I_{i+1}) \xrightarrow{\varphi_i/(p, I)} A^{1/p^i}/(p, I_i^{[p]}) \twoheadrightarrow A^{1/p^i}/(p, I_i) \end{align} $$

becomes a map of A-algebras, and we call it the ith Frobenius projection of the tower of prisms $(\{A^{1/p^i}\}, I)$ (or of the tower of rings $(\{A^{1/p^i}/I_i\}, \{t_{i, (p, I)}\})$ , see Lemma 3.14).

Observation 3.12. Let $(A, I)$ be a preprism. Through the isomorphism of towers of $\delta $ -rings $\{c_i^i\}$ (Lemma 3.5), we have a commutative diagram in the category of $\delta $ -rings

In particular, we can regard the Frobenius lift projection $\varphi _i\colon A^{1/p^{i+1}}\to A^{1/p^i}$ as the identity map $\operatorname {\mathrm {id}}_A\colon A_{i+1}\to A_i$ . Furthermore, the isomorphism of towers of rings $\{c_{i, (p, I)}^i\}$ gives a commutative diagram of rings

where the middle vertical isomorphism is induced from $c_i^i\colon A_i\to A^{1/p^i}$ . In particular, we can regard the ith Frobenius projection $\varphi _{i, (p, I)} \colon A^{1/p^{i+1}}/(p, I_{i+1}) \twoheadrightarrow A^{1/p^i}/(p, I_i)$ as the natural surjection of rings

$$ \begin{align*} \pi_i \colon A_{i+1}/(p, I^{[p^{i+1}]})A_{i+1} = A/(p, I^{[p^{i+1}]})A \twoheadrightarrow A/(p, I^{[p^i]})A = A_i/(p, I^{[p^i]})A_i \end{align*} $$

defined in Definition 3.9. The Frobenius maps F on $A^{1/p^{i+1}}/(p, I_{i+1})$ and $A_{i+1}/(p, I^{[p^{i+1}]})A_{i+1}$ factor through $\varphi _{i, (p, I)}$ and $\pi _i$ as follows:

(3.4)

Proof. This is clear by the construction of the Frobenius projection in Definition 3.9 (and Construction 3.11).

Notation 5.1. In this section, we follow the notation below:

• • Let V be an absolute unramified discrete valuation $\delta $ -ring of mixed characteristic $(0, p)$ .

• • Let $(A, I)$ be an orientable bounded Zariskian prism with an orientation $d \in I$ .

• • Let R be a p-torsion-free p-Zariskian $\delta $ -ring.

• • Write the Frobenius lift on A by $\varphi $ and the Frobenius lift on R by $\varphi _R$ .

• • Assume that both A and R are p-torsion-free $\delta $ -V-algebra.

In this case, we can identify $R \otimes _V I$ and $I (R \otimes _V A)$ as ideals of $R \otimes _V A$ and the tensor product $R \otimes _V A$ is flat over R and A. In particular, it is p-torsion-free. The tensor product $R \otimes _V A$ and its $(p, I)$ -ZariskizationFootnote ⁶ $(1 + (p, I))^{-1}(R \otimes _V A)$ inherit the unique $\delta $ -structure compatible with the maps from R and A. Here, we implicitly use $\varphi ((p, I)) = (p, d^p) \subseteq (p, I)$ in $R \otimes _V A$ and [Reference Bhatt and Scholze7, Lemma 2.15]. We take the flat $\delta $ - $(R, A)$ -algebra

(5.1)

The Frobenius lift on $P_{A, V}(R)$ is denoted by $\varphi _P$ , which is defined as the localization of $\varphi _R \otimes _{\varphi _V} \varphi $ on $R \otimes _V A$ .

Proof. We already know that $P_{A, V}(R)$ is a $\delta $ -ring and $p \in IP_{A, V}(R) + \varphi (I)P_{A, V}(R)$ . Since $P_{A, V}(R)$ is flat over A, the ideal $IP_{A, V}(R)$ is generated by a non-zero-divisor $1 \otimes d$ in $P_{A, V}(R)$ . The $p^{\infty }$ -torsion part of $P_{A, V}(R)$ is the base change of the $p^{\infty }$ -torsion part of A because of the flatness of R over V. The construction of $P_{A, V}(R)$ ensures that it is $(p, I)$ -Zariskian and therefore the pair $(P_{A, V}(R), IP_{A, V}(R))$ is an orientable bounded Zariskian prism.

If $R \otimes _V (A/\varphi ^i(I)A)$ is p-Zariskian, then the isomorphism $P_{A, V}(R)/\varphi ^i(I)P_{A, V}(R) \cong (1 + I)^{-1}(R \otimes _V (A/\varphi ^i(I)A))$ holds. It suffices to show that the p-Zariskian ring $R \otimes _V (A/\varphi ^i(I)A)$ is I-Zariskian but this follows from the equation $(p, \varphi ^i(I)) = (p, d^{p^i})$ on A.

Proof. By our assumption, we have injections $A \hookrightarrow A[1/t] \hookrightarrow S^{-1}A[1/t]$ . Let $a/st^n \in S^{-1}A[1/t]$ be an element satisfying that $(a/st^n)^p = a'/s' \in S^{-1}A$ . Then we have $(as'/t^n)^p = s^pa'(s')^{p-1} \in A$ and thus $as'/t^n \in A$ by the p-root closedness of A in $A[1/t]$ . This shows that $a/st^n = 1/ss' \cdot as'/t^n \in S^{-1}A$ and the lemma is proved.

Proof. By Theorem 4.5 and Lemma 5.2, it is enough to show that $p, d$ is a regular sequence on $P_{A, V}(R)$ and $P_{A, V}(R)/(p) \cong (1 + I)^{-1}(R/pR \otimes _{V/pV} A/pA)$ is p-root closed in $P_{A, V}(R)/(p)[1/d]$ . The first assertion follows from the regularity of $p, d$ on A and the flatness of $R \otimes _V A$ over A. In the second assertion, Lemma 5.3 implies that it is enough to show that $R/pR \otimes _{V/pV} A/pA$ is p-root closed in $(R/pR \otimes _{V/pV} A/pA)[1/d]$ . As in the proof of Theorem 4.5, this is equivalent to the injectivity of the pth power map $(R/pR \otimes _{V/pV} A/pA)/(d) \xrightarrow {x \mapsto x^p} (R/pR \otimes _{V/pV} A/pA)/(d^p)$ . This map can be factorized as the composition

$$ \begin{align*} R/pR \otimes_{V/pV} A/(p, d)A \xrightarrow{\operatorname{\mathrm{id}} \otimes \overline{\varphi}^{(0)}_{(p, I)}} R/pR \otimes_{V/pV} A/(p, d^p)A \xrightarrow{\operatorname{\mathrm{Frob}} \otimes \operatorname{\mathrm{id}}} R/pR \otimes_{V/pV} A/(p, d^p)A. \end{align*} $$

The former map is injective by the p-root closedness of $A/pA$ in $A/pA[1/d]$ and the flatness of R over V. Since V is an absolute unramified discrete valuation ring, $V/pV$ is a field and thus the latter map is injective by the reducedness of $R/pR$ .

By Lemma 5.2, we have a natural isomorphism $(1 + (p, I))^{-1}(R \otimes _V (A/\varphi ^i(I)A)) \cong R \otimes _V (A/\varphi ^i(I)A)$ . So we can deduce the last assertion from the first assertion.

Proof. The orientable bounded Zariskian prisms $((1 + (T))^{-1}\mathbb {Z}_{(p)}[T], (p-T))$ and $((1+(q-1))^{-1} \mathbb {Z}_{(p)}[q], ([p]_q))$ in Example 2.3 satisfy the assumption of Theorem 4.5. Applying Theorem 5.4 for , the isomorphisms $\mathbb {Z}[T]/(p-T^{p^i}) \cong \mathbb {Z}[p^{1/p^i}]$ and $\mathbb {Z}[q]/([p]_q) \cong \mathbb {Z}[\zeta _p]$ yield the perfectoid towers and their tilts are isomorphic to the tower $(\{R/pR[|T|]\}, \{F\})$ . If R is p-adically separated, Lemma 3.7 implies that the Frobenius lift $\varphi _R$ is injective. Since $\mathbb {Z} \hookrightarrow \mathbb {Z}[p^{1/p^i}]$ and $\mathbb {Z} \hookrightarrow \mathbb {Z}[\zeta _{p^i}]$ are flat, the transition maps in the towers are injective.

We next prove the last assertion. The isomorphism $R^{1/p^i}/pR^{1/p^i} \cong (R/pR)^{1/p^i}$ holds for each $i \geq 0$ and thus p is a prime element of $R^{1/p^i}$ . Since $\varphi _R$ is finite injective, the canonical map $R^{1/p^i} \hookrightarrow R^{1/p^{i+1}}$ is also finite injective. This gives a sequence of finite extensions of integral domains

$$ \begin{align*} R \hookrightarrow R^{1/p} \hookrightarrow R^{1/p^2} \hookrightarrow \cdots \hookrightarrow R^{1/p^i} \hookrightarrow \cdots. \end{align*} $$

This sequence is contained in a fixed absolute integral closure $R^+$ of R. By (5.2) and (5.3), we need to show that the canonical maps $R^{1/p^i}[T]/(p-T^{p^i}) \cong R^{1/p^i} \otimes _{\mathbb {Z}} \mathbb {Z}[p^{1/p^i}] \twoheadrightarrow R^{1/p^i}[p^{1/p^i}]$ and $R^{1/p^i}[q^{1/p^i}]/([p]_q) \cong R^{1/p^i} \otimes _{\mathbb {Z}_p} \mathbb {Z}_p[\zeta _{p^i}] \twoheadrightarrow R^{1/p^i}[\zeta _{p^i}]$ are isomorphisms. It is enough to show that $T^{1/p^i}-p$ and $[p]_q$ are irreducible polynomials in $R^{1/p^i}[T]$ and $R^{1/p^i}[q^{1/p^i}]$ , respectively. This follows from the fact that p is a prime element of $R^{1/p^i}$ and the Eisenstein criterion for $T^{1/p^i}-p$ and the variable transformation $((q+1)^p-1)/q$ of $[p]_q$ .

Proof. Set . First, we can show that K and $\mathbb {Q}[p^{1/p^i}]$ are linearly independent over $\mathbb {Q}$ in an algebraic closure $\overline {K}$ of K: If elements $x_0, \dots , x_i$ in K satisfy $x_0 + x_1 p^{1/p^i} + \cdots + x_{p^i-1} p^{(p^i-1)/p^i} = 0$ in $\overline {K}$ , then we may assume that $x_0, \dots , x_i$ are in R and so $x_0 + x_1 T + \cdots + x_{p^i-1} T^{p^i-1} = 0$ in $R[T]$ since we have an isomorphism $R[T]/(p-T^{p^i}) \cong R[p^{1/p^i}]$ as in the proof of Corollary 5.5. Therefore, $x_0 = \cdots = x_{p^i-1} = 0$ in R and so in K. This shows the linear independence. Especially, this implies that $\operatorname {\mathrm {Frac}}(R[p^{1/p^i}]) \cong K[p^{1/p^i}] \cong K \otimes _{\mathbb {Q}} \mathbb {Q}[p^{1/p^i}]$ . Since we have an isomorphism $R^{1/p^i} \otimes _{\mathbb {Z}} \mathbb {Z}[p^{1/p^i}] \cong R^{1/p^i}[p^{1/p^i}]$ , it suffices to consider the generic rank of the map $R \otimes _{\mathbb {Z}} \mathbb {Z}[p^{1/p^i}] \to (\varphi _*R) \otimes _{\mathbb {Z}} \mathbb {Z}[p^{1/p^{i+1}}]$ . Since the degree $[K(\varphi _*R): K(R)]$ is $\deg \varphi _R$ , we have

$$ \begin{align*} \operatorname{\mathrm{Frac}}((\varphi_*R) \otimes_{\mathbb{Z}} \mathbb{Z}[p^{1/p^{i+1}}]) \cong K(\varphi_*R) \otimes_{\mathbb{Q}} \mathbb{Q}[p^{1/p^{i+1}}] \cong K^{\oplus \deg \varphi_R} \otimes_{\mathbb{Q}} \mathbb{Q}[p^{1/p^{i+1}}] \end{align*} $$

and this is a $(p \cdot \deg \varphi _R)$ -dimensional $(K \otimes _{\mathbb {Q}} \mathbb {Q}[p^{1/p^i}])$ -vector space.

Proof. By Lemma 3.7, the Frobenius lift $\varphi _R$ is finite injective. Since is a prime ideal of R and R is Noetherian normal, taking the localization at p, we have a finite injective map $\varphi _{\mathfrak {p}} \colon R_{\mathfrak {p}} \hookrightarrow R_{\mathfrak {p}}$ between discrete valuation rings. In particular, this map $\varphi _{\mathfrak {p}}$ is a finite flat endomorphism of a discrete valuation ring. Therefore, $\varphi _{\mathfrak {p}, *}R_{\mathfrak {p}}$ is a finite free $R_{\mathfrak {p}}$ -module. Thus, the generic rank of $\varphi _{\mathfrak {p}}$ is the same as the generic rank of the Frobenius map on $K(R/pR) = R_{\mathfrak {p}}/pR_{\mathfrak {p}}$ which is p-power. Since the generic rank of $\varphi _R$ is the same as that of $\varphi _{\mathfrak {p}}$ , the generic rank of $\varphi _R$ is p-power.

Proof. Recall that the Frobenius lift $\varphi _R$ induces an endomorphism of schemes $\widetilde {F}_{\mathcal {X}} \colon \mathcal {X} \to \mathcal {X}$ and this is compatible with the Frobenius lift on $\operatorname {\mathrm {Spec}}(W(k))$ : $\varphi _R$ induces a morphism of schemes $U \to \mathcal {X} = \operatorname {\mathrm {Proj}}(R)$ from an open subscheme in $\mathcal {X}$ [Reference Görtz and Wedhorn9, (13.2.4)]. Since U contains the special fiber , U should be the whole $\mathcal {X}$ and thus $\widetilde {F}_{\mathcal {X}}$ is well-defined.

Taking the special fiber, the Frobenius map on $R/pR$ induces the Frobenius morphism $F_X \colon X \to X$ of a smooth k-scheme X (through the Frobenius map on k) and in particular this is a finite morphism. This says that the restriction of $\widetilde {F}_{\mathcal {X}} \colon \mathcal {X} \to \mathcal {X}$ to X has zero-dimensional fibers and this property extends to $\mathcal {X}$ . Consequently, $\widetilde {F}_{\mathcal {X}}$ is a finite morphism [22, Tag 02OG] and thus $\widetilde {F}_{\mathcal {X}}$ is flat by miracle flatness [22, Tag 00R4]. In particular, $\widetilde {F}_{\mathcal {X}}$ is finite locally free by [Reference Görtz and Wedhorn9, Proposition 12.19]. By [Reference Görtz and Wedhorn9, Proposition 13.37(2)], we have $R_f \cong R_{(f)}[T, T^{-1}]$ for any homogeneous element $f \in R_1$ . Since $\widetilde {F}_{\mathcal {X}}$ induces a finite free map $R_{(f)} \to R_{(\varphi (f))}$ between regular rings for some $f \in R_1$ , the generic rank $\deg \varphi $ is the same as the rank of the finite free map $R_f \to R_{\varphi (f)}$ induced from $\varphi $ . This is the same as the rank of the Frobenius map $(R/pR)_f \xrightarrow {F} (R/pR)_f$ since $D_+(f) \cap \mathcal {X}_{p=0} \neq \emptyset $ . Therefore, the generic degree $\deg \varphi $ is the p-power $[K(X)^{1/p}: K(X)]$ .

Proof. (1): Since Cohen–Macaulayness and Gorensteinness are stable under the quotient by a non-zero-divisor and the completion, it is enough to check such properties for the affine semigroup ring $\mathbb {F}_p[H]$ over $\mathbb {F}_p$ . By [Reference Goto, Suzuki and Watanabe10, Theorems (1) and (2)], such properties only depend on the simplicial monoid H and this proves (1).

(2): By [Reference Miller and Sturmfels20, Proposition 7.25], the normality of an affine semigroup ring $k[H]$ over a field k depends only on the monoid H. So the assumption implies that $\mathbb {Q}_p[H]$ is normal (in other words, the monoid H is normal). We know that the polynomial ring $\mathbb {Z}_p[\underline {t}]$ is integrally closed in $\mathbb {Q}_p[\underline {t}]$ and the equation $\mathbb {Z}_p[H] = \mathbb {Z}_p[\underline {t}] \cap \mathbb {Q}_p[H]$ holds. The normality of $\mathbb {Q}_p[H]$ leads to the normality of $\mathbb {Z}_p[H]$ . Moreover, in this case, the normality of $\mathbb {Q}_p[H]$ is stable under the completion due to [Reference Zariski23, Théorème 2]. So a similar argument shows that $\mathbb {Z}_p[|H|]$ is normal.

Using [Reference Hochster12, Theorem 1] (see also [Reference Miller and Sturmfels20, Corollary 13.43]), the normality of H implies that $\mathbb {Z}_p[H]$ and $\mathbb {Z}_p[|H|]$ are Cohen–Macaulay. The log-regularity follows from the definition of local log-regular rings: Normal affine semigroup H is fine, sharp, and saturated.

Proof. In (1), we have a bounded prism $(W[|\underline {T}|]/J, (d))$ . Taking the $(\underline {T})$ -adic completion, $p, d$ is also a regular sequence on $W[|\underline {T}|]/J$ . To apply Theorem 4.5, we need to show that $W[|\underline {T}|]/(p, J)$ is p-root closed in $W[|\underline {T}|]/(p, J)[1/f]$ . As in the proof of Theorem 4.5, this is equivalent to showing that the pth power map $W[|\underline {T}|]/(J, p, d) \xrightarrow {a \mapsto a^p} W[|\underline {T}|]/(J, p, d^p)$ is injective. This follows from the assumption and the faithfully flat property of the $(\underline {T})$ -adic completion.

Next, we show (2). As above, $W[|\underline {T}|]/J$ (resp. $W[\underline {T}]_{(p, \underline {T})}/J$ ) is p-torsion-free by taking the $(\underline {T})$ -adic completion (resp. $(p)$ -Zariskization). By using analytically unramified property of finitely generated k-algebras, $W[|\underline {T}|]/(p, J)$ is reduced (see, e.g., [Reference Swanson and Huneke21, Theorem 4.6.3, Proposition 9.1.3, and Theorem 9.2.2]). Also, the p-Zariskization $(1+(p))^{-1}W[\underline {T}]/(p, J) \cong W[\underline {T}]/(p, J)$ is reduced. By Corollary 5.5, we have a perfectoid tower arising from $(W[|\underline {T}|]/J, (p))$ (resp. $(W[\underline {T}]_{(p, \underline {T})}/J, (p))$ ) with injective transition maps. If we assume that $W[|\underline {T}|]/J$ (resp. $W[\underline {T}]_{(p, \underline {T})}/J$ ) is a normal domain and $W[|\underline {T}|]/(p, J)$ (resp. $W[\underline {T}]_{(p, \underline {T})}/(p, J)$ ) is an integral domain, then the generic extension of those transition maps has p-power degree.

Proof. In (1’), any distinguished element of a complete Noetherian local $\delta $ -ring $W[|\underline {T}|]/J$ with $\delta (T_i) = 0$ can be written as $p-f$ for some $f \in (T_1, \dots , T_n)$ up to unit by the proof of [Reference Ishizuka and Nakazato16, Lemma 5.1]. Other assertions follow from (1) and (2) in Proposition 6.9.

Proof. Set in $\mathbb {Z}[t_1, \dots , t_m]$ . Since $\delta (M_i) = 0$ holds for any $\phi $ -monomials $M_i$ , we can take a map of $\delta $ -rings $\mathbb {Z}[t_1, \dots , t_m] \to R$ sending $t_i$ to $M_i$ . If $\delta ^{n+1}(\overline {f})$ belongs to the ideal $(\overline {f}, \delta (\overline {f}), \dots , \delta ^n(\overline {f}))\mathbb {Z}[t_1, \dots , t_m]$ , then we have $\delta ^{n+1}(f) \in (f, \delta (f), \dots , \delta ^n(f))R$ and thus $\operatorname {\mathrm {ht}}^\delta (f) \leq n$ .

Proof. Under modulo $f = X_1^{n_1} + \cdots + X_m^{n_m}$ , we have

(6.7) $$ \begin{align} \delta(f) & \equiv -\sum_{\substack{0 \leq e_1, \dots, e_m \leq p-1 \\ e_1 + \cdots + e_m = p}} \frac{(p-1)!}{e_1! \dots e_m!}(-1)^{e_1}(X_2^{n_2} + \cdots + X_m^{n_m})^{e_1} X_2^{n_2e_2} \dots X_m^{n_me_m} \nonumber \\ & = -\sum_{\substack{0 \leq f_2, \dots, f_m, e_2, \dots, e_m \leq p-1 \\ f_2 + \cdots + f_m + e_2 \dots + e_m = p \\ f_2 + \cdots + f_m \leq p-1}} \frac{(p-1)!}{f_2! \dots f_m! e_2! \dots e_m!}(-1)^{f_2 + \cdots + f_m}X_2^{n_2(e_2+f_2)} \dots X_m^{n_m(e_m+f_m)} , \end{align} $$

which is an element of $\mathbb {Z}[X_2, \dots , X_m]$ . We will compute the coefficient of $X_2^{pn_2}$ and $X_2^{n_2(p-1)}$ in the summation (6.7).

In the case of $e_2+f_2 = p$ , the limitation of the sum in (6.7) implies that $e_3 = \cdots = e_m = 0$ , $f_3 = \cdots = f_m = 0$ , and $e_2> 0$ . So the coefficient of $X_2^{n_2p}$ is

$$ \begin{align*} \sum_{e_2=1}^{p-1} \frac{(p-1)!}{(p-e_2)!e_2!}(-1)^{p-e_2} & = \frac{1}{p} \left(\sum_{e_2=0}^p \frac{p!}{(p-e_2)!e_2!}(-1)^{p-e_2} - \left((-1)^{p-0} + (-1)^{p-p}\right)\right) \\ & = -\frac{(-1)^p+1}{p} \end{align*} $$

and this is $-1$ if $p = 2$ and $0$ if $p> 2$ .

Next one is $e_2+f_2 = p-1$ and we can assume that $e_i+f_i = 1$ for some $i> 2$ . Then $e_j = f_j = 0$ for $j \neq 2, i$ and there are two cases: $e_i = 0$ and $f_i = 1$ , or $e_i = 1$ and $f_i = 0$ . In the former case, the coefficient of $X_2^{n_2(p-1)}X_i^{n_i}$ is

$$ \begin{align*} \sum_{e_2=1}^{p-1} \frac{(p-1)!}{(p-1-e_2)!1!e_2!}(-1)^{(p-1-e_2)+1} & = -\left(\sum_{e_2=0}^{p-1} \frac{(p-1)!}{(p-1-e_2)!e_2!}(-1)^{p-1-e_2} - (-1)^{p-1}\right) \\ & = (-1)^{p-1}. \end{align*} $$

In the latter case, the coefficient of $X_2^{n_2(p-1)}X_i^{n_i}$ is

$$ \begin{align*} \sum_{e_2=0}^{p-1} \frac{(p-1)!}{(p-1-e_2)!e_2!1!}(-1)^{p-1-e_2} = 0. \end{align*} $$

Therefore, the terms in (6.7) of degree $n_2(p-1)$ in $X_2$ can be summed up as $(-1)^{p}X_2^{n_2(p-1)}(X_3^{n_3} + \cdots + X_m^{n_m})$ . Consequently, the summation (6.7) can be written as

(6.8)

for some polynomials $\beta _i^{(m)}$ in $X_3, \dots , X_m$ and set and . In particular, $\beta ^{(m)}_{p-1}$ is a non-zero element and so is $\beta ^{(m)}$ . The last term $\beta ^{(m)}_0$ is the case that $e_2+f_2 = 0$ in (6.7) and this is the same as $\beta ^{(m-1)}(X_3^{n_3}, \dots , X_{m}^{n_m})$ for any $m \geq 4$ . This shows the assertions (1) and (2).

We show the next assertion (3). Let $g = g_1f + g_2\beta ^{(m)}$ be a polynomial in $\Lambda [X_1, \dots , X_m]$ with $g_1, g_2 \in \Lambda [X_1, \dots , X_m]$ and the degree of $X_1$ in g is strictly less than $n_1$ . Write the polynomials ${g_1 = a_NX_1^N + a_{N-1}X_1^{N-1} + \cdots + a_1X_1 + a_0}$ and $g_2 = b_MX_1^M + b_{M-1}X_1^{M-1} + \cdots + b_1X_1 + b_0$ such that $a_i$ and $b_j$ are in $\Lambda [X_2, \dots , X_m]$ and $a_N$ and $b_M$ are non-zero. Since g has the degree of $X_1$ strictly less than $n_1$ , we can deduce that $N+n_1 = M$ . For simplicity, set and . So we have $g = g_1 (X_1^{n_1} + \alpha ) + g_2 \beta $ , namely,

(6.9) $$ \begin{align} g & = (a_NX_1^{N+n_1} + \cdots + a_{N-n_1}X_1^{N-n_1+n_1} + \cdots + a_0X_1^{n_1}) + (\alpha a_N X_1^N + \cdots + \alpha a_0) + \\ & \quad+ (\beta b_MX_1^M + \cdots + \beta b_NX_1^{N} + \cdots + \beta b_0). \nonumber \end{align} $$

It suffices to show that $a_i$ are divisible by $\beta $ in $\Lambda [X_2, \dots , X_m]$ for all $0 \leq i \leq N$ . If $n_1> N$ , the above equation shows

(6.10) $$ \begin{align} a_iX_1^{i+n_1} + \beta b_{i+n_1}X_1^{i+n_1} = 0 \end{align} $$

for all $0 \leq i \leq N$ since the degree of $X_1$ in g is strictly less than $n_1$ . This implies that $a_i$ is divisible by $\beta $ for all $0 \leq i \leq N$ .

If $n_1 \leq N$ , the same vanishing (6.10) holds for any $N-n_1+1 \leq i \leq N$ and thus $a_i$ is divisible by $\beta $ for all $N-n_1+1 \leq i \leq N$ . The equation (6.9) gives

(6.11) $$ \begin{align} a_{i-n_1}X_1^i + \alpha a_iX_1^i + \beta b_iX_1^i = 0 \end{align} $$

for any $n_1 \leq i \leq N$ because of $n_1 \leq N$ . Whether $N - n_1 + 1$ is smaller than or greater than $n_1$ , setting $a_i$ as $0$ for $i < 0$ , we have the same equation (6.11) for any $N - n_1 + 1 \leq i \leq N$ . Then the divisibility of $a_i$ shows that $a_{i-n_1}$ is divisible by $\beta $ for any $N-n_1+1 \leq i \leq N$ , that is, $a_i$ is divisible by $\beta $ for any $N-2n_1+1 \leq i \leq N-n_1$ . Using (6.11) again, $a_{i-n_1}$ is divisible by $\beta $ for any $N-2n_1+1 \leq i \leq N-n_1$ , that is, $a_i$ is divisible by $\beta $ for any $N-3n_1+1 \leq i \leq N-2n_1$ . Repeating this argument, $a_i$ is divisible by $\beta $ for all $0 \leq i \leq N$ . Consequently, in both cases, $g_1$ is contained in $(\beta )$ in $\Lambda [X_1, \dots , X_m]$ .

Proof. For the first statement, it is enough to show that f is a non-zero-divisor on $\mathbb {F}_p[X_1, \dots , X_m]/(\beta ^{(m)})$ since $\delta (f)$ and $\beta ^{(m)}$ are equivalent under modulo f by Lemma 6.15(1). If a polynomial $g_1$ in $\mathbb {F}_p[X_1, \dots , X_m]$ satisfies $g_1f \in (\beta ^{(m)})$ , then $g_1$ is contained in $(\beta ^{(m)})$ by using Lemma 6.15(3).

For the initial ideal, take $g \in (f, \delta (f)) = (f, \beta ^{(m)})$ . Dividing by f, we may assume that the degree of $X_1$ in g is strictly less than $n_1$ . By Lemma 6.15(3) again, g is contained in $(\beta ^{(m)})$ in $\Lambda [X_1, \dots , X_m]$ and thus its initial term is generated by the initial term of $\beta ^{(m)}$ , which is $X_2^{2n_2}$ if $p = 2$ and $X_2^{n_2(p-1)}X_3$ if $p> 2$ .