A calculation of the perfectoidization of semiperfectoid rings

We show that perfectoidization can be (almost) calculated by using $p$-root closure in certain cases, including the semiperfectoid case. To do this, we focus on the universality of perfectoidization and uniform completion, as well as the $p$-root closed property of integral perfectoid rings. Through this calculation, we establish a connection between a classical closure operation ``$p$-root closure'' used by Roberts in mixed characteristic commutative algebra and a more recent concept of ``perfectoidization'' introduced by Bhatt and Scholze in their theory of prismatic cohomology.


Introduction
Let p be a prime number.The method of perfectoidization, introduced by Bhatt and Scholze in [BS22], is an application of the theory of prismatic cohomology to commutative algebra.This yields a universal integral perfectoid ring over a ring, such as a semiperfectoid ring, which is a derived p-complete ring that can be written as a quotient of an integral perfectoid ring.This method can be seen as a generalization of the perfect closure of positive characteristic rings.See Section 2 for an explanation of the terminology used in perfectoid theory.
In contrast to these applications, they are not yet widely used in commutative algebra.
One problem is that many abstract theories, including homotopy theory, have been used, and therefore perfectoidization has a mysterious ring structure.To the best of the author's knowledge, perfectoidization has only been explicitly calculated in [Rei20, §2.3.1] and in the proof of [Din22,Theorem 4.4].
1.1.p-root closure.In this paper, we give an explicit description of the perfectoidization of semiperfectoid rings by using p-root closure.Before explaining our first main theorem, we recall the notion of p-root closure.
Definition 1.1 ([Rob08]).Let R be a p-torsion-free ring.We say that containing R. Focusing on the case of "in R[1/p]", Roberts provided an explicit discription of the p-root closure C(R) as follows: The term "in R[1/p]" is omitted in this paper because only this case is considered.
Here is a brief mention of the history of (p-)root closure.The notion of p-root closure is a special case of total n-root closure introduced in commutative algebra by Anderson, Dobbs and Roitman [ADR90].Previously, n-root closedness was used, for example, by Angermüller [Ang83], Anderson [And82], Watkins [Wat82] and Brewer, Costa and Mc-Crimmon [BCM79].Furthermore, its origin can be traced back to Sheldon's definition of root closedness in [She71].
In the context of commutative algebra in mixed characteristic, Roberts provided the above explicit description of p-root closure and applied this even before perfectoid rings appeared.Most recently, (total) p-root closure has been renamed p-integral closure by Česnavičius and Scholze in [CS23] and is found to be more closely related to the perfectoid theory.
1.2.Main Theorem.Under this notation, our main theorems can be stated in the following forms.For simplicity, we only state the theorems in the case of p-torsion-free rings.
Readers interested in the general cases may refer to the referenced statements in each theorem, in conjunction with Section 1.5.
Theorem 1.2 ((Theorem 5.7; p-torsion-free case)).Let R be a p-torsion-free ring which satisfies the following conditions: (1) The p-adic completion R of R has a map from some integral perfectoid ring.
(2) The perfectoidization ( R) perfd of R is an (honest) integral perfectoid ring. 1 (3) The p-adic completion C(R) of the p-root closure C(R) is an integral perfectoid ring.
Then we have an isomorphism This result clarifies the ring structure of perfectoidization by using p-root closure, which is a quite explicit closure operation.The left-hand side is constructed abstractly by homotopy theory, but it can be described as a p-root closure followed by a p-adic completion, which are only ring-theoretic operations.
1.3.Applications.Let R be a p-torsion-free ring such that its p-adic completion R is a semiperfectoid ring, that is, R is a quotient of some integral perfectoid ring.In applications of Theorem 1.2, it is crucial that R satisfies the assumptions of the theorem: (1) The semiperfectoid ring R has a surjective map from some integral perfectoid ring by the definition of semiperfectoid rings.
( So we can provide an explicit description of the perfectoidization of such R as follows. Theorem 1.3 ((Corollary 5.9; p-torsion-free case)).Let R be a p-torsion-free ring such that its p-adic completion R becomes a semiperfectoid ring.Then we have an isomorphism ( R) perfd ∼ = C(R).In particular, any p-torsion-free and (classically) p-adically complete semiperfectoid ring S has an isomorphism S perfd ∼ = C(S).
In the study of commutative rings in mixed characteristic, semiperfectoid rings often appear as in [IS23].An application of Theorem 1.3 is as follows.
Theorem 1.4 ((Construction 6.1 and Corollary 6.2)).Let (R 0 , m, k) be a complete Noetherian local domain of mixed characteristic (0, p) with perfect residue field k and let p, x 2 , . . ., x n be any system of generators of the maximal ideal m such that p, x 2 , . . ., x d 1 As explained in the rest of Section 2, the perfectoidization of an algebra over some integral perfectoid ring is only a complex.So this assumption states that ( R) perfd gives an honest ring (see Theorem 2.9).
forms a system of parameters of R 0 .Choose compatible sequences of p-power roots inside the absolute integral closure R + 0 , the integral closure of R 0 in the algebraic closure of the fraction field of R 0 .Set a subring R ∞ of R + 0 as Then the p-adic completion R ∞ of R ∞ becomes a p-torsion-free semiperfectoid ring by Cohen's structure theorem (see Construction 6.1 for details).Then ( R ∞ ) perfd is isomorphic 1.4.Strategy of Proof.Let us comment on the strategy of the proof of Theorem 1.2.
Our proof is attributed to some universalities and the following principle of "rigidity lemma" that makes sense in certain situations (for example Lemma 5.2).
Lemma.Let f : R → R be an endomorphism of a ring R. Assume that R has a "good" map S → R from some ring S. Then if f is a map of S-algebras, f is exactly the identity map.
In the proof of Theorem 1.2, two maps of rings ϕ : C(R) → ( R) perfd and ψ : ( R) perfd → C(R) are obtained through the universality of uniform completions and perfectoidizations, respectively.By using the lemma mentioned above, we can show that ψ • ϕ and ϕ • ψ are identity maps.The former is a consequence of the universality of uniform completions (Proposition 3.7), while the latter is a consequence of the use of "almost elements", which is a concept from almost mathematics (Definition 4.3 and Lemma 5.2).
1.5.p-torsion case.While the aforementioned theorems only deal with p-torsion-free rings, we can show that similar statements hold in general.
For this purpose, we introduce the following symbol.Let R be a ring.The p-torsion-free quotient R ptf is defined as the quotient ring Then R ptf is a p-torsion-free ring and we have a canonical map R ։ The general case of main theorems are obtained by substituting C(R) and ( R) perfd for C(R ptf ) and (( R) perfd ) ptf , respectively.Furthermore, the symbols (−) perfd and (−) ptf are often interchangeably because of Corollary 5.3.
To show the general version of Theorem 1.2 (i.e., Theorem 5.7), we need to pass from the possibly p-torsion case to the p-torsion-free case as in [And20].With this in mind, we show that any integral perfectoid ring can be canonically modified into a p-torsionfree integral perfectoid ring (see Theorem 4.9) by using pre-perfectoid pairs as defined in Section 4. Finally, I would like to express my deepest gratitude to the anonymous referee.The benefit of the referee is everywhere.In particular, the referee gave me the idea of focusing on the "rigidity lemma" (Section 1.4) and precise advice which helps the author to make a clear and readable representation in the Introduction.

Perfectoid Rings
In this section, we recall and fix some definitions of perfectoid objects.
Definition 2.1 ([BMS18, Definition 3.5]).Let S be a (non-zero) ring.Then S is an integral perfectoid ring if the following conditions hold: (1) There exists an element π ∈ S such that S is π-adically complete and π p divides p in S.
(3) The kernel of θ : This π ∈ S is called a perfectoid element in this paper.Here, we do not require that π is a non-zero-divisor in S.
Recently, integral perfectoid rings are simply called perfectoid rings.To avoid confusion with perfectoid Tate rings defined later in Definition 2.5, we do not use the term perfectoid rings, but only integral perfectoid rings and perfectoid Tate rings.
Lemma 2.2 (([BMS18, Lemma 3.9])).Let S be an integral perfectoid ring and let π ∈ S be a perfectoid element.Then S has compatible sequences of p-power roots {(uπ) 1/p j } j≥0 and {(vp) 1/p j } j≥0 of uπ and vp where u and v are unit elements in S.
We fix the element ̟ := (vp) 1/p of S. Then ̟ becomes a perfectoid element of S.
Without loss of generality, we can assume that a perfectoid element π has a compatible sequence of p-power roots {π 1/p j } j≥0 .
We next check that ̟ is a perfectoid element of S. Note that the conditions (2) and (3) in Definition 2.1 are independent of the choice of a perfectoid element.Since ̟ p = vp divides p in S, it suffices to show that S is p-adically complete and this is also clear (see Remark 2.3 ([INS23, Theorem 3.52]).Let S be a ring and let π be an element of S.
Then S is an integral perfectoid ring with a perfectoid element π ∈ S if and only if π ∈ S satisfies the following: (1) S is π-adically complete and π p divides p in S.
(2) The p-th power map S/πS we recall the definition of semiperfectoid rings.
Definition 2.4 ([BS22, Notation 7.1]).A ring S is a semiperfectoid ring if it is a derived p-complete ring that is isomorphic to a quotient of an integral perfectoid ring.
We next explain perfectoid Tate rings.See Notation (Section 1.6) at the end of the Introduction for the basic terminology of Tate rings.

Definition 2.5 ([BMS18, Fon13]
).Let A be a complete Tate ring (more generally, let A be a Banach ring).Then A is a perfectoid Tate ring if the following conditions hold: (1) A is uniform, that is, the set of all power-bounded elements A • is bounded in A.
(2) There exists a pseudo-uniformizer π ∈ A such that π p divides p in A • and the This π ∈ A is again called a perfectoid element.
The following lemma establishes the connection between integral perfectoid rings and perfectoid Tate rings.
Lemma 2.6 (([BMS18, Lemma 3.20])).Let A be a Tate ring and let A + 0 be a ring of integral elements in A. If A is a perfectoid Tate ring, then A + 0 is an integral perfectoid ring.In particular, A • is an integral perfectoid ring.
Conversely, if A + 0 is an integral perfectoid ring and bounded in A, the Tate ring A is a perfectoid Tate ring.
Remark 2.7.Let A be a perfectoid Tate ring and let π ∈ A be a perfectoid element.Then π is a non-zero-divisor in A. In general, an integral perfectoid ring is not necessarily isomorphic to the set of all power-bounded elements in some perfectoid Tate ring.
For convenience, we summarize a brief overview of the properties of perfectoidization.
Let S be an integral perfectoid ring and let R be a derived p-complete S-algebra.The perfectoidization R perfd of R is defined by using the prismatic cohomology ∆ R/S .Note that R perfd is typically a commutative algebra object in D ≥0 (S) and has a map R → R perfd in D(S).The complex R perfd is concentrated in degree 0 in the following cases: • If R can be written as a quotient of S, that is, R is a semiperfectoid ring, then R perfd is an integral perfectoid ring and furthermore the map R → R perfd is surjective.
• If S → R is an integral map, R perfd is an integral perfectoid ring.
Furthermore, the following property plays an essential role in this paper.
Theorem 2.9 (([BS22, Corollary 8.14])).If R perfd is concentrated in degree 0, it becomes an integral perfectoid ring.In this case, the map R → R perfd is the universal map to integral perfectoid rings.Namely, every map R → R ′ to an integral perfectoid ring R ′ uniquely factors through R → R perfd .

Uniform Completion
We use the notion of the uniform completion of Tate rings.In this section, we review the uniform completion outlined in [IS23].
Definition 3.1.A Tate ring A is uniform if the set of all power-bounded elements A • is bounded.
Any Tate ring has the structure of a seminormed ring as follows (see [NS22, Definition 2.26] for more details).
Definition 3.2.Let A := A 0 [1/t] be a Tate ring.Fix a real number c > 1.Then, we can define a seminorm The seminorm defines a seminormed ring (A, • A 0 ,t,c ).The topology of the seminormed ring (A, ) is equal to the topology of the Tate ring In particular, the topology induced from the norm • A 0 ,t,c does not depend on the choices of A 0 , t, and c.
So we write the seminorm • A 0 ,t,c as • for simplicity.
The spectral seminorm attached to this seminorm • is defined as By Fekete's subadditivity lemma, we have f sp ≤ f .Definition 3.4.Let A be a Tate ring.Fix a pair of definition (A 0 , (t)) of A and a ring of integral elements A + 0 of A such that A 0 ⊆ A + 0 .We define the following terminology.(1) The uniformization of A with respect to (A 0 , (t)) and A + 0 is the Tate ring (2) The uniform completion of A with respect to (A 0 , (t)) and A + 0 is the completion of the Tate ring A + 0 [1/t].This is in fact the Tate ring A + 0 [1/t] where A + 0 is the t-adic completion of A + 0 .
Remark 3.5.At first glance, the above definitions depend on the choices of (A 0 , (t)) and A + 0 .The motivation for these definitions is that we wanted to define them "functorially".Furthermore, even if we take a different pair of definition (A ′ 0 , (t ′ )) of A and a different ring of integral elements A ′ + 0 such that A 0 ⊆ A ′ + 0 , the uniformization of A with respect to (A ′ 0 , (t ′ )) and A ′ + 0 is isomorphic to the uniformization of A with respect to (A 0 , (t)) and [IS23,Lemma 5.5] and [NS22, Lemma 2.3]).In particular, the same statement is true for the uniform completion.So the next definitions are well-defined.Definition 3.6.Let A be a Tate ring.The uniformization A u of A (resp., uniform completion A u of A) is the uniformization (resp., uniform completion) of A with respect to a pair of definition (A 0 , (t)) and a ring of integral elements A + 0 of A such that A 0 ⊆ A + 0 .By the above Remark 3.5, these definitions are independent of the choices of (A 0 , (t)) and

Its isomorphism is obtained by the identity map on the abstract ring
Recall that the canonical map of Tate rings i : We record some lemmas as follows.
Lemma 3.8.Let A be a Tate ring.Then, the completion A u• of the set of all powerbounded elements A u• of A u is isomorphic to the set of all power-bounded elements (A u ) • of A u as topological ring.
Proof.Fix a pair of definition (A 0 , (t)) of A and a ring of integral elements A + 0 of A such that A 0 ⊆ A + 0 .By [NS22, Lemma 2.3], we have an inclusion t(A + 0 ) * A ⊆ A + 0 where (A + 0 ) * A is the complete integral closure of A + 0 in A. By [NS22, Proposition 2.4], the canonical map

is an isomorphism of Tate rings and this map induces an isomorphism of topological rings A
) has a ring of definition A + 0 (resp, A + 0 ) and the complete integral closure is equal to the set of all power-bounded elements by [NS22, Lemma 2.13], we have an isomorphism of topological

Lemma 3.9 (([IS23, Proposition 5.6])). Assume that a Tate ring
• is a ring of definition of A u , we have f p n ≤ c for a fixed c > 1 by Definition 3.2.In particular, f p n 1/p n ≤ c 1/p n for any n ∈ Z >0 .Taking the limit n → ∞,

Some Ring-Theoretic Properties of Pre-Perfectoid Pairs
Let R be an integral perfectoid ring and let π ∈ R be a perfectoid element.Our goal in this section is to convert a situation where π is a zero-divisor into a situation where it is a non-zero-divisor, following the approach of [And20, §2.3.2].Our argument is based on [And18,And20] and is similar to [Sch12,Bha17].If it is sufficient to consider only πtorsion-free rings (resp., p-torsion-free rings), the symbol (−) πtf defined in Definition 4.1 (resp., (−) ptf ) can be removed.
For the sake of generality, we define pre-perfectoid pairs as follows.
Definition 4.1.Let (S, π) be a pair such that S is a ring and π is an element of S which has a compatible sequence of p-power roots {π 1/p j } j≥0 in S and π p divides p in S. If the pth power map S/πS a →a p −−−→ S/π p S is isomorphism, we call such a pair (S, π) pre-perfectoid pair.
For a pre-perfectoid pair (S, π), the π-torsion-free quotient S πtf of S is defined as the quotient ring where S[π ∞ ] is the ideal of all π ∞ -torsion elements of S. Note that S ։ S πtf is an isomorphism if and only if π is a non-zero-divisor of S. In the case of S πtf is equal to the p-torsion-free quotient S ptf of S defined in Section 1.5.
For example, an integral perfectoid ring R and a perfectoid element π of R form a pre-perfectoid pair (R, π) because of Remark 2.3 (2).
Definition 4.2.Let (S, π) be a pre-perfectoid pair.An S-module M is called (π) For any S-module M , the set of almost elements of M is defined as surjective and (π) 1/p ∞ -almost injective.Furthermore, the inclusion Proof.If π is a non-zero-divisor of S, this lemma is similar to [Dos23, Proposition 2.16 (b)].We only have to make the same proof as [Sch12, Lemma 5.6], being careful that S is not necessarily π-torsion-free in our case.
First, we show that S * /πS * a →a p −−−→ S * /π p S * is injective.Let t ∈ S * be an element in the kernel of the p-th power map.There exists some t ′ ∈ S * such that By definition of S * , multiplying π 1/p n by the equation for each n ∈ Z >0 , we have Moreover, π 1/p n+1 t and π 1/p n t ′ are elements of S πtf .Then, there exist some elements s n+1 and s ′ n in S such that s n+1 /1 = π 1/p n+1 t and s ′ n /1 = π 1/p n t ′ in S πtf ⊆ S[1/π].By the above equation (4.3), we have In particular, π 1/p m+1 s n+1 is in the kernel of the p-th power map S/πS → S/π p S, which is zero by assumption of (S, π), and so we have Proof of (4.4).By definition, we have (S πtf ) * = S * and then the second equality is clear.Any element of π(S πtf ) * can be written as πt by using some element t ∈ (S πtf ) * .Because of π 1/p n (πt) = π(π 1/p n t) ∈ πS πtf for any n ∈ Z >0 , we have πt ∈ (πS πtf ) * and so π(S πtf ) * ⊆ (πS πtf ) * .
Conversely, take any element x ∈ (πS πtf ) * ⊆ S[1/π].Then, there exists t n ∈ S πtf such that π 1/p n x = πt n ∈ πS πtf for each n ∈ Z >0 .Since S[1/π] is π 1/p n -torsion-free, we have Second, we show that For each n ∈ Z >0 , we have π 1/p n x ∈ S πtf and thus there exists an s n ∈ S such that π 1/p n x = s n /1 ∈ S πtf .Since S/πS a →a p −−−→ S/π p S is surjective, there exists some s ′ n ∈ S such that (s ′ n ) p − s n is in π p S. Then, we have Theorem 4.9.For any pre-perfectoid pair (S, π), the induced pair (S * , π) is again a pre-perfectoid pair.
In particular, for any integral perfectoid ring R and a perfectoid element π ∈ R (not necessarily a non-zero-divisor), R * = π −1/p ∞ R is a π-torsion-free integral perfectoid ring with a perfectoid element π.
Remark 4.10.By Lemma 2.2, any integral perfectoid ring R has a compatible sequence of p-power roots {̟ 1/p j } j≥0 of ̟ ∈ R such that ̟ p is some unit multiple of p in R. Then R forms a pre-perfectoid pair (R, ̟).In particular, this pre-perfectoid pair (R, ̟) satisfies all the statements in this section.
where the vertical maps are canonical ones.Since (A 0 ) perfd, * is also an integral perfectoid ring by Theorem 4.9 and (A 0 ) perfd is a universal integral perfectoid ring over A 0 , the composite f • c is nothing but the canonical map c : (A 0 ) perfd → (A 0 ) perfd, * .For any x ∈ (A 0 ) perfd, * , there exists an element a ∈ (A 0 ) perfd such that c(a) = px in (A 0 ) perfd, * .Then, we have pf p-torsion-free, we have f (x) = x and we are done.
As a consequence of this lemma, we can show the following relation between (−) ptf and (−) perfd .
Corollary 5.3.Let A 0 be a derived p-complete algebra over some integral perfectoid ring.Assume that (A 0 ) perfd and (A ptf 0 ) perfd are (honest) integral perfectoid rings.Then the canonical map (A 0 ) perfd → (A ptf 0 ) perfd induces the isomorphism Since the above isomorphism prevents confusion when ((A 0 ) perfd ) ptf is written as (A 0 ) ptf perfd , we use this symbol (−) ptf perfd in the following.
In the proof of [Din22, Theorem 4.4], Dine states that a quotient of a perfectoid Tate ring by some ideal has the perfectoidization that is isomorphic to its uniform completion.
We reformulate the proof for the situation of integral perfectoid rings as follows.
Since A 0 is the p-adic completion of R, η : A u → (A 0 ) perfd [1/p] is a unique extension of A 0 → (A 0 ) ptf perfd .By Lemma 4.7 and the proof of Proposition 5.1, η induces the unique map (A u ) • → (A 0 ) perfd, * taken in Proposition 5.1 and this extends ϕ.
If R is p-torsion-free, A 0 = R is also p-torsion-free and so is ( In particular, any semiperfectoid ring satisfies the assumptions of Theorem 5.7 and so we have the following corollary. Corollary 5.9.Let R be a ring such that the p-adic completion R of R becomes a semiperfectoid ring which contains Z as a subring.Then ( R) ptf perfd is isomorphic to the p-adic completion C(R ptf ) of C(R ptf ).If R is p-torsion-free, we have ( R) perfd ∼ = C(R).
Proof.First, the semiperfectoid ring R has a surjective map from some integral perfectoid ring by the definition of semiperfectoid rings (Definition 2.4).Second, The perfectoidization ( R) perfd of R is an (honest) integral perfectoid ring by [BS22, Corollary 7.3 and Proposition 8.5].Finally, the p-adic completion C(R ptf ) of the p-root closure C(R ptf ) is an integral perfectoid ring by Remark 5.8.So R satisfies all assumptions of Theorem 5.7, and this completes the proof.

Connections between p-root Closure and Perfectoidization
We recall a mixed characteristic analog of the perfection of rings, which was introduced in [IS23].This construction includes an example from [Rob08, Section 4] which demonstrates a good behavior of p-root closure from the perspective of Fontaine rings.Construction 6.1.Let (R 0 , m, k) be a complete Noetherian local domain of mixed characteristic (0, p) with perfect residue field k and let p, x 2 , . . ., x n be any system of generators of the maximal ideal m such that p, x 2 , . . ., x d forms a system of parameters of R 0 .Choose compatible sequences of p-power roots (6.1) {p 1/p j } j≥0 , {x

)
The perfectoidization ( R) perfd of R is an (honest) integral perfectoid ring by [BS22, Corollary 7.3 and Proposition 8.5] or the rest of Section 2. (3) The p-adic completion C(R) of the p-root closure C(R) is an integral perfectoid ring by virtue of [CS23, Proposition 2.1.8](see Remark 5.8).

Lemma 3. 3
(([NS22, Lemma 2.29])).Let A = A 0 [1/t] be a uniform Tate ring.Then A • is equal to the unit disk A • sp ≤1 by the spectral seminorm in A. Next, we define the uniformization and uniform completion of Tate rings as in [Bha17, Exercise 7.2.6] and [KL15, Definition 2.8.13].
A 0 ) perfd by [Ma+22, Lemma A.2]. Then C(R ptf ) = C(R) is isomorphic to ( R) perfd .Remark 5.8.Let R be a (not necessarily p-adically complete) ring containing a compatible sequence of p-power roots {̟ 1/p j } j≥0 of ̟ ∈ R such that ̟ p is some unit multiple ofp in R. If the Frobenius map R/pR F − → R/pRis surjective, the p-adic completion C(R) is an integral perfectoid ring by [CS23, Proposition 2.1.8].Compare Remark 5.5 above.
almost by definition.This completes the proof.