Finite étale extension of Tate rings and decompletion of perfectoid algebras
aa r X i v : . [ m a t h . A C ] M a r FINITE ´ETALE EXTENSION OF TATE RINGS AND DECOMPLETION OFPERFECTOID ALGEBRAS
KEI NAKAZATO AND KAZUMA SHIMOMOTO
Abstract.
In this paper, we examine the behavior of ideal-adic separatedness and completenessunder certain ring extensions using trace map. Then we prove that adic completeness of a basering is hereditary to its ring extension under reasonable conditions. We aim to give many resultson ascent and descent of certain ring theoretic properties under completion. As an application, wegive conceptual details to the proof of the almost purity theorem for Witt-perfect rings by Davisand Kedlaya. Witt-perfect rings have the advantage that one does not need to assume that therings are complete and separated.
Contents
1. Introduction 12. Notation and preliminaries 32.1. Integrality and almost integrality 32.2. Tate rings 62.3. Preuniform rings and pairs 62.4. Non-archimedean seminorms 93. Witt-perfect and perfectoid algebras 123.1. Almost ring theory and semiperfect rings 123.2. Witt-perfect and perfectoid algebras 153.3. Almost Witt-perfect and almost perfectoid algebras 184. Finite ´etale extension 194.1. Finite ´etale extsnsion and completeness 194.2. Studies on preuniform pairs and the condition ( ∗ ) 215. The almost purity theorem for Witt-perfect rings 245.1. Almost ´etale ring map 245.2. Proof of the almost purity theorem 276. Appendix: A historical remark on the (almost) purity theorem 29References 301. Introduction
In the basic part of perfectoid geometry, one of the most fundamental tools that is frequentlyused is the
Almost purity theorem , which was first proved by Faltings for certain big algebras overdiscrete valuation rings coming from smooth algebras and by Scholze for perfectoid algebras over aperfectoid field. By definition, an (integral) perfectoid algebra is a p -torsion free, complete ring A Key words and phrases.
Adic-completion, almost purity, ´etale extension, Witt-perfect algebra.2010
Mathematics Subject Classification : 13A18, 13B22, 13B40, 13F35, 13J10. such that the Frobenius map on A / ( p ) is surjective with cokernel πA for which π p = pu for some u ∈ A × . One drawback is that one needs to work with p -adically complete ring, which prevents usfrom taking infinite integral extensions of p -adically complete rings directly. Roughly speaking, a Witt-perfect condition on a p -torsion free ring is defined in the same way as for perfectoid algebras,except that it need not be p -adically complete. This class of rings had been introduced by Davisand Kedlaya in [7] and [8]. However, a difficulty lurks in dealing with Witt-perfect rings, due tothe lack of tilting correspondence for those rings.The present note stems from authors’ endeavor to reaching deeper understanding of the papers[7] and [8]. Our aim is to prove some basic results and explain their consequences which are ofgreat importance in the situation where one wants to avoid taking p -adic completion. We alsomake extensive studies on the comparisons between (almost) Witt-perfect algebra and (almost)perfectoid algebras. It is clear from Andr´e’s work [1] that almost perfectoid algebras naturallyshow up in certain applications. All rings are assumed to be commutative with a unity. For a finiteprojective ring extension A ֒ → B , denote by Tr B/A : B → A the trace map; see the book [12] forthe construction. Let us state the main results; see Proposition 4.1, Proposition 4.2, Theorem 4.9and Theorem 5.6 below. Main Theorem 1.
Let A be a ring and let B be a finite ´etale A -algebra. Let A ⊂ A be a subringwith an ideal I ⊂ A . Let B be an A -subalgebra of B such that B = S n ≥ ( B : I n ) . Assumethat there exists an integer c > such that Tr B/A ( tm ) ∈ A for every t ∈ I c and every m ∈ B .(1) If A is I -adically separated, then so is B .(2) If A is I -adically complete, then so is B . Main Theorem 2.
Let ( R, I ) be a basic setup and let f : A → B be an R -algebra homomorphismwith an element t ∈ A . Denote by c A and c B the t -adic completions of A and B , respectively.Let b f : c A → c B be the R -algebra homomorphism induced by f . Assume that the morphism ofpairs f : ( A , ( t )) → ( B , ( t )) satisfies ( ∗ ) (cf. Definition 4.3). Then the following assertions hold.(1) The natural c A [ t ] -algebra homomorphism B [ t ] ⊗ A [ t ] c A [ t ] → c B [ t ] is an isomorphism.(2) b f : ( c A , ( t )) → ( c B , ( t )) also satisfies ( ∗ ) (cf. Definition 4.3).(3) The following conditions are equivalent. ( a ) B is I -almost finitely generated and I -almost projective over A . ( b ) c B is I -almost finitely generated and I -almost projective over c A .(4) The following conditions are equivalent. ( a ) f : A → B is I -almost finite ´etale. ( b ) b f : c A → c B is I -almost finite ´etale. An important consequence that follows from the above results is the almost purity theoremfor Witt-perfect rings; see Theorem 5.8. This statement was originally established by Davis andKedlaya which ultimately relies on the almost purity theorem by Kedlaya-Liu; see papers [7] and[8] and [19].
Corollary 1.1 (Almost purity) . Let A be a Witt-perfect ring and f : A → B be a ring map. Put I := √ pA . Assume that the morphism f : ( A , ( p )) → ( B , ( p )) satisfies ( ∗ ) (cf. Definition 4.3), A is integrally closed in A [ p ] , ( A ) + B [ p ] ⊂ B , and ( A , I ) is a basic setup. Then the followingassertions hold.(1) B is also Witt perfect.(2) f : A → B is I -almost finite ´etale. INITE ´ETALE EXTENSION OF TATE RINGS AND DECOMPLETION OF PERFECTOID ALGEBRAS 3
The structure of this paper goes as follows.In §
2, we give a review on Tate rings with their topological structure. As a preliminary to thestudy of uniform Banach rings, we introduce the notion of preuniformity for both Tate rings andpairs of rings; see Definition 2.12 and Definition 2.14. This notion is important when one needsto distinguish uniform Banach rings and uniform non-Banach rings and moreover, our definitionis of algebraic nature that is appealing to algebraists. Thus, their properties are studied withconnections to Banach rings.In §
3, we will use Fontaine’s version of perfectoid rings that was introduced in [11] as a gener-alization of Scholze’s perfectoid algebras; see Definition 3.9. We study its relation to Witt-perfectrings.In §
4, we study finite ´etale extensions over a Tate ring. In particular, the reader will find thata trace map becomes an essential tool for testing certain topological structure such as completeor separated, on module-finite algebras over Tate rings. The main results in this section areProposition 4.1, Proposition 4.2 and Theorem 4.9. We also prove some basic results on (complete)integral closure and their behavior under completion. We believe that some of these results areknown to experts. However, as they do not seem to be documented in existing literatures, we tryto give detailed proofs with maximal generality.In §
5, by specializing the results obtained in §
3, we complete the proof of the almost puritytheorem for Witt-perfect rings by reducing to the case of perfectoid rings by Kedlaya-Liu [19]; seeTheorem 5.6 and Theorem 5.8. A review on some basic definitions of almost ring theory is alsogiven, borrowing from Gabber-Ramero’s monograph [15].In §
6, we give some historical remarks on the almost purity theorems. The reason for the inclusionof this appendix is to help commutative algebraists to understand core ideas. No new results areproved here.Let us point out that the almost purity theorem without completion has also been establishedby Gabber-Ramero in [16], under the name of formal perfectoid rings . Although their treatmentis quite general, we think that our approach is short and concise. We plan to apply the result ofthis paper to the construction of almost Cohen-Macaulay algebras after establishing a variant ofAndr´e’s perfectoid abhyankar’s lemma in the forthcoming paper [22].2.
Notation and preliminaries
For the definition of perfectoid algebras, we follow the original version by Scholze [24] and itsessential extension by Fontaine [11]. There is, however, a more general version as introduced inthe paper [4]. A pair is meant to be a pair (
A, I ) consisting of a ring A and an ideal I ⊂ A . A morphism of pairs f : ( A, I ) → ( B, J ) is a ring map f : A → B such that I n ⊂ f − ( J ) for some n >
0. Let (
A, I ) be a pair, M be an A -module, and N be an A -submodule of M . Then the I -saturation of N in M is defined to be the A -submodule N I -sat = { m ∈ M | for any x ∈ I, there exists some n > x n m ∈ N } . In particular, for the ideal (0) ⊂ A , the I -saturation (0) I -sat denotes the ideal in A consisting of all I -torsion elements.2.1. Integrality and almost integrality.
Here we discuss the notion of integrality and almostintegrality.
Definition 2.1.
Let A ⊂ B be a ring extension. K.NAKAZATO AND K.SHIMOMOTO (1) An element b ∈ B is integral over A , if P ∞ n =0 A · b n is a finitely generated A -submodule of B .The set of all elements, denoted as C , of B that are integral over A forms an A -subalgebraof B . If A = C , then A is called integrally closed in B .(2) An element b ∈ B is almost integral over A , if P ∞ n =0 A · b n is contained in a finitely generated A -submodule of B . The set of all elements, denoted as C , of B that are almost integralover A forms an A -subalgebra of B , which is called the complete integral closure of A in B .If A = C , then A is called completely integrally closed in B .This definition can be extended to any ring homomorphism A → B in a natural way: Let A be a ring, let B be an A -algebra and let b ∈ B be an element. Then we say that b is integral (resp. almost integral ) over A , if b is integral (resp. almost integral) over the image of A in B .Unlike integral closure, the complete integral closure of an integral domain in its field of fractions isnot necessarily completely integrally closed in the same field of fractions; see [17] for such examples. Notation : For a ring A and an A -algebra B , we denote by A + B (resp. A ∗ B ) the A -subalgebra of B consisting of all elements that are integral (resp. almost integral) over A .We often use the following results. Lemma 2.2.
Let A be a ring with a nonzero divisor t . Let B be a t -torsion free A -algebra suchthat the induced A [ t ] -algebra B [ t ] is module-finite. Then one has t ( A ) ∗ B ⊂ ( A ) + B .Proof. Put A := A [ t ], B := B [ t ] and B ′ := ( A ) + B . Pick b ∈ ( A ) ∗ B . Then there exists a finitelygenerated A -submodule N ⊂ B such that b n belongs to N for every n >
0. On the other hand,since B is module-finite over A , we have B = B ′ [ t ]. Thus, t l N is contained in B ′ for some l > tb ) l (= t l b l ) lies in B ′ and therefore, so does tb by the definition of B ′ . Hence tb isintegral over A , as desired. (cid:3) Proposition 2.3.
Let A be a ring with a nonzero divisor t . Denote by c A the t -adic completionof A . Put A := A [ t ] , A ′ := c A [ t ] , A + := ( A ) + A , and A ◦ := ( A ) ∗ A .(1) Suppose that there exists some c ≥ for which t c A + ⊂ A (resp. t c A ◦ ⊂ A ). Then thefollowing assertions hold.(a) One has t c ( c A ) + A ′ ⊂ c A (resp. t c ( c A ) ∗ A ′ ⊂ c A ).(b) Denote by c A + and c A ◦ the t -adic completions. Then the inclusion map A ֒ → A + (resp. A ֒ → A ◦ ) induces an isomorphism A ′ ∼ = −→ c A + [ t ] (resp. A ′ ∼ = −→ c A ◦ [ t ] ) whose restrictionto ( c A ) + A ′ (resp. ( c A ) ∗ A ′ ) yields an isomorphism ( c A ) + A ′ ∼ = −→ c A + (resp. ( c A ) ∗ A ∼ = −→ c A ◦ ).(2) Conversely, if there exists some c ≥ for which t c ( c A ) + A ′ ⊂ c A (resp. t c ( c A ) ∗ A ′ ⊂ c A ), thenone has t c A + ⊂ A (resp. t c A ◦ ⊂ A ). To prove this, let us verify a fundamental lemma.
Lemma 2.4.
Let A be a ring and let I ⊂ A be a finitely generated ideal. Let f : N ֒ → M bean injective homomorphism between A -modules. Denote by c M and c N the I -adic completions of M and N , respectively. Let b f : c N → c M be the c A -linear map induced by f . Assume that thecokernel of f is annihilated by I m for some m ≥ . Then, b f is also injective and the cokernel of b f is annihilated by I m c A .Proof of Lemma 2.4. Consider the exact sequence: 0 → N f −→ M → L → L being thecokernel of f . By assumption, the A -module L is killed by I m . For an arbitrary n >
0, we have
INITE ´ETALE EXTENSION OF TATE RINGS AND DECOMPLETION OF PERFECTOID ALGEBRAS 5 the induced exact sequence:(2.1) 0 → N / ( I n M ∩ N ) → M /I n M → L /I n L → . Since lim ←− n N / ( I n M ∩ N ) ∼ = 0 and I m + n M ∩ N ⊂ I n N , the following sequence induced by (2 . → c N → c M → c L ∼ = L → , where c L denotes the I -adic completion of L . In particular, the cokernel of b f is killed by I m c A .This yields the assertion. (cid:3) Moreover, we need the following result from [2]; see also [26, Tag 0BNR].
Lemma 2.5 (Beauville-Laszlo) . Let A be a ring with a nonzero divisor t ∈ A and let c A be the t -adic completion. Then t is a nonzero divisor of c A and one has the commutative diagram: (2.2) A ψ −−−−→ c A ι y y ι ′ A [ t ] −−−−→ ψ t c A [ t ] that is cartesian. In other words, we have A ∼ = A [ t ] × c A [ t ] c A . Now let us start to prove Proposition 2.3.
Proof of Proposition 2.3.
Notice that c A , c A + and c A ◦ are t -torsion free (cf. Lemma 2.5).We first prove the assertion (1). By assumption, t c A + ⊂ A (resp. t c A ◦ ⊂ A ) for some c ≥ A ֒ → A + (resp. A ֒ → A ◦ ), we find that theinduced map c A → c A + (resp. c A → c A ◦ ) is injective and its cokernel is killed by t c . Therefore,we have a canonical c A -isomorphism A ′ ∼ = −→ c A + [ t ] (resp. A ′ ∼ = −→ c A ◦ [ t ]). Moreover, c A + (resp. c A ◦ )is integrally closed (resp. completely integrally closed) in c A + [ t ] (resp. c A ◦ [ t ]) by [3, Lemma 5.1.2](resp. [3, Lemma 5.1.3]). Hence we have inclusions c A ⊂ ( c A ) + A ′ ⊂ c A + (resp. c A ⊂ ( c A ) ∗ A ′ ⊂ c A ◦ ).Thus, we also have inclusions t c ( c A ) + A ′ ⊂ t c c A + ⊂ c A (resp. t c ( c A ) ∗ A ′ ⊂ t c c A ◦ ⊂ c A ), which yieldsthe assertion ( a ). In particular, { t n c A } n ≥ gives a fundamental system of open neighborhoods of0 ∈ ( c A ) + A ′ (resp. 0 ∈ ( c A ) ∗ A ′ ). Hence ( c A ) + A ′ (resp. ( c A ) ∗ A ′ ) is t -adically complete and separated.Thus, by the universal property of completion (cf. [14, Proposition 7.1.9 in Chapter 0]), we obtainthe A + -linear map (resp. A ◦ -linear map) c A + → ( c A ) + A ′ (resp. c A ◦ → ( c A ) ∗ A ′ ) and the composite c A + → ( c A ) + A ′ ֒ → c A + (resp. c A ◦ → ( c A ) ∗ A ′ ֒ → c A ◦ ) is the identity. Therefore ( c A ) + A ′ ֒ → c A + (resp.( c A ) ∗ A ′ ֒ → c A ◦ ) is an isomorphism, which yields the assertion ( b ).Next we show the assertion (2). We consider the commutative diagram (2.2). Keeping thenotation as above, assume that t c ( c A ) + A ′ ⊂ c A (resp. t c ( c A ) ∗ A ′ ⊂ c A ) for some c ≥
0. Pick an element x ∈ A + (resp. y ∈ A ◦ ). Then one can check that ψ t ( x ) ∈ c A [ t ] (resp. ψ t ( y ) ∈ c A [ t ]) is integral(resp. almost integral) over c A , because the diagram (2.2) commutes. Hence by assumption, ψ t ( t c x )(resp. ψ t ( t c y )) comes from c A . Thus, letting B (resp. C ) be the A -subalgebra of A [ t ] generatedby all elements of t c A + (resp. t c A ◦ ), we find that the composite map B ֒ → A [ t ] ψ t −→ c A [ t ] (resp. The complete integral closedness of A ◦ in A is a not trivial issue. However, this is easily checked from thehypothesis t c ( A ) ∗ A ⊂ A . We will discuss this condition as preuniformity in the following section. K.NAKAZATO AND K.SHIMOMOTO C ֒ → A [ t ] ψ t −→ c A [ t ]) factors through c A . Therefore, Lemma 2.5 implies that B ⊂ A (resp. C ⊂ A ). Consequently, we have t c A + ⊂ A (resp. t c A ◦ ⊂ A ), as wanted. (cid:3) Corollary 2.6.
Keep the notation as in Proposition 2.3. Then ( A ) + A = A (resp. ( A ) ∗ A = A ) ifand only if ( c A ) + A ′ = c A (resp. ( c A ) ∗ A ′ = c A ). Tate rings.
We first recall basic terms on Tate rings.
Definition 2.7 (Boundedness) . Let A be a topological ring. We say that a subset S ⊂ A is bounded , if for every open neighborhood U of 0 ∈ A there exists some open neighborhood V of0 ∈ A such that V · S ⊂ U , and we consider the empty set as being bounded. Definition 2.8 (Tate ring) . A topological ring A is called Tate , if there is an open subring A ⊂ A together with an element t ∈ A such that the topology on A induced from A is t -adic and t becomes a unit in A . A is called a ring of definition and t is called a pseudouniformizer and thepair ( A , ( t )) is called a pair of definition . Denote by A ◦ the set of powerbounded elements of A and by A ◦◦ the set of all topologically nilpotent elements of A . Then A ◦◦ is an ideal of A ◦ and A ◦ is a subring of A .Any Tate ring comes from a pair of a ring and a nonzero divisor in it as follows. Lemma 2.9.
The following assertions hold:(1) Let A be a ring with a nonzero divisor t , and put A := A [ t ] . Equip A with the lineartopology defined by { t n A } n ≥ . Then A is equipped with the structure as a Tate ring with aring of definition A and a pseudouniformizer t ∈ A .(2) Conversely, let A be a Tate ring with a ring of definition A and a pseudouniformizer t ∈ A . Then one has A = A [ t ] .Proof. (1) is easy to check. (2) is due to [18, Lemma 1.5]. (cid:3) Definition 2.10.
Let ( A , I ) be a pair. If I is generated by a nonzero divisor t ∈ A , then wecall the Tate ring A [ t ] in Lemma 2.9(1) the Tate ring associated to ( A , I ) . The notion of integrality is useful for describing important subrings of a Tate ring. The followinglemma should be well-known, but we insert its proof.
Lemma 2.11.
Let A be a Tate ring with a ring of definition A and a pseudouniformizer t ∈ A .(1) The complete integral closure of A in A coincides with A ◦ . In particular, if A ◦ is bounded,then A ◦ is completely integrally closed in A .(2) One has tA ◦ ⊂ ( A ) + A ⊂ A ◦ .Proof. Let us prove (1). For an element a ∈ A , a is almost integral over A if and only if thereexists some c > t c a m ∈ A for every m >
0. Here the latter condition is equivalentto the condition that a belongs to A ◦ . Hence A ◦ is the complete integral closure of A in A . Thesecond statement is clear, because any open and bounded subring of A forms a ring of definition.Next we prove (2). Pick a ∈ A ◦ . Then one has ( ta ) c = t c a c ∈ A for some c >
0. Hence ta ∈ ( A ) + A ,as desired. (cid:3) Preuniform rings and pairs.
We will discuss (pre)uniformity of Tate rings in many contextslater. Here we give the definition.
Definition 2.12 (Preuniform rings) . Let A be a Tate ring. We say that A is preuniform if A ◦ isbounded. We say that A is uniform if it is preuniform and complete and separated. INITE ´ETALE EXTENSION OF TATE RINGS AND DECOMPLETION OF PERFECTOID ALGEBRAS 7
Let us recall the following fact.
Lemma 2.13.
A separated preuniform Tate ring is reduced. In particular, a uniform Tate ring isreduced.Proof.
Let A be a separated preuniform Tate ring. Then, since A ◦ is bounded, we can take A ◦ asa ring of definition of A . Pick a pseudouniformizer t ∈ A ◦ of A . Then A ◦ is t -adically separatedbecause A is separated. Let f ∈ A be such that f k = 0 for some k >
0. Then for an integer n > t − n f is nilpotent, and therefore we get t − n f ∈ A ◦ . Hence f ∈ t n A ◦ . Since A ◦ is t -adicallyseparated and n is arbitrary, it follows that f = 0. (cid:3) We define preuniformity also for pairs that induce Tate rings.
Definition 2.14 (Preuniform pairs) . Let ( A , I ) be a pair of a ring A and an ideal I ⊂ A .(1) We say that ( A , I ) is preuniform , if A has a nonzero divisor t with the following property: • I = tA , and there exits some c > t c ( A ) + A [ t ] ⊂ A .(2) We say that ( A , I ) is uniform , if it is preuniform and A is I -adically complete andseparated.The above definition is derived from the following fact. Lemma 2.15.
Let A be a ring with a nonzero divisor t ∈ A and let A be the Tate ring associatedto ( A , ( t )) . Then A is preuniform (resp. uniform) if and only if the pair ( A , ( t )) is preuniform(resp. uniform).Proof. It follows immediately from Lemma 2.11(2). (cid:3)
Finitely generated modules over a Tate ring.
One can define a canonical topology on a finitelygenerated module over a Tate ring.
Lemma 2.16.
Let A be a Tate ring and let M be a finitely generated A -module. Take a ringof definition A ⊂ A , a pseudouniformizer t ∈ A and a finite generating set S of M over A .Let M ⊂ M be the A -submodule generated by S . Equip M with the linear topology defined by { t n M } n> .(1) The topology on M is independent of the choices of A , t and S .(2) For every finitely generated A -submodule N of M such that M = N [ t ] , the inducedtopology on N coincides with the t -adic topology.(3) Let f : M → N be a homomorphism of A -modules, where N is finitely generated. Equip N with the topology defined above. Then f is continuous.Proof. Since M = S n> ( M : ( t n )), for every finitely generated A -submodule N of M , thereexists some c > t c N ⊂ M . Hence (2) and (3) are easy to see. To show (1), let usconsider another data: ( A ′ , t ′ , S ′ , M ′ ). Pick an integer m >
0. Then it suffices to check that thereexists some m ′ > t ′ m ′ M ′ ⊂ t m M holds. Let N ′ ⊂ M be the A ′ -submodule generatedby S . Then t ′ c M ′ ⊂ N ′ for some c >
0, as M ′ is finitely generated. Meanwhile, since t m A ⊂ A is open, there exists some c > t ′ c A ′ ⊂ t m A and so t ′ c N ′ ⊂ t m M . Hence by putting m ′ := c + c , we obtain t ′ m ′ M ′ ⊂ t m M , as wanted. (cid:3) Notice that one can set M = A in Proposition 2.16, and the resulting topology on A coincideswith the original one. Now we can give a canonical Tate ring structure to any module-finite algebraextension of a Tate ring. K.NAKAZATO AND K.SHIMOMOTO
Lemma 2.17.
Let A be a Tate ring and let B be a module-finite A -algebra. Equip B with thetopology as in Lemma 2.16. Then B is equipped with the structure as a Tate ring with the followingproperty: • for every ring of definition A and every pseudouniformizer t ∈ A of A , there exists a ringof definition B of B that is an integral A -subalgebra of B with finitely many generatorsand t ∈ B is a pseudouniformizer of B .Proof. Take a system of generators x , . . . , x r of the A -module B . Multiplying each x i by a powerof t if necessary, we may assume that they are integral over A . Let B ⊂ B be an A -subalgebragenerated by x , . . . , x r . As B = B [ t ], we can introduce a Tate ring structure into B with a ringof definition B and a pseudouniformizer t ∈ B as in Lemma 2.9(1). Meanwhile, since each x i isintegral over A , B is a module-finite A -algebra. Hence the topology on B coincides with the onedefined by setting M = B in Lemma 2.16. (cid:3) If a finitely generated module M over a Tate ring admits a structure of a finitely generatedmodule over another Tate ring, then one can consider two canonical topologies on M . In thefollowing situation, these topologies coincide. Lemma 2.18.
Let A be a Tate ring and let B be a module-finite A -algebra. Equip B with thecanonical structure as a Tate ring as in Lemma 2.17. Let M be a finitely generated B -module.Then the following two topologies: • the canonical topology on M as a finitely generated A -module; • the canonical topology on M as a finitely generated B -module;coincide.Proof. Let A be a ring of definition of A and let t ∈ A be a pseudouniformizer of A . Then we cantake a ring of definition B of B that is finitely generated over A and satisfies B = B [ t ]. Let M be a finitely generated B -submodule of M such that M = M [ t ]. Then, also as an A -module, M is finitely generated and satisfies M = M [ t ]. Hence the assertion follows. (cid:3) Remark 2.19.
Let A be a ring with a nonzero divisor t ∈ A , and put A := A [ t ]. Let M be an A -module and let M ⊂ M be an A -submodule such that M = M [ t ]. Equip M with the lineartopology defined by { t n M } n ≥ (this situation already occurred in the above two lemmas). Nowlet us consider the completion M ′ of M (i.e. M ′ = lim ←− n M/t n M ). One applies the same operationsto both A and A . That is, c A := lim ←− n A /t n A and A ′ := b A = lim ←− n A/t n A . By [6, Chaptires III,Paragraph 6.5, Proposition 6, and Chapitres II, Paragraph 3.9, Corollaire 1], one checks that c A and b A are rings. Now since M is t -torsion free, so is the t -adic completion c M (cf. [25, Lemma4.2]). We equip ( c M )[ t ] with the linear topology defined by { t n c M } n ≥ . Then ( c M )[ t ] is completeand separated. Since M and c M are t -torsion free, one can check that the map( c M )[ 1 t ] → M ′ , ( m n mod t n M ) n ≥ t h (cid:0) m n + h t h mod t n M (cid:1) n ≥ (2.3)is well-defined. Indeed, consider the exact sequence 0 → M → M → M/M →
0. Then it inducesanother exact sequence 0 → M /t n M → M/t n M → M/M →
0. Taking inverse limits, withrespect to n ∈ N , we obtain an exact sequence0 → c M → M ′ → M/M → . (2.4)Since we are assuming the condition M = S n ≥ ( M : ( t n )), tensoring A [ t ] with the sequence(2.4), it follows that (2.3) is an isomorphism of A -modules. Next we observe that (2.3) gives an INITE ´ETALE EXTENSION OF TATE RINGS AND DECOMPLETION OF PERFECTOID ALGEBRAS 9 isomorphism of topological A -modules. The map (2.3) is a continuous A -homomorphism, and themap M → c M induces the continuous A -homomorphism M → ( c M )[ t ]. Moreover, the resultingdiagram ( c M )[ t ] / / M ′ M c c ●●●●●●●●● > > ⑥⑥⑥⑥⑥⑥⑥⑥ commutes. Hence it follows from the universality of completion (cf. [14, Proposition 7.1.9 in Chapter0]) that the map (2.3) is an A -isomorphism that is also a homeomorphism.Next let us consider a base extension of a module-finite algebra over a Tate ring. Then one maydefine two types of canonical topologies on it, but they are the same. Lemma 2.20.
Let A be a Tate ring, let ( A , ( t )) be a pair of definition of A and let A → A ′ be acontinuous ring map between Tate rings. Let B be a module-finite A -algebra, and set B ′ := B ⊗ A A ′ .Equip B (resp. B ′ ) with the canonical topology by regarding it as a module-finite A -algebra (resp. A ′ -algebra). Let B be a ring of definition of B that is an A -subalgebra of B , and let A ′ be aring of definition of A ′ that is an A -subalgebra of A ′ . Let B ′ be the image of the natural map B ⊗ A A ′ → B ′ . Then ( B ′ , ( t )) is a pair of definition of B ′ .Proof. Since A = A [ t ], A ′ = A ′ [ t ], and B = B [ t ], we have B ′ = B ′ [ t ]. Thus by Lemma 2.9(1), itsuffices to check that { t n B ′ } n ≥ forms a fundamental system of neighborhoods of 0 ∈ B ′ . Now wemay assume that B is generated by finitely many elements x , . . . , x s ∈ B as an A -submodule of B (such a ring of definition exists by the proof of Lemma 2.17). Put x ′ i := x i ⊗ A A ′ for i = 1 , . . . , s .Then B ′ is generated by x ′ , . . . , x ′ s as an A ′ -submodule of B ′ . Thus, since B ′ = B ′ [ t ], the assertionfollows. (cid:3) Non-archimedean seminorms.
Here we give a brief review and some new notation on non-archimedean seminorms. Our basic references are [5, Chapter 1], [14, Chapter 2, Appendix C] and[19, Chapter 2].
Definition 2.21.
Let A be a commutative ring.(1) A function || · || : A → R ≥ is called a ( non-archimedean ) seminorm on A , if it satisfies thefollowing conditions.(a) || || = 0.(b) || f − g || ≤ max {|| f || , || g ||} for every f, g ∈ A .(c) || || ≤ || f g || ≤ || f |||| g || for every f, g ∈ A .A seminorm || · || on A is called a norm , if it satisfies the following condition.(a’) For f ∈ A , one has || f || = 0 if and only if f = 0.(2) Let || · || and || · || ′ be seminorms on A . We say that || · || and || · || ′ are equivalent (or || · || is equivalent to || · || ′ ), if there exist real numbers C, C ′ > || f || ′ ≤ C || f || ≤ C ′ || f || ′ for every f ∈ A . We mean by || · || ∼ || · || ′ that || · || is equivalent to || · || ′ .(3) Let || · || be a seminorm on A . We say that f ∈ A is powermultiplicative with respect to || · || ,if || f n || = || f || n holds for every n >
0. We say that f ∈ A is multiplicative with respect to || · || , if || f g || = || f |||| g || holds for every g ∈ A . We say that || · || is powermultiplicative (resp. multiplicative ), if any f ∈ A is powermultiplicative (resp. multiplicative) with respect to || · || .Here we list some basic facts on seminorms. Lemma 2.22.
Let A be a ring and let || · || , || · || ′ , and || · || ′′ be seminorms on A .(1) If || · || ∼ || · || ′ and || · || ′ ∼ || · || ′′ , then || · || ∼ || · || ′′ .(2) If || · || and || · || ′ are powermultiplicative and || · || ∼ || · || ′ , then || · || = || · || ′ .(3) Let u be a unit in A . Suppose that || · || is not identically zero. Then || u || 6 = 0 . Moreover, u is multiplicative with respect to || · || if and only if || u − || = || u || − .Proof. The assertions (1) and (2) are easy to check. (3) follows from [5, § (cid:3) Let A be a ring, and let || · || be a seminorm on A . We put F r := { f ∈ A | || f || ≤ r } for everypositive real number r . Then F forms a subring of A , and each F r forms an F -submodule of A .Hence one can define the linear topology on A such that the filtration { F r } r> forms a fundamentalsystem of open neighborhoods of 0 ∈ A . Moreover, the ring A equipped with this topology is atopological ring. For this topological ring A , a subset S ⊂ A is bounded (Definition 2.7) if andonly if S ⊂ F r for some r >
0. If two seminorms || · || and || · || ′ on A are equivalent, then theydefine the same topology on A (but the converse does not necessarily hold). The following class ofseminorms is useful for characterization of Tate rings. Definition 2.23.
Let A be a ring with a nonzero divisor t ∈ A . We say that a seminorm || · || on A [ t ] is associated to ( A , ( t )), if there exists a real number c > || · || is equivalentto the seminorm || · || A , ( t ) ,c defined by || f || A , ( t ) ,c := ( c min { m ∈ Z | t m f ∈ A } ( f / ∈ T ∞ n =0 t n A )0 ( f ∈ T ∞ n =0 t n A ) . Lemma 2.24.
Let A be a topological ring, let t ∈ A be an element and let A be a subring of A .Then the following conditions are equivalent. ( a ) A is a Tate ring with a ring of definition A and a pseudouniformizer t ∈ A . ( b ) t ∈ A is a unit, and there exists some seminorm || · || on A with the following properties: • the topology on A is induced by || · || ; • || t || < , and t is multiplicative with respect to || · || ; • A = { f ∈ A | || f || ≤ } .Proof. To see ( a ) ⇒ ( b ), it is enough to consider the seminorm || · || A , ( t ) , . Conversely, if ( b ) issatisfied, then t n A = { f ∈ A | || f || ≤ || t || n } for every n > a ) from ( b ). (cid:3) For a ring A and a seminorm || · || on A , we define a function || · || sp : A → R ≥ by || f || sp := inf n ≥ || f n || n ( f ∈ A ) . || · || sp has the following properties. Lemma 2.25 ([5, § . Let A be a ring and let || · || be a seminorm on A .(1) One has || f || sp = lim n →∞ || f n || n and || f || sp ≤ || f || for every f ∈ A and || · || sp is apowermultiplicative seminorm on A .(2) If f ∈ A is multiplicative with respect to || · || , then f is also multiplicative with respect to || · || sp . INITE ´ETALE EXTENSION OF TATE RINGS AND DECOMPLETION OF PERFECTOID ALGEBRAS 11
We call || · || sp the spectral seminorm associated to || · || . Using spectral seminorms, we obtainthe following characterization of preuniformity. Lemma 2.26.
Let A be a ring with a nonzero divisor t ∈ A .(1) t ∈ A is multiplicative with respect to the seminorm || · || A , ( t ) ,c for every c > .(2) The following conditions are equivalent. ( a ) ( A , ( t )) is preuniform. ( b ) For any seminorm || · || associated to ( A , ( t )) , one has || · || ∼ || · || sp (or equivalently, || · || is equivalent to a powermultiplicative seminorm).Proof. The assertion (1) is clear. Let us prove (2). To see ( b ) ⇒ ( a ), it is enough to considerthe topology induced by a powermultiplicative seminorm. Here we show ( a ) ⇒ ( b ). Since A ◦ isbounded with respect to (the topology induced by) || · || , we may assume that || · || = || · || A ◦ , ( t ) , .Then t is multiplicative with respect to || · || and || · || sp . Thus, it suffices to show the existence ofsome constants C, C ′ > C || a || ≤ || a || sp ≤ C ′ || a || for an arbitrary a ∈ A ◦ \ tA ◦ . Now a n / ∈ t n A ◦ holds for an arbitrary n >
0; otherwise, ( t − a ) l would belong to A ◦ for some l > a ∈ tA ◦ as A ◦ is integrally closed in A [ t ]. Hence || t || < || a n || n and therefore, || t || ≤ || a || sp ≤
1. Thus we have 2 − || a || ≤ || a || sp ≤ || a || , as wanted. (cid:3) Moreover, if A is a valuation ring V and V [ t ] is a field, then || · || sp is a multiplicative norm. Lemma 2.27.
Let V be a valuation ring and assume that there is a nonzero element t ∈ V forwhich V is t -adically separated. Let || · || : V [ t ] → R ≥ be a seminorm associated to ( V, ( t )) . Thenthe spectral seminorm || · || sp is multiplicative.Proof. Since V is t -adically separated, it follows that K := V [ t ] is the field of fractions of V by [14,Proposition 6.7.2]. In view of Lemma 2.22(1), Lemma 2.22(2) and Lemma 2.26(2), we may assumethat || · || = || · || V, ( t ) ,c for a real number c >
1. It suffices to prove that any f ∈ K is multiplicativewith respect to || · || sp . As clearly 0 ∈ K is multiplicative, we assume that f = 0. By Lemma2.22(3), we are reduced to showing that || f − || sp = || f || − . Pick an arbitrary g ∈ K × and put λ ( g ) := min { m ∈ Z | t m g ∈ V } . Since V is a valuation ring, we have t − ( λ ( g ) − V ⊂ gV ⊂ t − λ ( g ) V .It implies that t λ ( g ) V ⊂ g − V ⊂ t λ ( g ) − V . Therefore, we find that || g || − = c − λ ( g ) ≤ || g − || ≤ c − ( λ ( g ) − = c || g || − . Thus we have ( || f n || n ) − ≤ || f − n || n ≤ c n ( || f n || n ) − for every n >
0. Taking the limits, we obtain || f || − = || f − || sp , as wanted. (cid:3) Now let us recall
Banach rings . Definition 2.28 (Banach rings) . (1) A Banach ring is a ring R equipped with a norm || · || such that R is complete with respectto the topology defined by || · || .(2) Let R and S be Banach rings and denote by || · || R and || · || S the norms on R and S ,respectively. We say that a ring homomorphism ϕ : R → S is bounded , if one has || ϕ ( f ) || S ≤|| f || R for every f ∈ R (notice that this property is stable under composition).(3) Let R be a Banach ring. A Banach R -algebra is a Banach ring S equipped with a boundedhomomorphism ϕ : R → S .(4) Let A be a ring with a nonzero divisor t ∈ A that is t -adically complete and separated.We say that a Banach ring R is associated to ( A , ( t )), if the underlying ring R is equal to A [ t ] and the norm on R is associated to ( A , ( t )). From now on, we view a Banach ring also as a (complete and separated) topological ring byconsidering the topology defined by the norm. Then we can say that a Banach ring associated to( A , ( t )) in Definition 2.28(3) and the Tate ring associated to ( A , ( t )) have the same topologicalring structure. Moreover, we use the following notation: for a Banach ring R with a norm || · || , weequip the ring R [ T p ∞ ] with the norm ||·|| Gauss defined by || P h ∈ Z [ p ] r h T h || Gauss := sup h ∈ Z [ p ] {|| r h ||} (where r h ∈ R ), and denote by R h T p ∞ i the resulting Banach ring obtained by completion. Lemma 2.29.
Let A be a ring with a nonzero divisor t ∈ A and let B be a t -torsion free A -algebra. Assume that A and B are t -adically complete and separated. Then there exist Banachrings R and S with the following properties: • R is associated to ( A , ( t )) and S is associated to ( B , ( t )) ; • the ring homomorphism R → S induced by the homomorphism A → B is bounded.If further ( A , ( t )) and ( B , ( t )) are uniform, then one can take R so that the norm on R ispowermultiplicative.Proof. To see the first assertion, it suffices to take a real number c > A [ t ] and B [ t ]with the norms || · || A , ( t ) ,c and || · || B , ( t ) ,c , respectively. To check the last assertion, it is enough toreplace || · || A , ( t ) ,c and || · || B , ( t ) ,c with their respective spectral norms. (cid:3) Witt-perfect and perfectoid algebras
Almost ring theory and semiperfect rings.
We start with basic part of the theory ofalmost rings and modules. We say that a pair (
R, I ) is a basic setup , if I is an ideal of a ring R , I = I and I is a flat R -module. An R -module map f : M → N is said to be I -almost injective (resp. I -almost surjective ), if the kernel (resp. cokernel) of f is annihilated by I . A generalreference is the book [15]. We will be mainly concerned with the following cases: • ( R, I ) = ( Z [ T p ∞ ] , ( T ) p ∞ ). • ( R, I ) = ( K ◦ h T p ∞ i , T p ∞ K ◦◦ K ◦ h T p ∞ i ), where K is a perfectoid field. • ( R, I ) = ( A , I ), where A is a ring of definition of a Tate ring A (see Example 3.1 below). Example . Fix a prime number p >
0. Let A be a ring with a sequence of nonzero divisors { t n } n ≥ such that A is integrally closed in A := A [ t ] and assume that for every n ≥ t pn +1 = t n u n for some unit u n ∈ A × . Denote by I the ideal p ( t ) ⊂ A . Then the pair ( A , I ) isa basic setup. Let us observe it. Pick x ∈ I . Then we have x p m = t a for some m > a ∈ A .Thus, since t p m m = t u for some unit u ∈ A , an equality ( xt m ) p m = au − holds in A . Hence x lies in t m A , because A is integrally closed in A . Consequently, we have I = lim −→ n t n A . Therefore, wefind that I = I p and I is flat over A .We give definitions of (almost) semiperfect rings that include both classes of Witt-perfect andperfectoid algebras. Definition 3.2 (Almost semiperfect ring) . Let A be a ring and fix a prime number p > A is perfect (resp. semiperfect ), if the Frobenius endomorphism on A / ( p ) isbijective (resp. surjective). The flatness of I is important for developing a solid theory of almost rings and modules. INITE ´ETALE EXTENSION OF TATE RINGS AND DECOMPLETION OF PERFECTOID ALGEBRAS 13 (2) Assume that A is a Z [ T p ∞ ]-algebra with a basic setup ( Z [ T p ∞ ] , ( T ) p ∞ ). We say that A is ( T ) p ∞ -almost semiperfect , if the Frobenius endomorphism on A / ( p ) is ( T ) p ∞ -almostsurjective.First we give several lemmas on (almost) semiperfect rings for later use. For an element t ina ring A , we say that A is t -adically Zariskian , if t is contained in the Jacobson radical of A ;see [14] and [27] for details on Zariskian geometry. Notice that for any ring, one can define atrivial Z [ T p ∞ ]-algebra structure by assigning the unity to each T pn . Over such a Z [ T p ∞ ]-algebra,( T ) p ∞ -almost semiperfectness is equivalent to semiperfectness. Lemma 3.3.
Let A be a ring with a nonzero divisor t such that p ∈ t p A . Assume that A is t -adically Zariskian and A / ( t p ) is semiperfect. Then there exists a sequence { t n } n ≥ in A suchthat t = t and for every n ≥ , we have t pn +1 = t n u n for some unit u n ∈ A × .Proof. We carry out the proof by induction. Put t := t . Then, since A / ( t p ) is semiperfect, wefind a , b ∈ A for which a p = t + t p b = t (1 + t p − b ). Here 1 + t p − b is a unit in A , because A is t -adically Zariskian. Hence we can take t = a . Next pick an integer m > n ≤ m −
1. Take a m , b m ∈ A for which t m = a pm + t p b m . Now( t p m m ) = ( t ) as ideals by assumption and therefore, we have c m ∈ A for which t = t m c m . Then itholds that a pm = t m (1 + t p − m c pm b m ) and 1 + t p − m c pm b m ∈ A × . Hence we can take t m +1 = a m , whichcompletes the proof. (cid:3) Lemma 3.4.
Let A be a Z [ T p ∞ ] -algebra with a basic setup ( Z [ T p ∞ ] , ( T ) p ∞ ) . Then the followingconditions are equivalent.(1) A is ( T ) p ∞ -almost semiperfect.(2) There is a semiperfect Z [ T p ∞ ] -algebra B with the following property: A / ( p ) is ( T ) p ∞ -almost isomorphic to B / ( p ) as Z [ T p ∞ ] -algebras.Proof. First we assume (1). We denote by g pn the image of T pn in A / ( p ) for every n ≥ A ♭ := lim ←− x x p A / ( p ), together with a ring map Z [ T p ∞ ] → A ♭ that assigns ( g, g p , g p , . . . ) ∈ A ♭ to T . Let Φ A : A ♭ → A / ( p ) be the projection map definedby the rule ( a , a , a , . . . ) a . Then the induced map A ♭ / Ker(Φ A ) ֒ → A / ( p ) is a ( T ) p ∞ -almost isomorphism and A ♭ / Ker(Φ A ) is semiperfect. Hence the implication (1) ⇒ (2) follows.The converse is easy to check. (cid:3) For a surjective ring map A ։ B with B semiperfect, clearly the semiperfectness does not liftto A in general. On the other hand, in the situations we deal with later, the following assertionholds. Lemma 3.5.
Let A be a Z [ T p ∞ ] -algebra with a nonzero divisor ̟ such that p ∈ ̟ p A . Assumethat ̟ admits a p -th root ̟ p ∈ A . Then the following assertions hold.(1) A / ( ̟ ) is ( T ) p ∞ -almost semiperfect if and only if A / ( ̟ p ) is ( T ) p ∞ -almost semiperfect.(2) Equip A with the ̟ -adic topology and assume further that ( p ) ⊂ A is closed and A iscomplete and separated. Then A / ( ̟ p ) is ( T ) p ∞ -almost semiperfect if and only if A / ( p ) is ( T ) p ∞ -almost semiperfect. Proof. If A / ( ̟ p ) (resp. A / ( p )) is ( T ) p ∞ -almost semiperfect, then clearly so is A / ( ̟ ) (resp. A / ( ̟ p )). Thus, it suffices to prove the inverse implications. Fix an arbitrary integer n > g pk := T pk · ∈ A for every k ≥
0. Assume that A / ( ̟ ) is ( T ) p ∞ -almost semiper-fect. Pick an element a ∈ A , and put a ′ := g pn a . Then by assumption, there exist some a , b ∈ A such that g p − pn +1 a = a p + ̟b . Multiplying both sides by g pn +1 , we obtain a ′ = g pn +1 a p + ̟ ( g pn +1 b ). Similarly, we can find some a , b ∈ A such that g pn +1 b = g pn +2 a p + ̟ ( g pn +2 b ).This procedure yields the following assertion: if a system of elements a , . . . , a m ∈ A satis-fies a ′ ≡ P mi =0 g pn + i +1 a pi ̟ i mod ( g pn + m +1 ̟ m +1 ), then there exists some a n +1 ∈ A for which a ′ ≡ P m +1 i =0 g pn + i +1 a pi ̟ i mod ( g pn + m +2 ̟ m +2 ). Hence by axiom of choice, we obtain a sequence { a m } m ≥ in A such that a ′ ≡ P mi =0 g pn + i +1 a pi ̟ i mod ( g pn + m +1 ̟ m +1 ) for every m ≥
0. In par-ticular, we have a ′ ≡ p − X i =0 g pn + i +1 a pi ̟ i ≡ (cid:18) p − X i =0 g pn + i +2 a i ̟ ip (cid:19) p mod ( ̟ p ) , which yields (1). To prove (2), we equip A with the ̟ -adic topology, and assume further that( p ) ⊂ A is closed and A is complete and separated. Set b m := P mi =0 g pn + i +2 a i ̟ ip ( m ≥
0) and b := lim m →∞ b m ∈ A . Then P mi =0 g n + i +1 a pi ̟ i − b pm ∈ ( p ) for every m . Hence it follows that a ′ − b p = lim m →∞ m X i =0 g n + i +1 a pi ̟ i − lim m →∞ b pm = lim m →∞ (cid:16) m X i =0 g n + i +1 a pi ̟ i − b pm (cid:17) ∈ ( p ) , because ( p ) ⊂ A is a closed ideal and so (2) follows. (cid:3) Corollary 3.6.
Let A be a ring with a nonzero divisor ̟ such that p ∈ ̟ p A and A is integrallyclosed in A [ ̟ ] and let g ∈ A be an element. Assume that A admits compatible systems of p -power roots ̟ pn , g pn ∈ A . Then the following conditions are equivalent. ( a ) The Frobenius endomorphism on A / ( ̟ p ) is ( g ) p ∞ -almost surjective. ( b ) The Frobenius endomorphism on A / ( ̟ p ) is ( ̟g ) p ∞ -almost surjective.Proof. ( a ) ⇒ ( b ) is clear. To show the converse, we assume ( b ). Fix an arbitrary integer n >
0, andput ̟ − pk := ( ̟ pk ) − for every k ≥
1. Pick a ∈ A . Then there exist elements b, c ∈ A such that ̟ pn g pn a = b p + ̟ p c and therefore, g pn a = ( ̟ − pn +1 b ) p + ̟ ( ̟ − pn ̟ p − c ) and ̟ − pn ̟ p − c belongsto A . Moreover, ̟ − pn +1 b belongs to A , because ( ̟ − pn +1 b ) p ∈ A and A is integrally closed in A [ ̟ ]. Thus we see that the Frobenius endomorphism on A / ( ̟ ) is ( g ) p ∞ -almost surjective andLemma 3.5 implies ( a ), as wanted. (cid:3) Lemma 3.5 also implies the following fact. Here notice that A + and A ◦ are essentially differentin general. Lemma 3.7.
Let A be a Tate ring with a topologically nilpotent unit t ∈ A . Let A + be a Z [ T p ∞ ] -algebra that is an open and integrally closed subring of A ◦ . Put ̟ := t p and assume that p ∈ ̟ p A + .Then the following conditions are equivalent. ( a ) A ◦ / ( ̟ p ) is ( T ) p ∞ -almost semiperfect. In [24], the distinction between A + and A ◦ does not cause any serious issue. INITE ´ETALE EXTENSION OF TATE RINGS AND DECOMPLETION OF PERFECTOID ALGEBRAS 15 ( b ) A + / ( ̟ p ) is ( T ) p ∞ -almost semiperfect.Proof. Fix an arbitrary integer n > g pn := T pn · ∈ A + . Assume that A ◦ / ( ̟ p ) is ( T ) p ∞ -almost semiperfect. Pick a ∈ A + . Then g pn a = b p + ̟ p c for some b, c ∈ A ◦ . Since ̟ p − c ∈ A + , wefind that b p ∈ A + which gives b ∈ A + . Hence A + / ( ̟ ) is ( T ) p ∞ -almost semiperfect, which impliesthat A + / ( ̟ p ) is also ( T ) p ∞ -almost semiperfect in view of Lemma 3.5. Hence ( a ) ⇒ ( b ) holds.To show the converse, assume that A + / ( ̟ p ) is ( T ) p ∞ -almost semiperfect. Pick d ∈ A ◦ . Then ̟d ∈ A + and thus g pn ̟d = e p + ̟ p f for some e, f ∈ A + . Then g pn d = ( et ) p + ̟ p − f . Since A ◦ isintegrally closed in A , we obtain et ∈ A ◦ . Hence A ◦ / ( ̟ ) is ( T ) p ∞ -almost semiperfect. Therefore, A ◦ / ( ̟ p ) is semiperfect in view of Lemma 3.5, as required. (cid:3) Proposition 3.8.
Let A be a p -torsion free semiperfect ring and assume that a finite group G acts on A as ring automorphisms with | G | invertible on A . Then the ring of invariants A G isalso semiperfect.Proof. Let A ♭ := lim ←− x x p A / ( p ). Then there is an exact sequence: A ♭ → A / ( p ) →
0. As | G | is invertible on A , the Reynolds operator: σ ( x ) := | G | P g ∈ G g ( x ) gives a section of the naturalinclusion ( A / ( p )) G ֒ → A / ( p ). Moreover, the G -action extends to A ♭ that is compatible with theabove exact sequence. So we get a surjection ( A ♭ ) G ։ ( A / ( p )) G . It is easy to see that ( A ♭ ) G is aperfect ring. Applying the functor of G -invariants to the short exact sequence: 0 → A p −→ A → A / ( p ) →
0, we have an isomorphism A G / ( p ) ∼ = ( A / ( p )) G . As A G / ( p ) is the surjective image ofthe perfect ring ( A ♭ ) G , it follows that A G is semiperfect. (cid:3) Witt-perfect and perfectoid algebras.
We now recall Fontaine’s perfectoid rings; see [11]for reference.
Definition 3.9 (Fontaine) . We say that a Banach ring R is perfectoid , if R is uniform and R hasa topologically nilpotent unit ̟ such that p ∈ ̟ p R ◦ and R ◦ / ( ̟ p ) is semiperfect. We call such ̟ ∈ R a perfectoid pseudouniformizer of R .Notice that for a Banach ring R , it only depends on the topological ring structure whether R isperfectoid or not. Definition 3.10.
Let R be a perfectoid Banach ring.(1) A perfectoid field is a perfectoid Banach ring K that is a field and whose topology can bedefined by a multiplicative norm on K .(2) A perfectoid R -algebra is a Banach R -algebra that is perfectoid.Notice that our definition of a perfectoid field and a perfectoid K -algebra (where K is a perfectoidfield) coincides with Scholze’s original one. To see this, it suffices to check the following. Lemma 3.11.
Let K be a field equipped with a norm || · || .(1) The following conditions are equivalent. ( a ) K is a perfectoid field. ( b ) K is complete, K ◦ is a non-Noetherian valuation ring of rank , || · || is equivalent toan absolute value associated to K ◦ , || p || < and K ◦ / ( p ) is semiperfect.(2) Suppose that K is a perfectoid field, and let R be a Banach K -algebra. Then the followingconditions are equivalent. ( a ) R is a perfectoid K -algebra. ( b ) R is uniform and for every ̟ ∈ K such that || p || ≤ || ̟ || < , R ◦ / ( ̟ ) is semiperfect.Proof. (1): Let us assume ( b ) first. Then by [24, Lemma 3.2], we have some ̟ ∈ K ◦ for which p ∈ ̟ p K ◦ . Hence it is easily seen that ( a ) holds. Next we assume ( a ) conversely. Then K iscomplete and K ◦ is a valuation ring of rank 1 by assumption. Now || · || ∼ || · || sp by Lemma 2.26(2),and || · || sp coincides with an absolute value associated to K ◦ by Lemma 2.27. Moreover, || · || sp isnot discretely valued in view of [21, Lemma 1.1(iii)]. The remaining part follows from Lemma 3.5,because ( p ) ⊂ K ◦ is closed with respect to the topology defined by || · || sp .(2): ( b ) ⇒ ( a ) is clear. Now we deduce ( b ) from ( a ). Since K is a perfectoid field, p ∈ K is atopologically nilpotent unit or equal to zero. The same assertion also holds for p ∈ R , as K → R is continuous. Hence ( p ) ⊂ R ◦ is closed with respect to the induced topology from R . Thus, ( a )implies that R ◦ / ( p ) is semiperfect in view of Lemma 3.5. Therefore ( a ) ⇒ ( b ) holds. (cid:3) Any perfectoid pseudouniformizer can be replaced so that the following statement holds.
Lemma 3.12.
Let A be a perfectoid Banach ring and let A + be an open and integrally closedsubring of A . Then the following assertions hold.(1) There exists some ̟ ′ ∈ A + such that p ∈ ̟ ′ p A + and A + / ( ̟ ′ p ) is semiperfect.(2) Assume further that p ∈ A is a topologically nilpotent unit. Then A + / ( p ) is semiperfect,and there exists some t ∈ A + such that p = t p u for some unit u ∈ A + . Moreover, thereexists some s ∈ A + such that s p ≡ p mod p A + .Proof. (1): Let ̟ ∈ A ◦ be a perfectoid pseudouniformizer of A . Then by Lemma 3.3, we find some ̟ ∈ A ◦ such that ̟ = ̟ p u for some unit u ∈ A ◦ (notice that ̟ is a topologically nilpotentunit in A ). Hence p ∈ ̟ p ̟ p − A ◦ and since ̟ p − A ◦ ⊂ A + , we have p ∈ ̟ p A + . Moreover, since A ◦ / ( ̟ p ) is semiperfect, A + / ( ̟ p ) is also semiperfect by Lemma 3.7. So letting ̟ ′ := ̟ completes(1).(2): Next assume further that p ∈ A is a topologically nilpotent unit. Then, since ( p ) ⊂ A + isclosed with respect to the ̟ -adic topology, A + / ( p ) is semiperfect in view of Lemma 3.5(2). Toprove the existence of t ∈ A + and a unit u ∈ A + such that p = t p u , we take a ∈ A + for which p = ̟ p a . Then we have b, c ∈ A + that satisfy a = b p + pc and so p (1 − ̟ p c ) = ( ̟ b ) p . Thus, since u := 1 − ̟ p c ∈ A + is a unit, it suffices to take t = ̟ b . Finally, let us prove the existence of s ∈ A + as in the assertion. Take d, e ∈ A + such that u = d p + pe . Then p = t p ( d p + pe ) and therefore, p = ( td ) p + pt p e . Here notice that t p = pu − ∈ pA + . Thus we have ( td ) p ≡ p (mod p A + ), aswanted. (cid:3) Example . We exhibit a specific example of perfectoid ring in the sense of Fontaine, but does notfit into the original definition of perfectoid algebras by Scholze. Fix a prime number p >
0. For aninteger n >
0, let R n := Z p [[ T ]][ p pn , T pn ], denote by R n [( pT ) pn ] the R n -subalgebra R n [( T pn ) − p pn ]of R n [( T pn ) − ] = R n [ T ] and let R ∞ [( pT ) p ∞ ] be the ring S n> R n [( pT ) pn ]. Since R n is a regular localring with a regular system of parameters p pn , T pn , one can easily show that R n [( pT ) pn ] is completelyintegrally closed in R n [ T ]. Hence R ∞ [( pT ) p ∞ ] is completely integrally closed in R ∞ [( pT ) p ∞ ][ T ].Denote by d R ∞ h ( pT ) p ∞ i the T -adic completion of R ∞ [( pT ) p ∞ ], and let A be a Banach ring associatedto (cid:0)d R ∞ h ( pT ) p ∞ i , ( T ) (cid:1) . Then A is uniform and A ◦ = d R ∞ h ( pT ) p ∞ i by Lemma 2.6. Hence A is aperfectoid ring in the sense of Fontaine by putting ̟ = T p . Indeed, we have p ∈ ̟ p A ◦ .Fix a prime number p > Witt-perfect rings due to Davis andKedlaya as in [7] and [8].
INITE ´ETALE EXTENSION OF TATE RINGS AND DECOMPLETION OF PERFECTOID ALGEBRAS 17
Definition 3.14 (Witt-perfect ring) . We say that a p -torsion free ring A is Witt-perfect , if theWitt-Frobenius map F : W p n ( A ) → W p n − ( A ) is surjective for all n > Lemma 3.15 ([7, Theorem 3.2]) . For a prime number p > , assume that A is a p -torsion freering. Then the following statements are equivalent.(1) A is a Witt-perfect ring.(2) The Frobenius endomorphism on A / ( p ) is surjective and for every a ∈ A , one can find b ∈ A such that b p ≡ pa (mod p A ) . Proposition 3.16.
Let A be a p -torsion free ring. Denote by c A the p -adic completion of A .(1) The following conditions are equivalent. ( a ) A is Witt-perfect and integrally closed (resp. completely integrally closed) in A [ p ] . ( b ) For any Banach ring R associated to ( c A , ( p )) (cf. Definition 2.28(4)), R is perfectoidin the sense of Fontaine and c A is open and integrally closed in R (resp. R ◦ = c A ).(2) Assume further that A is a p -adically separated valuation ring. Then the following condi-tions are equivalent. ( a ) A is Witt-perfect and of rank . ( b ) For any Banach ring K associated to ( c A , ( p )) , K is a perfectoid field and K ◦ = c A .Proof. (1): Let R be a Banach ring associated to ( c A , ( p )). First we assume that A is Witt-perfect and integrally closed in A [ p ]. Then c A is integrally closed in R by Corollary 2.6, c A / ( p )is semiperfect, and there is some ̟ ∈ c A such that ̟ p ≡ p mod p c A : In particular, p = ̟ p u holds for some unit u ∈ c A , because c A is p -adically Zariskian. Thus we also have some t ∈ c A forwhich t p ≡ ̟ (mod p c A ) (and so ̟ = t p u ′ for some unit u ′ ∈ c A ). Hence by Lemma 3.7, R ◦ / ( p )is semiperfect. Therefore, R is perfectoid. If further A is completely integrally closed in A [ p ],then c A is completely integrally closed in R by Corollary 2.6, and so R ◦ = c A . Consequently weobtain the implication ( a ) ⇒ ( b ). Conversely, we then assume the condition ( b ) (i.e. R is perfectoidand c A is open and integrally closed in R ). Then A is integrally closed in A by Corollary 2.6.Moreover, c A / ( p ) ∼ = A / ( p ) is semiperfect and there is some t ∈ c A for which t p ≡ p (mod p c A )by Lemma 3.12(2). Hence A is Witt-perfect. If further R ◦ = c A , then A is completely integrallyclosed in A [ p ] by Corollary 2.6. Consequently we find that ( b ) implies ( a ), as required.(2): First we assume ( a ). Then c A is a valuation ring of rank 1 with the fraction field c A [ p ].Thus, for a norm || · || on c A [ p ] associated with ( c A , ( p )), the spectral norm || · || sp is multiplicativeby Lemma 2.27. We equip c A [ p ] with the norm || · || sp . Then by the assertion (1), we find that c A [ p ] is perfectoid (and so it is a perfectoid field) and ( c A [ p ]) ◦ = c A . Hence ( b ) follows. Next weassume ( b ), conversely. Then in view of (1), A is Witt-perfect and completely integrally closed in A [ p − ]. Therefore, the value group is of rank 1. Hence ( a ) follows. (cid:3) Corollary 3.17.
Let A be a p -torsion free Witt-perfect ring that is integrally closed in A [ p ] .Denote by c A the p -adic completion of A . Denote by I and I ′ the ideals p ( p ) ⊂ A and p ( p ) ⊂ c A , respectively. Then one has I ′ = I c A and I = I . Proof.
By Lemma 3.15, there is a sequence { ̟ n } n ≥ in A such that ̟ = p , ̟ p ≡ ̟ mod ( p ),and ̟ pn +1 ≡ ̟ n mod ( p ) for every n ≥
0. By induction on n , we have ̟ pn +1 = ̟ n u n for some unit u n ∈ c A , because I ′ is contained in the Jacobson radical of c A . Let R be a Banach ring associatedto ( c A , ( p )). Then R ◦◦ = I ′ . Moreover, by Lemma 2.26(2) and Lemma 3.16(1), we may assumethat the norm on R is powermultiplicative. Hence I ′ is generated by { ̟ n } n ≥ . Therefore, one has I ′ = I c A and I ′ = I ′ . In particular, I c A = I c A . Next pick an element x ∈ I . Then by theequality stated just now, we have x = P ri =1 y i α i for some y i ∈ I and α i ∈ c A ( i = 1 , . . . , r ). Take a i ∈ A for which a i ≡ α i (mod p c A ) ( i = 1 , . . . , r ). Then we have x − P ri =1 y i a i ∈ p c A ∩ A = p A . Therefore, x ∈ I , as wanted. (cid:3) Almost Witt-perfect and almost perfectoid algebras.
Let us recall Andr´e’s almostperfectoid algebras (cf. Definition 3.5.2 and Proposition 3.5.4 in [1]).
Definition 3.18 (Almost perfectoid K h T p ∞ i -algebra) . Let K be a perfectoid field and let R bea uniform Banach K h T p ∞ i -algebra. Let m := T p ∞ K ◦◦ K ◦ h T p ∞ i be an ideal of K ◦ h T p ∞ i . Wesay that R is an almost perfectoid K h T p ∞ i -algebra , if the Frobenius endomorphism on R ◦ / ( p ) is m -almost surjective.We then introduce the following class of rings to establish a variant of Lemma 3.16 fitting foralmost mathematics. Definition 3.19 (Almost Witt-perfect ring) . Let A be a p -torsion free ring with an element g ∈ A admitting a compatible system of p -power roots g pn ∈ A . Then we say that A is ( g ) p ∞ -almostWitt-perfect , if the following conditions are satisfied.(1) The Frobenius endomorphism on A / ( p ) is ( g ) p ∞ -almost surjective.(2) For every a ∈ A and every n >
0, there is an element b ∈ A such that b p ≡ pg pn a (mod p A ). Proposition 3.20.
Let V be a p -adically separated p -torsion free valuation domain and let A bea p -torsion free V [ T p ∞ ] -algebra. Put g pn := T pn · ∈ A for every n ≥ and denote by b V and c A the p -adic completions of V and A , respectively. Then the following conditions are equivalent. ( a ) V is a Witt-perfect valuation domain of rank and A is ( g ) p ∞ -almost Witt-perfect andintegrally closed (resp. completely integrally closed) in A [ p ] . ( b ) There exist a perfectoid field K and an almost perfectoid K h T p ∞ i -algebra R with the fol-lowing properties: • K is a Banach ring associated to ( b V , ( p )) , the norm on K is multiplicative and K ◦ = b V ; • R is a Banach ring associated to ( c A , ( p )) , and c A is open and integrally closed in R (resp. R ◦ = c A ); • the bounded homomorphism K h T p ∞ i → R is induced by the ring map V [ T p ∞ ] → A .Proof. We first prove ( a ) ⇒ ( b ). We denote by || · || (resp. || · || ) the norm || · || b V , ( p ) ,p on b V [ p ] (resp.the norm || · || c A , ( p ) ,p on c A [ p ]), and let || · || , sp (resp. || · || , sp ) be the associated spectral seminorm.Notice that || · || , sp (resp. || · || , sp ) is equivalent to || · || b V , ( p ) ,p (resp. || · || c A , ( p ) ,p ) by Corollary 2.6and Lemma 2.26(2). We then equip b V [ p ] with || · || , sp (resp. c A [ p ] with || · || , sp ), and denote by INITE ´ETALE EXTENSION OF TATE RINGS AND DECOMPLETION OF PERFECTOID ALGEBRAS 19 K (resp. R ) the resulting Banach ring. Then the ring map V [ T p ∞ ] → A induces a bounded map K h T p ∞ i → R . Moreover, K is a perfectoid field with K ◦ = b V by Proposition 3.16(2), and c A forms an open and integrally closed subring of R by Corollary 2.6. Let ̟ ∈ K ◦ be a perfectoidpseudouniformizer that satisfies p = ̟ p u for some unit u ∈ K ◦ and admits a compatible systemof p -power roots (such an element ̟ exists by Lemma 3.12 and [4, Lemma 3.9]). Then, since theFrobenius endomorphism on c A / ( p ) is ( g ) p ∞ -almost surjective, the Frobenius endomorphism on R ◦ / ( p ) is ( ̟g ) p ∞ -almost surjective by Corollary 2.6, Corollary 3.6 and Lemma 3.7. So we concludethat R is an almost perfectoid K h T p ∞ i -algebra. The remaining part follows from Corollary 2.6and Lemma 2.11.Next we prove ( b ) ⇒ ( a ). Assume ( b ). In view of Corollary 2.6 and Proposition 3.16(2), it sufficesto show that A is ( g ) p ∞ -almost Witt-perfect. Let ̟ ∈ K ◦ be a perfectoid pseudouniformizer withthe property mentioned above. Since K ◦ = b V and the map K → R carries b V into c A , thereis some unit u ∈ c A such that p = ̟ p u . Moreover, the Frobenius endomorphism on R ◦ / ( p ) is( ̟g ) p ∞ -almost surjective by assumption. Hence by Corollary 3.6 and Lemma 3.7, the Frobeniusendomorphism on c A / ( p ) ∼ = A / ( p ) is ( g ) p ∞ -almost surjective. Thus it is enough to check thecondition (2) in Definition 3.19 for a = 1. Fix an integer n >
0. Then we have g pn p = ̟ p ( g pn u ),and there exist some b, c ∈ c A such that g pn u = b p + pc . Thus we have g pn p = ( ̟b ) p + ̟ p pc =( ̟b ) p + p u − c , which yields ( ̟b ) p ≡ pg pn (mod p c A ). Since c A / ( p ) ∼ = A / ( p ), the assertionfollows. (cid:3) Finite ´etale extension
Finite ´etale extsnsion and completeness.
Let A be a ring, let I ⊂ A be an ideal, andlet M be an A -module. Assume that A is I -adically complete and separated. Then one may askthe question: Is M also I -adically complete and separated? As is well known, if A is Noetherianand M is a finitely generated A -module, then the question is affirmative; see [20]. However, inthe absence of Noetherian property, this question is subtle and often require a careful argument(or even counterexamples exist). Now we consider the following case: there exists a ring extension A ⊂ A such that M is an A -submodule of some finite projective A -algebra B . Then in somesituations, the completeness is ensured by a condition on the trace map (even if M or I is notfinitely generated). A detailed account of the trace map for finite projective ring extension is foundin [12]. Proposition 4.1.
Let A be a ring and let B be a finite ´etale A -algebra. Let A ⊂ A be a subringwith an ideal I ⊂ A . Let B be an A -subalgebra of B such that B = S n ≥ ( B : I n ) . Assumethat there exists an integer c > such that Tr B/A ( tm ) ∈ A for every t ∈ I c and every m ∈ B .(1) If A is I -adically separated, then so is B .(2) If A is I -adically complete, then so is B .Proof. First note that since B is finite ´etale over A , the A -homomorphism B → Hom A ( B, A ); b Tr B/A ( b · )(4.1)is an isomorphism by [12, Corollary 4.6.8].Let us prove (1). Pick m ∈ T ∞ n =0 I n B . Since (4.1) is injective, it suffices to show thatTr B/A ( mx ) = 0 for an arbitrary element x ∈ B . Take l > I l x ⊂ B . Then for every n >
0, we have m ∈ I n + c + l B and thus, there exist t n,i ∈ I n , u n,i ∈ I c + l , and m n,i ∈ B ( i = 1 , . . . , r ) such that m = P ri =1 t n,i u n,i m n,i . HenceTr B/A ( mx ) = r X i =1 t n,i Tr B/A ( u n,i xm n,i ) ∈ I n A for every n >
0. Thus, since A is I -adically separated, we have Tr B/A ( mx ) = 0, as desired.Next we prove (2). Since B is a finite projective A -module, there exist A -homomorphisms s : B → A ⊕ d and π : A ⊕ d → B such that π ◦ s is the identity. Let us equip A (resp. A ⊕ d , resp. B ) with the topology such that { I n A } n ≥ (resp. { I n A ⊕ d } n ≥ , resp. { I n B } n ≥ ) forms a systemof fundamental open neighborhoods of 0. We consider these topologies in what follows. Since (4.1)is surjective, there exist b , . . . , b d ∈ B such that s ( x ) = (Tr B/A ( b x ) , . . . , Tr B/A ( b d x )) for every x ∈ B . Let { m n } n ≥ be a Cauchy sequence in B with respect to the I -adic topology. Then foreach i = 1 , . . . , d , { Tr B/A ( b i m n ) } n ≥ forms a Cauchy sequence in A . Hence by assumption, each { Tr B/A ( b i m n ) } n ≥ converges to some a i ∈ A . Then { s ( m n ) } n ≥ converges to ( a , . . . , a d ) ∈ A ⊕ d .Thus, { m n } n ≥ = { π ( s ( m n )) } n ≥ converges to π (( a , . . . , a d )) ∈ B . Since π (( a , . . . , a d )) − m n ∈ B for n ≫
0, we have π (( a , . . . , a d )) ∈ B . Hence the assertion follows. (cid:3) Proposition 4.2.
Let A be a ring with a nonzero divisor t and put A = A [ t ] . Let B be a finite´etale A -algebra. Let B ⊂ B be an A -subalgebra for which B = B [ t ] . Assume that there existssome l > such that Tr B/A ( t l b ) ∈ A for every b ∈ B . Denote by c A and c B the t -adic completionsof A and B , respectively. Then the natural A -algebra homomorphism B ⊗ A c A → c B inducesan isomorphism: ( B ⊗ A c A ) / (0) t − sat ∼ = −→ c B (4.2) (where (0) t -sat denotes the ( t ) -saturation of the ideal (0) ⊂ B ⊗ A c A ). In particular, the natural A -algebra homomorphism ( B ⊗ A c A )[ 1 t ] → ( c B )[ 1 t ] is an isomorphism.Proof. Since c B is t -torsion free, the map ϕ : B ⊗ A c A → c B induces a commutative diagram: B ⊗ A c A π / / ϕ * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ ( B ⊗ A c A ) / (0) t − sat e ϕ (cid:15) (cid:15) c B where π is the canonical projection map. We prove that e ϕ is an isomorphism. First we show that( B ⊗ A c A ) / (0) t − sat is t -adically complete and separeated. Let us apply Proposition 4.1 by setting I = ( t ). Put A ′ := ( c A )[ t ]. Notice that ( B ⊗ A c A ) / (0) t − sat is isomorphic to the c A -subalgebra C ⊂ ( B ⊗ A c A )[ t ] that is the image of B ⊗ A c A → ( B ⊗ A c A )[ t ], and ( B ⊗ A c A )[ t ] isidentified with the finite ´etale A ′ -algebra B ⊗ A A ′ . Since we haveTr B ⊗ A A ′ /A ′ ( b ⊗ A A ′ ) = Tr B/A ( b ) ⊗ A A ′ ( ∀ b ∈ B ) , it follows that every element c ∈ C satisfies Tr B ⊗ A A ′ /A ′ ( t l c ) ∈ c A . Hence C is t -adically completeand separated by Proposition 4.1, and therefore so is ( B ⊗ A c A ) / (0) t − sat , as wanted. Now notethat the t -adic completion \ B ⊗ A c A of B ⊗ A c A is naturally isomorphic to c B . Then by the INITE ´ETALE EXTENSION OF TATE RINGS AND DECOMPLETION OF PERFECTOID ALGEBRAS 21 universality of completion [14, Proposition 7.1.9 in Chapter 0], the isomorphism \ B ⊗ A c A ∼ = −→ c B factors as \ B ⊗ A c A b π −→ ( B ⊗ A c A ) / (0) t − sat e ϕ −→ c B , where the map b π is surjective, because π is so. Since e ϕ ◦ b π is an isomorphism, it follows that b π isan isomorphism. Thus, e ϕ is also an isomorphism. Finally, It readily follows that ( B ⊗ A c A )[ t ] → ( c B )[ t ] is an isomorphism. (cid:3) As the condition on the trace map in Proposition 4.2 is subtle, we will unravel some verifiablehypotheses on ring maps with a desired trace map in the next subsection.4.2.
Studies on preuniform pairs and the condition ( ∗ ) . Here we establish several basic prop-erties of preuniform pairs (cf. Definition 2.14). We especially investigate the following condition.
Definition 4.3.
Let f : ( A , I ) → ( B , J ) be a morphism of pairs. Then we say that f satisfies” the condition ( ∗ )”, if I = tA and J = tB for some t ∈ A and f satisfies the following axioms:(a) ( A , I ) is preuniform.(b) The ring map A [ t ] → B [ t ] induced by f is finite ´etale.(c) B is t -torsion free, and B ⊂ ( A ) ∗ B [ t ] . Example . Let V be a valuation ring with a non-zero element t ∈ V for which V is t -adicallyseparated. Set K := Frac( V ). Let L be a finite separable extension of K . Let W be the integralclosure of V in L . Then the morphism ( V, ( t )) → ( W, ( t )) satisfies ( ∗ ).A morphism ( A , ( t )) → ( B , ( t )) satisfying ( ∗ ) has the following good properties. In particular,the condition imposed on the trace map of Proposition 4.2 may be realized by it. Proposition 4.5.
Let f : ( A , ( t )) → ( B , ( t )) be a morphism of pairs that satisfies ( ∗ ) . Put A := A [ t ] and B := B [ t ] . Then the following assertions hold.(1) There exists an integer c > such that Tr B/A ( t c B ) ⊂ A .(2) There exist an integer l > , a finite free A -module F , and A -homomorphisms B → F → B whose composition is multiplication by t l . In particular, t l B is contained in a finitelygenerated A -submodule of B .(3) One has ( A ) ∗ B = ( B ) ∗ B .(4) The pair ( B , ( t )) is preuniform. To prove this, we need the following lemma.
Lemma 4.6.
Let A be a ring with a nonzero divisor t , and put A = A [ t ] . Let B be a finite ´etale A -algebra. Assume that A is integrally closed in A . Then for every integral element b ∈ B over A , Tr B/A ( b ) belongs to A .Proof of Lemma 4.6. A proof of the lemma in the special case that t is a prime number is given in[8, Lemma 2.6], but the same proof is valid for the general case. (cid:3) Here notice the following fact.
Lemma 4.7.
Keep the notation as in Proposition 4.5. Put A + := ( A ) + A and B + := ( A ) + B . Let f : A → B be the ring map induced by f , and let f + : A + → B + be the ring map such that f | A + factors through f + . Then B = B + [ t ] , and the morphism f + : ( A + , ( t )) → ( B + , ( t )) satisfies ( ∗ ) .Proof of Lemma 4.7. It is clear because B is integral over A and A = A [ t ]. (cid:3) Now let us start to prove Proposition 4.5.
Proof of Proposition 4.5.
We define the morphism f + : ( A + , ( t )) → ( B + , ( t )) as in Lemma 4.7.Then by Lemma 4.6, we have Tr B/A ( B + ) ⊂ A + . Thus, since ( A , ( t )) is preuniform, there existssome c > B/A ( t c B + ) ⊂ A . Meanwhile, we have tB ⊂ t ( A ) ∗ B ⊂ B + by Lemma2.2. Thus it holds that Tr B/A ( t c +1 B ) ⊂ A , which yields the assertion (1).Let us prove (2). Since B is finite projective over A , we have A -homomorphisms π : A ⊕ d → B and s : B → A ⊕ d such that π ◦ s is the identity. Now since (4.1) is surjective, there exist b , . . . , b d ∈ B such that s ( x ) = (Tr B/A ( b x ) , . . . , Tr B/A ( b d x )) for every x ∈ B . Hence by (1), there exists l > t l s ) | B factors through an A -homomorphism s l : B → A ⊕ d . Now there also exists aninteger l > t l π ) | A ⊕ d factors through an A -homomorphism π l : A ⊕ d → B . Then π l ◦ s l is multiplication by t l + l , which yields the claim.Next we prove (3). The containment ( A ) ∗ B ⊂ ( B ) ∗ B is easy to see. Let us show the reverseinclusion. Pick an element b ∈ B and assume that b is almost integral over B . Then, since B = B [ t ], there exists some m > t m ( P ∞ n =0 B · b n ) ⊂ B . Hence by the assertion (2), t l + m ( P ∞ n =0 B · b n ) is contained in a finitely generated A -submodule of B . Therefore, P ∞ n =0 A · b n is contained in a finitely generated A -submodule of B . Hence b ∈ B is almost integral over A , asdesired.Finally we prove (4). By the assertion (3), we have ( B ) + B ⊂ ( B ) ∗ B ⊂ ( A ) ∗ B . Hence in viewLemma 2.2, we have t ( B ) + B ⊂ t ( A ) ∗ B ⊂ ( A ) + B . On the other hand, by Lemma 4.7 and the assertion(2), there exists some l ′ > t l ′ ( A ) + B is contained in a finitely generated ( A ) + A -submodule N of ( A ) + B . Moreover, since ( A , ( t )) is preuniform and B = B [ t ], we have t c ′ N ⊂ B for some c ′ >
0. Thus, t c ′ + l ′ +1 ( B ) + B is contained in B . This yields the assertion. (cid:3) Corollary 4.8.
Let A be a preuniform Tate ring. Let f : A → B be a finite ´etale ring map. Equip B with the canonical structure as a Tate ring (cf. Lemma 2.17). Then the following assertionshold.(1) B is also preuniform. In particular, for any ring of definition A of A , ( A ) + B and ( A ) ∗ B are rings of definition of B .(2) For any topologically nilpotent unit t ∈ A , the morphism ( A ◦ , ( t )) → ( B ◦ , ( t )) satisfies ( ∗ ) .(3) B ◦ = ( A ◦ ) ∗ B .(4) If f is injective, then f − ( B ◦ ) = A ◦ .(5) If A is uniform, then so is B .Proof. The assertions (1), (2) and (3) are immediate consequences of Lemma 2.2. Lemma 2.17 andProposition 4.5. The assertion (5) follows from (1) and Proposition 4.1. Let us prove (4). Assumethat f is injective. Since the containment A ◦ ⊂ f − ( B ◦ ) is clear, it suffices to show the reverseinclusion. Pick a ∈ A for which f ( a ) ∈ B ◦ . Then, since B ◦ = ( A ◦ ) ∗ B by the assertion (3), there issome l > t l a n ∈ A ◦ for every n >
0. Hence we have a ∈ ( A ◦ ) ∗ A = A ◦ , as wanted. (cid:3) In the next theorem, we show that the condition ( ∗ ) is stable under completion. This propertyis quite important for our study (cf. Corollary 4.10 and Theorem 5.6). The theorem itself can beinterpreted as a practical form of Proposition 4.2. Theorem 4.9.
Keep the notation as in Proposition 4.5. Denote by c A and c B the t -adic comple-tions of A and B , respectively. Let b f : ( c A , ( t )) → ( c B , ( t )) be the morphism of pairs induced by f . Put A ′ := c A [ t ] and B ′ := c B [ t ] . Then the following assertions hold.(1) The natural A ′ -algebra homomorphism B ⊗ A A ′ → B ′ is an isomorphism. INITE ´ETALE EXTENSION OF TATE RINGS AND DECOMPLETION OF PERFECTOID ALGEBRAS 23 (2) b f : ( c A , ( t )) → ( c B , ( t )) also satisfies ( ∗ ) .Proof. (1) is a consequence of Proposition 4.2 and Proposition 4.5(1). Let us prove (2). ByProposition 2.3, the morphism b f : ( c A , ( t )) → ( c B , ( t )) satisfies (a) in Definition 4.3. Moreover, asa corollary of the assertion (1), one finds that b f also satisfies (b). Hence the remaining part is toshow that c B is contained in ( c A ) ∗ B ′ . To carry out this, we take a finitely generated A -submodule N ⊂ B such that t l B ⊂ N for some l > c N the t -adic completion of N . Then by Lemma 2.4, c N is viewed as an c A -submodule of c B such that(4.3) t l c B ⊂ c N . By applying the topological Nakayama’s lemma [20, Theorem 8.4], one finds that(4.4) c N is a finitely generated c A -module , because c N /t c N ∼ = N /tN is finitely generated over c A /t c A ∼ = A /tA . Combining (4 .
3) and (4 . c B is contained in a finitely generated c A -submodule of B ′ . In particularit holds that c B ⊂ ( c A ) ∗ B ′ , as wanted. (cid:3) Corollary 4.10.
Let ( A , ( t )) be a preuniform pair with the associated Tate ring A . Let c A bethe t -adic completion of A and let A be the Tate ring associated to ( c A , ( t )) . Suppose that B isa finite ´etale A -algebra and denote by B the finite ´etale A -algebra B ⊗ A A . Equip B and B withthe canonical structure as a Tate ring (cf. Lemma 2.17) respectively. Then the following assertionshold.(1) Let c B ◦ be the t -adic completion of B ◦ . Then the natural ring map ϕ : B → c B ◦ [ t ] is anisomorphism which induces an isomorphism B ◦ ∼ = −→ c B ◦ .(2) For the natural map ψ : B → B , it holds that ψ − ( B ◦ ) = B ◦ .Proof. We first prove (1). Let c A ◦ be the t -adic completion of A ◦ and put A ′ := c A ◦ [ t ] and B ′ := c B ◦ [ t ]. Then the natural ring map A → A ′ is an isomorphism by Lemma 2.4, and it induces c A ◦ ∼ = −→ A ◦ by Corollary 2.6. Since the morphism ( A ◦ , ( t )) → ( B ◦ , ( t )) satisfies ( ∗ ) by Corollary4.8(2), ϕ is an isomorphism by Theorem 4.9(1). Thus, ϕ induces an isomorphism of c A ◦ -algebras( A ◦ ) ∗B ∼ = −→ ( c A ◦ ) ∗ B ′ . Therefore it suffices to show that(4.5) B ◦ = ( A ◦ ) ∗B and c B ◦ = ( c B ◦ ) ∗ B ′ = ( c A ◦ ) ∗ B ′ . By Theorem 4.9(2), the morphism ( c A ◦ , ( t )) → ( c B ◦ , ( t )) also satisfies ( ∗ ). Hence (4.5) follows fromProposition 4.5(3) and Corollary 4.8(3), as wanted.Let us prove the assertion (2). Let B ′ denote the subring ψ − ( B ◦ ) of B . By Corollary 4.8(3),we have B ◦ = ( A ◦ ) ∗ B and B ◦ = ( A ◦ ) ∗B = ( c A ◦ ) ∗B . On the other hand, we have ψ (( A ◦ ) ∗ B ) ⊂ ( A ◦ ) ∗B ⊂ ( c A ◦ ) ∗B . Thus it follows that B ◦ ⊂ B ′ . We then prove the reverse inclusion. By the assertion (1), we havethe commutative diagram: B ◦ ∼ = −−−−→ c B ◦ y y B ∼ = −−−−→ ϕ c B ◦ [ t ] where the vertical arrows denote the inclusion maps. Hence the map ( ϕ ◦ ψ ) | B ′ factors through c B ◦ .Thus by Lemma 2.5, we have B ′ ⊂ B ◦ , as wanted. (cid:3) Finally, we remark the following.
Lemma 4.11.
Let ( A , ( t )) be a preuniform pair and let A ֒ → B be an integral ring extension suchthat B is t -torsion free. Denote by c A and c B the t -adic completions of A and B , respectively.Then c A → c B is an injective t -torsion free map.Proof. Let us prove the injectivity of the homomorphism c A → c B . Put A = A [ t ] and choose aninteger c > t c ( A ) + A ⊂ A . Then it suffices to prove that A ∩ t n + c B ⊂ t n A for every n >
0. Let x ∈ A ∩ t n + c B . Then one can write x = t n + c b for b ∈ B and hence b = xt n + c ∈ B ∩ A. As B is integral over A , we have t c b ∈ A . Hence we have x = t n ( t c b ) ∈ t n A , as required. (cid:3) The almost purity theorem for Witt-perfect rings
Almost ´etale ring map.
First we review the definition of almost ´etale ring maps and thestatement of the almost purity theorem, due to Davis and Kedlaya. We refer the reader to [15,Definition 3.1.1] for the following definitions. For the results concering classical ´etale ring maps,the recent book [12] is a good reference.
Definition 5.1.
Fix a basic setup (
R, I ) and let A → B be an R -algebra homomorphism.(1) A → B is called I -almost weakly unramified , if the diagonal map B ⊗ A B → B is I -almostflat.(2) A → B is called I -almost unramified , if the diagonal map B ⊗ A B → B is I -almost projective.(3) A → B is called I -almost weakly ´etale , if A → B is I -almost flat and I -almost weaklyunramified.(4) A → B is called I -almost ´etale , if A → B is I -almost flat and I -almost unramified.(5) A → B is called I -almost finite ´etale , if it is I -almost ´etale and B is an I -almost finitelypresented A -module.So far, we have been using the symbol ( A , I ) to emphasize that A is a ring of definitionof some Tate ring and I is its idempotent ideal. From now on, we will use ( A, I ) and ( A , I )interchangeably not to make the writing heavy. Before going further, we clarify the relationshipbetween ”almost integrality” and ” I -almost integrality” in a special case. Lemma 5.2.
Let A be a ring with a sequence of nonzero divisors { t n } n ≥ such that for every n ≥ we have t k n n +1 = t n u n for some k n ≥ and some unit u n ∈ A × . Put A := A [ t ] . Then, a ∈ ( A ) ∗ A implies that t n a ∈ ( A ) + A for every n ≥ . Moreover, the converse holds true if ( A , ( t )) is preuniform.Proof. Let us choose an element a ∈ A and let n > c n := Q ≤ i ≤ n − k i . Thenby assumption, t c n n = t v n for some unit v n ∈ A × . Suppose that a ∈ ( A ) ∗ A . Then there existssome d > t d a m ∈ A for every m ≥
1. Thus, we have ( t n a ) c n d ∈ A and therefore, t n a ∈ ( A ) + A . This yields the first assertion. To show the converse, we suppose that ( A , ( t )) ispreuniform and t m a ∈ ( A ) + A for every m ≥
1. Then we have t a c n = ( t n a ) c n v − n ∈ ( A ) + A . Thus,since c n → ∞ ( n → ∞ ), it is easy to check t a m ∈ ( A ) + A for every m ≥
1. Here by assumption,there exists some e > t e ( A ) + A ⊂ A . Hence we have a ∈ ( A ) ∗ A , as wanted. (cid:3) INITE ´ETALE EXTENSION OF TATE RINGS AND DECOMPLETION OF PERFECTOID ALGEBRAS 25
Corollary 5.3.
Let ( R, I ) be a basic setup and let A be an R -algebra. Assume that I admitsa sequence of elements { t n } n ≥ such that A is t -torsion free, IA is generated by the set of all t n ∈ A ( n ≥ and for every n ≥ we have t k n n +1 = t n u n for some k n > and some unit u n ∈ A × .Put A := A [ t ] . Then the following assertions hold.(1) Let A ′ ⊂ A be a subring containing ( A ) + A . If one has A ′ ⊂ ( A ) ∗ A , then the inclusion map ( A ) + A ֒ → A ′ is an I -almost isomorphism (in particular, ( A ) + A ֒ → ( A ) ∗ A is an I -almostisomorphism). Moreover, the converse holds true if ( A , ( t )) is preuniform.(2) Equip A with a linear topology so that A is the Tate ring associated to ( A , ( t )) . Let A → B be a module-finite extension of Tate rings as in Lemma 2.17. Then one has ( A ) + B ⊂ B ◦ and the inclusion map is an I -almost isomorphism.Proof. The assertion (1) follows from Lemma 5.2 immediately. To show (2), take a ring of definition B ⊂ B as in Lemma 2.17. Then B ◦ = ( B ) ∗ B and ( A ) + B = ( B ) + B , because B is integral over A .Hence the assertion follows from (1). (cid:3) In the situation of Corollary 5.3, one can obtain an ”almost” variant of Corollary 2.6.
Lemma 5.4.
Keep the notation and the assumption as in Corollary 5.3. Denote by c A the t -adiccompletion of A and put A ′ := c A [ t ] . Then the following conditions are equivalent. (a) The inclusion map A ֒ → ( A ) + A (resp. A ֒ → ( A ) ∗ A ) is I -almost isomorphic. (b) The inclusion map c A ֒ → ( c A ) + A ′ (resp. c A ֒ → ( c A ) ∗ A ′ ) is I -almost isomorphic.Proof. The proof of the lemma follows immediately from Proposition 2.3 and Corollary 5.3, becausethe sequence of ideals { t n A } n ≥ defines the same adic topology on A . (cid:3) Moreover, notice the following.
Lemma 5.5 ([24, Lemma 5.3(i)]) . Keep the notation and the assumption as in Corollary 5.3. Let M be an I -almost flat A -module. Then ( M a ) ∗ is t -torsion free.Proof. By the assumption on I and A , any I -almost zero element in Hom R ( I, M ) must be zero.Thus the argument in the proof of [24, Lemma 5.3(i)] still works under our setting. (cid:3) Next we prove a key result in this section. This is an important consequence of studies in § . Theorem 5.6.
Let ( R, I ) be a basic setup and let f : A → B be an R -algebra homomorphismwith an element t ∈ A . Denote by c A and c B the t -adic completions of A and B , respectively.Let b f : c A → c B be the R -algebra homomorphism induced by f . Assume that the morphism ofpairs f : ( A , ( t )) → ( B , ( t )) satisfies ( ∗ ) . Then the following assertions hold.(1) The following conditions are equivalent. ( a ) B is I -almost finitely generated and I -almost projective over A . ( b ) c B is I -almost finitely generated and I -almost projective over c A .(2) The following conditions are equivalent. ( a ) f : A → B is I -almost finite ´etale. ( b ) b f : c A → c B is I -almost finite ´etale. For proving this, we use the following lemma.
Lemma 5.7.
Keep the notation as in Theorem 5.6. Assume that f : ( A , ( t )) → ( B , ( t )) satisfies ( ∗ ) . Assume further that for every ε ∈ I and for every integer n > there exist a finite free A -module F and A -homomorphisms B → F → B whose composition is congruent to multiplicationby ε modulo ( t n ) . Then B is I -almost finitely generated and I -almost projective over A . Proof of Lemma 5.7.
Fix ε ∈ I arbitrarily. By Proposition 4.5(2), there exist an integer l > A -module F , and A -linear maps π l : F → B , s l : B → F such that π l ◦ s l ismultiplication by t l . On the other hand, by assumption we have A -linear maps π ε,l : F → B and s ε,l : B → F for some finite free A -module F , such that π ε,l ◦ s ε,l is congruent to multiplicationby ε modulo ( t l ). Thus one can define the A -linear maps f ε : B → B , x t l (cid:0) εx − ( π ε,l ◦ s ε,l )( x ) (cid:1) and s ε : B → F ⊕ F , x (cid:0) ( s l ◦ f ε )( x ) , s ε,l ( x ) (cid:1) . Consider the A -linear map π ε : F ⊕ F → B , ( a , a ) π l ( a ) + π ε,l ( a ) . Then π ε ◦ s ε is multiplication by ε . Hence the assertion follows. (cid:3) Now let us complete the proof of Theorem 5.6.
Proof of Theorem 5.6.
Put A := A [ t ], A ′ := c A [ t ], B := B [ t ], and B ′ := c B [ t ]. Notice that themorphism b f : ( c A , ( t )) → ( c B , ( t )) satisfies ( ∗ ) by Theorem 4.9. In particular, B ′ is finite ´etaleover A ′ .(1): We first show ( a ) ⇒ ( b ). Assume that ( a ) is satisfied. Then for an arbitrary ε ∈ I , thereexist A -linear maps π ε : A ⊕ d → B and s ε : B → A ⊕ d such that π ε ◦ s ε is multiplication by ε . Let b π ε : ( c A ) ⊕ d → c B and b s ε : c B → ( c A ) ⊕ d be the induced c A -linear maps. Then we have the naturalcommutative diagram of A -linear maps: B s ε −−−−→ A ⊕ d π ε −−−−→ B y y yc B −−−−→ b s ε ( c A ) ⊕ d −−−−→ c π ε c B , where the vertical maps become isomorphisms after base extension along A → A /t n A for anarbitrary n >
0. Hence b π ε ◦ b s ε is multiplication by ε modulo ( t n ). Thus, we can apply Lemma 5.7to this situation. Therefore ( b ) is satisfied.Next we show ( b ) ⇒ ( a ). Assume that ( b ) is satisfied. Like the proof of the inverse implication,it suffices to construct A -linear maps B → A ⊕ d and A ⊕ d → B whose composition B → B ismultiplication by ε modulo ( t n ) for an arbitrary ε ∈ I and an arbitrary n >
0. By assumption,there exist c A -linear maps b π ε : ( c A ) ⊕ d → b B and b s ε : c B → ( c A ) ⊕ d such that b π ε ◦ b s ε is multiplicationby ε . These maps induce A -linear maps π ε : A ⊕ d / ( t n ) → B / ( t n ) and s ε : B / ( t n ) → A ⊕ d / ( t n )such that π ε ◦ s ε is multiplication by ε . Since there exists a A -linear map π ε : A ⊕ d → B thatis a lift of π ε , we are reduced to constructing a A -linear map B → A ⊕ d that is a lift of s ε . Byinverting t , b s ε is extended to a A ′ -linear map b s ′ ε : B ′ → A ′⊕ d . Since B ′ is finite ´etale over A ′ , thereexist b b , . . . , b b d ∈ B ′ such that b s ′ ε ( x ) = (Tr B ′ /A ′ ( b b x ) , . . . , Tr B ′ /A ′ ( b b d x )) for every x ∈ B ′ (and thus,for each i = 1 , . . . , d , Tr B ′ /A ′ ( b b i x ) ∈ c A if x ∈ c B ). For an integer k >
0, we take b , . . . , b d ∈ B such that b b i ≡ b i mod t k c B for each i (cf. Remark 2.19). Pick m ∈ B . ThenTr B ′ /A ′ ( b i m ) = Tr B ′ /A ′ ( b b i m ) + t k (Tr B ′ /A ′ ( t k x i )) ( i = 1 , . . . , d ) INITE ´ETALE EXTENSION OF TATE RINGS AND DECOMPLETION OF PERFECTOID ALGEBRAS 27 for some x i ∈ c B . Now by Proposition 4.5(1), we have Tr B ′ /A ′ ( t l c B ) ⊂ c A for some l >
0. Thus wemay assume that Tr B ′ /A ′ ( t k x i ) ∈ c A ( i = 1 , . . . , d ) by increasing k if necessary. Hence we have(5.1) Tr B ′ /A ′ ( c i m ) ≡ Tr B ′ /A ′ ( b b i m ) mod t k c A ( i = 1 , . . . , d ) . In particular, Tr B ′ /A ′ ( b i m ) ∈ c A . Meanwhile, since B ⊗ A A ′ ∼ = B ′ by Theorem 4.9, the diagram of B -linear maps B Tr B/A −−−−→ A y y B ′ −−−−−→ Tr B ′ /A ′ A ′ commutes. Hence Tr B/A ( b i m ) ∈ A in view of Lemma 2.5. Therefore, one can define the A -linearmap s ε : B → A ⊕ d , m (cid:0) Tr B/A ( b m ) , . . . , Tr B/A ( b d m ) (cid:1) , which is a lift of s ε in view of (5.1).(2): First we assume that ( a ) is satisfied. Then c B is an I -almost finitely generated projective c A -module in view of the assertion (1). Hence by [15, Remark 2.4.12 and Proposition 2.4.18], itfollows that c B is an I -almost flat and I -almost finitely presented c A -module. On the other hand,since A ′ → B ′ is unramified by Theorem 4.9 and c A / ( t ) → c B / ( t ) is I -almost unramified by [15,Lemma 3.1.2], it follows from [15, Theorem 5.2.12], together with the fact that c B is an I -almostfinitely presented c A -module, that c A → c B is I -almost unramified. Thus we find that ( b ) issatisfied. The inverse implication ( b ) ⇒ ( a ) can be shown similarly. (cid:3) Proof of the almost purity theorem.
Now we are ready to prove the almost purity theoremby Davis and Kedlaya. In this subsection, we fix a prime number p >
0, and for any ring R , wedenote by b R the p -adic completion of R . Moreover for a ring map f : R → S , we denote by b f thering map b R → b S induced by f . Theorem 5.8 (Almost purity) . Let A be a Witt-perfect ring and f : A → B be a ring map.Put I := √ pA . Assume that the morphism f : ( A , ( p )) → ( B , ( p )) satisfies ( ∗ ) , A is integrallyclosed in A [ p ] , ( A ) + B [ p ] ⊂ B , and ( A , I ) is a basic setup . Then the following assertions hold.(1) B is also Witt perfect.(2) f : A → B is I -almost finite ´etale.Proof. Let A and B be Banach rings associated to complete Tate rings ( c A , ( p )) and ( c B , ( p ))respectively such that the map A → B induced by b f is bounded. Then by Proposition 3.16 andCorollary 3.17, A is perfectoid and A ◦◦ = I c A . Moreover, since ( c A , ( p )) → ( c B , ( p )) satisfies ( ∗ )by Theorem 4.9, A → B is finite ´etale, c B ⊂ ( c A ) ∗B , and ( c A ) ∗B = ( c B ) ∗B by Proposition 4.5(3).Meanwhile, we also have p ( B ) ∗ B = p ( A ) ∗ B ⊂ ( A ) + B ⊂ B by Lemma 2.2 and Proposition 4.5(3),which implies that p ( c B ) ∗B ⊂ c B by Proposition 2.3. Thus we find that p c B ⊂ p ( c A ) ∗B = p ( c B ) ∗B ⊂ c B . For example, this assumption is realized if A admits { p pn } n ≥ , A is p -adically Zariskian, or A is an algebraover a p -torsion free Witt-perfect valuation domain of rank 1 (cf. Example 3.1 and Lemma 3.3). Hence, the topology on B coincides with the canonical topology on c B [ p ] as a finitely generated A -module by Corollary 4.8(1). Thus, in view of Corollary 5.3, the maps c A ֒ → A ◦ and c B ֒ → B ◦ are I -almost isomorphic. Moreover, by almost purity theorem by Kedlaya-Liu [19, Theorem 3.6.21 and5.5.9], it follows that A ◦ → B ◦ is I -almost finite ´etale (and therefore so is b f ) and B is perfectoid. Inparticular, B is Witt-perfect by Lemma 3.7. Now the assertion (2) follows from Theorem 5.6. (cid:3) Remark 5.9.
Here is a way to check almost flatness for the extension A → B . Suppose that A [ p ] → B [ p ] is flat and A / ( p ) → B / ( p ) is I -almost flat. Then since p is a nonzero divisor onboth A and B , a simple discussion using the short exact sequence: 0 → A p −→ A → A / ( p ) → A i ( B , A / ( p )) = 0 for i >
0. By applying [15, Lemma 5.2.1] (see also [13, Lemma1.2.5] for the absolute version), one sees that B is a I -almost flat A -module.Let ( R, I ) be a basic setup and let A be an R -algebra. Let us denote by Alg a ( A ) the quotientcategory of the category of A -algebras Alg ( A ) by the Serre subcategory of objects in Alg ( A ) thatare I -almost zero. Then we define F . Et a ( A ) to be the full subcategory of Alg a ( A ) that consistof I -almost finite ´etale A -algebras. We also define F . Et ( A ) to be the category of finite ´etale A -algebras. Corollary 5.10.
Let A be a p -torsion free algebra over a p -torsion free Witt-perfect valuationdomain V of rank . Assume that A is Witt-perfect and integrally closed in A [ p ] . Put I := √ pA .Consider the basic setup ( A , I ) and the diagram of functors: (5.2) F . Et a ( A ) Φ −−−−→ F . Et a ( c A ) Φ y y Φ F . Et ( A [ p ]) −−−−→ Φ F . Et ( c A [ p ]) where Φ , Φ , Φ and Φ are given by the association B B ⊗ A a ( c A ) a , B ( B ) ∗ [ p ] , B B ⊗ A [ p ] c A [ p ] , and B ′ ( B ′ ) ∗ [ p ] , respectively (cf. [15, Lemma 3.1.2(i)] ). Then the followingassertions hold.(1) Φ and Φ yield equivalences of categories. Moreover, for an I -almost finite ´etale A -algebra C , the natural A -algebra homomorphism C ⊗ A c A → c C is I -almost isomorphic.(2) Assume that A is henselian along the ideal ( p ) . Then Φ and Φ yield equivalences ofcategories.Proof. We can equip b V [ p ] and c A [ p ] with norms so that b V [ p ] is a perfectoid field and c A [ p ] isa perfectoid b V [ p ]-algebra by Proposotion 3.20. Thus by the almost purity theorem for perfectoid b V [ p ]-algebras [24, Theorem 4.17, Theorem 5.2, Proposition 5.22, and Theorem 7.9] and Proposition4.5(3), Φ admits a quasi-inverse Ψ : F . Et ( c A [ p ]) → F . Et a ( c A ) given by the association B ′ ( c A ) ∗ aB ′ . More precisely, in view of [24, Lemma 5.6], for any B ′ ∈ ob( F . Et a ( c A )) we have(5.3) ( c A ) ∗ Φ ( B ′ ) = ( B ′ ) ∗ . On the other hand, by Theorem 5.8, we can also define a functor Ψ : F . Et ( A [ p ]) → F . Et a ( A )given by the association B ( A ) ∗ aB . Let us show that Ψ gives a quasi-inverse of Φ . Thenontrivial part is to prove the existence of a functorial isomorphism B ≃ (Ψ ◦ Φ )( B ) for B ∈ ob( F . Et a ( A )). We let C be the I -almost finite ´etale A -algebra ( B ) ∗ . Then C is p -torsion free INITE ´ETALE EXTENSION OF TATE RINGS AND DECOMPLETION OF PERFECTOID ALGEBRAS 29 by Lemma 5.5, and thus c A → c C is also I -almost finite ´etale by Theorem 5.6. Put C := C [ p ]and C ′ := c C [ p ]. Since C ′ = Φ (( c C ) a ), we have ( c C ) ∗ C ′ = ( c A ) ∗ C ′ = (( c C ) a ) ∗ by (5.3). Thereforethe inclusion map c C ֒ → ( c C ) ∗ C ′ is I -almost isomorphic. Hence by Lemma 5.4, C ֒ → ( C ) ∗ C isalso I -almost isomorphic. Applying the functor of almostification ( · ) a , we obtain an isomorphism C a ∼ = (Ψ ◦ Φ )( B ), which yields the desired isomorphism. Next we apply the functor Φ ◦ ( · ) a to the natural map ϕ : C ⊗ A c A → c C . Then we obtain the natural map ( C ⊗ A c A )[ p ] → c C [ p ],which is an isomorphism by Theorem 4.9. Thus, since Φ is fully faithful, we find that ϕ is I -almostisomorphic. Hence the assertion (1) follows. If further A is henselian along the ideal ( p ), then Φ yields an equivalence of categories due to [15, Proposition 5.4.53], and therefore (1) implies thatΦ yields an equivalence of categories (because (5.2) is essentially commutative). It completes theproof. (cid:3) Appendix: A historical remark on the (almost) purity theorem
In this appendix, we will provide some background history around the almost purity theorem aswell as its classical version, which is known as the
Purity over Noetherian local rings . Let us beginwith the definition of purity for schemes.
Definition 6.1 (Purity) . Let X be a scheme together with an open subset U ⊂ X . Let Et ( Y )denote the category of ´etale finite Y -schemes. If the restriction functor: F . Et ( X ) → F . Et ( U ); Y Y ′ := Y × X U is an equivalence of categories, then we say that ( X, U ) is a pure pair .To make this definition work, one is requited to put more conditions, such as normality. Let usrecall the following classical result due to Grothendieck.
Theorem 6.2 (Grothendieck) . Assume that ( R, m ) is a Noetherian local ring and let X := Spec( R ) and U := Spec( R ) \ { m } . Then the following assertions hold:(1) If R is a regular local ring with dim R ≥ , then ( X, U ) is pure.(2) If R is a complete intersection with dim R ≥ , then ( X, U ) is pure.Proof. The proof of the first statement is in [26, Tag 0BMA], while the second statement is in [26,Tag 0BPD]. (cid:3)
These results are of use to the proof of the so-called
Zariski-Nagata’s purity theorem whose proofis found in [26, Tag 0BMB].
Theorem 6.3 (Purity of branch locus) . Assume that f : X → S is a finite dominant morphism ofNoetherian integral schemes such that S is regular and X is normal. Then the ramification locusof f is of pure codimension one. The almost analogue of pure pair has been suggested in Gabber-Ramero’s treatise [Definition14.4.1][16], in which case the almost purity theorem takes the form of X = Spec( R ) and U =Spec( R [ p ]) for a certain big ring R . To the best of authors’ knowledge, the first appearance ofthe almost purity theorem is Tate’s work on p -divisible groups over local fields [28] and its higher-dimensional analog was studied by Faltings [9], who also outlined ideas of almost ring theory.Faltings gave another proof in [10], where he made an effective use of the Frobenius action oncertain local cohomology modules combined with his normalized length. We refer the reader toOlsson’s notes [23]. An ultimate version of the almost purity was proved by Scholze in [24], usingadic spaces. Acknowledgement .
The authors are grateful to Professor K. Fujiwara for encouragement andcomments on this paper.
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