Another proof of the almost purity theorem for perfectoid valuation rings
aa r X i v : . [ m a t h . A C ] J a n ANOTHER PROOF OF THE ALMOST PURITY THEOREM FORPERFECTOID VALUATION RINGS
SHINNOSUKE ISHIRO AND KAZUMA SHIMOMOTO
Abstract.
The almost purity theorem is central to the geometry of perfectoid spaces and hasnumerous applications in algebra and geometry. This result is known to have several different proofsin the case that the base ring is a perfectoid valuation ring. We give a new proof by exploiting thebehavior of Faltings’ normalized length under the Frobenius map. Introduction
The almost purity theorem is a key to the many foundational results of perfectoid spaces andit has been used to solve many outstanding problems. Recall that the almost purity theorem forperfectoid valuation rings was an intermediary step toward Scholze’s proof of the almost puritytheorem for general perfectoid rings with the help of algebraization theorems in the style of Elkikand Gabber-Ramero. Now let us turn our attention to the prehistory of the appearance of Scholze’sperfectoid spaces. The normalized length was introduced in [5] and played a role in the proof ofFaltings’ approach to the almost purity theorem. However, to the best of authors’ knowledge,the normalized length was used only to calculate the local cohomology of certain big rings withFrobenius action (see also [16]). Our aim is to give an application of the normalized length to offera new proof to the almost purity theorem for perfectoid valuation rings.
Theorem 1.1.
Let V be a perfectoid valuation ring with field of fractions K , and let W be theintegral closure of V in a finite separable extension of fields K → L .(1) The Frobenius endomorphism on W/pW is surjective.(2) V → W is faithfully flat and almost finite ´etale. An outline of two different proofs is given in [14, Theorem 3.7], where the first proof usesramification theory [7, Proposition 6.6.2 and Theorem 6.6.12], while the second proof makes areduction to the case of an absolutely integrally closed perfectoid valuation rings. Our new proofrelies on fundamental properties of the normalized length (see [15] and [16]). The reason for aninclusion of this proof is that the authors think that the almost purity theorem has fundamentalimportance and it is still worth recollecting classical approaches. The structure of this paper goesas follows.In §
2, we give a brief review on valuation rings and some properties of valuation topology.In §
3, we begin with almost ring theory. Some results on almost projective modules that arenot explicit in [7] are proved, which are of independent interest. Then we introduce the normalizedlength over a valuation ring of rank one. The most important aspect is contained in the
Frobeniuspull-back formula ; see Theorem 3.9.In §
4, we introduce perfectoid valuation rings with their tilts . The details of the proof of themain result are given.
Key words and phrases.
Almost purity, Frobenius pull-back formula, normalized length, valuation ring.2020
Mathematics Subject Classification : 13A18, 13A35, 13B40, 14G45. In §
5, we prove a result on the flatness of the Frobenous map on valuation rings in mixedcharacteristic. This result arises from the analysis of the main theorem given in §
4. An openquestion is also formulated. 2.
Valuation rings
Let K be a field and V be its subring. Then V is called a valuation ring if x ∈ V or x − ∈ V forevery nonzero element x ∈ K . As an another definition of a valuation ring, one can use a valuationof K . Proposition 2.1.
Suppose that V is a henselian local domain with field of fraction K , and that L/K is an algebraic extension and let W be the integral closure of V in L . Then W is a localdomain.Proof. See [6, Proposition 10.1.6] for the proof. (cid:3)
Proposition 2.2.
Let K be a field, let v be a valuation on K , let V be a valuation ring of v andlet L/K be an algebraic extension. Let W be the integral closure of V in L . If w is a valuation of L that dominates v , then the valuation ring of L associated to w is isomorphic to the localizationof W m for some maximal ideal m ⊂ W . In particular, W is itself a valuation ring under the samehypothesis on V as in Proposition 2.1.Proof. See [3, Chapter VI, §
10, Proposition 6] for the proof. (cid:3)
The valuation ring V is called microbial if it has a height-one prime ideal. Microbial valuationrings are characterized as follows. Proposition 2.3.
Let K be a field, let v be a valuation and let V be its valuation ring. Then thefollowing are equivalent.(1) V is a microbial valuation ring.(2) There exist a nonzero element x ∈ K such that a limit of the sequence of ( x n ) n ≥ is . Suchan element is called a ”topologically nilpotent element”.(3) The valuation topology defined by v coincides with the I -adic topology for some ideal I of V .Proof. This is well-known and the proof is found in [12, Theorem I.1.5.4]. (cid:3)
Remark 2.4. If ̟ ∈ K × is topologically nilpotent, then we have K = V [ ̟ ], and one can take I = ( ̟ ) in Proposition 2.3(3). Thus, V is complete with respect to the valuation topology definedby v if and only if V is ̟ -adically complete.We often use the following notation: Let A be a ring and let M be an A -module. Fix a prime p > pM = 0. Then M [ F ] is defined by setting M [ F ] = M as additive groupsand the A -module structure on M is given through the restriction of scalars via the Frobeniusendomorphism on A/pA . 3.
Almost ring theory
Basic notions.
Our references of almost ring theory are [7] and [9]. We call (
V, I ) a basicsetup if V is a commutative ring and I is an ideal such that I = I and I ⊗ V I is flat. Almost ringtheory works under a basic setup ( V, I ). Definition 3.1.
Let (
V, I ) be a basic setup, let R and S be V -algebras, and let M be a V -module.(1) M is called I -almost zero if IM = 0. We denote this by M ≈ NOTHER PROOF OF THE ALMOST PURITY THEOREM FOR PERFECTOID VALUATION RINGS 3 (2) M is called I -almost finitely generated if for every ǫ ∈ I , there exists a finitely generated V -module M ǫ and a V -module maps f ǫ : M ǫ → M such that both kernel and cokernel of f ǫ are annihilated by ǫ .(3) M is called I -almost finitely presented if for every ǫ ∈ I , there exists a finitely presented V -module M ǫ and a V -module maps f ǫ : M ǫ → M such that both kernel and cokernel of f ǫ are annihilated by ǫ .(4) M is called I -almost flat if Tor Vi ( M, N ) ≈ V -module N and i > M is called I -almost projective if Ext iV ( M, N ) ≈ V -module N and i > R → S is called I -almost unramified if S is an I -almost projective S ⊗ R S -module.(7) R → S is called I -almost ´etale if it is I -almost unramified and S is an I -almost flat R -module.(8) R → S is called I -almost finte ´etale if it is I -almost ´etale and S is an I -almost finitelypresented R -module. Lemma 3.2.
Let ( V, I ) be a basic setup and let M be a V -module. Then M is an I -almost finitelygenerated module if and only if for every finitely generated ideal I ⊂ I , there exists a finitelygenerated submodule M ⊂ M such that I M ⊂ M .Proof. This is [7, Proposition 2.3.10]. (cid:3)
The following proposition is an almost analogue of Nakayama’s lemma.
Proposition 3.3.
Let V be a commutative ring and let ̟ ∞ = S n> ( ̟ pn ) be an ideal of V forsome regular element ̟ ∈ V (so that ( V, ̟ ∞ ) is a basic setup). Let M be a V -module which is ̟ -adically separated. If M/̟M is a ̟ ∞ -almost finitely generated V /̟V -module, then M is a ̟ ∞ -almost finitely generated V -module.Proof. Since
M/̟M is I -almost finitely generated, there exists a finitely generated submodule M ′ ⊆ M/̟M such that ̟ pn ( M/̟M ) ⊆ M ′ for any fixed n >
0. After writing M ′ = ( V /̟V ) ω n + · · · + ( V /̟V ) ω n r , we have ̟ pn M ⊂ V ω n + · · · + V ω n r + ̟M ⊆ V ω n + · · · + V ω n r + \ k> ̟ ( pn + kpn ) M. Since M is ̟ -adically separated, we have T k> ̟ ( pn + kpn ) M = 0. So ̟ pn M ⊂ V ω n + · · · + V ω n r ,as desired. (cid:3) Proposition 3.4.
Let ( V, I ) be a basic setup and let M be an I -almost finitely generated V -module.Then M is almost projective if and only if for any ǫ ∈ I , there exist n ( ǫ ) ∈ N and a V -linear map M v ǫ −→ V n ( ǫ ) u ǫ −→ M such that v ǫ ◦ u ǫ = ǫ id M .Proof. We refer the reader to [7, Lemma 2.4.15] for the proof. (cid:3)
Let V → W be a ring extension. An element x ∈ W is called almost integral over V if thereexists a finitely generated V -submodule M ⊂ W such that P ∞ n =0 Rx n ⊂ M . We say that W is almost integral over V if all elements of W are almost integral over V . Now the following lemmaholds. We say that V is completely integrally closed in W if every element of W that is almostintegral over V belongs to V . Lemma 3.5.
Let ( V, I ) be a basic setup and suppose that W is a V -algebra such that W is an I -almost finitely generated V -module and suppose that one of the following conditions holds. S.ISHIRO AND K.SHIMOMOTO (1) V is an integral domain, or(2) there is a regular element ̟ ∈ V such that I = S n> ( ̟ pn ) .Then W is almost integral over V .Proof. Since W is I -almost finitely generated V -module, for all ǫ ∈ I there exist a finitely gen-erated V -module M ǫ and a V -homomorphism f ǫ : M ǫ → M such that Ker( f ǫ ) and Coker( f ǫ ) areannihilated by ǫ . This implies that for every an element x ∈ W , ǫx ∈ Im( f ǫ ). Now ǫ ( P ∞ n =0 V x n ) = P ∞ n =0 V ǫx n ⊂ Im( f ǫ ). Thus P ∞ n =0 V x n ⊂ ǫ Im( f ǫ ) ∼ = Im( f ǫ ) ⊆ W . (cid:3) Lemma 3.6.
Let V be a valuation ring of rank one. If V → W is an almost integral extensionand W is an integral domain, then V → W is an integral extension.Proof. Notice that Q ( V ) → Q ( W ) is an algebraic field extension. Let U be the integral closure of V in Q ( W ) and let M-Spec( U ) denote the set of all maximal ideals. Then for any maximal ideal m ∈ M-Spec( U ), U m is a valuation ring of rank one. Now we can write U = T m ∈ M-Spec( U ) U m by[11, Theorem 4.7]. Since every U m is completely integrally closed in Q ( U m ), U is also completelyintegrally closed in Q ( U ). This implies that U = W and hence, every element of W is integral over V . (cid:3) Normalized length over a valuation ring of rank one.
Let us briefly recall normalizedlength that is defined in the category of torsion modules over a valuation ring of rank one. Solet V be a valuation ring of rank one with a unique maximal ideal m V , which is assumed to beeither discrete or non-discrete. Denote by Mod { m V } the category of V -modules with support atthe maximal ideal { m V } . For M ∈ Mod { m V } , one can define a well-behaved length λ ∞ ( M ) ∈ R ≥ ∪ {∞} . If M does not belong to Mod { m V } , then put λ ∞ ( M ) = ∞ for completeness of logic. Now we recallthe definition and some properties. For more details, see [8], [9], [15] and [16].First of all, we start with finitely generated torsion V -modules. Let M ∈ Mod { m V } be afinitely generated module. Then the 0-th Fitting ideal F ( M ) a is well-defined, and we obtainthe isomorphism Div( V a ) ∼ = c Γ V where I
7→ | I | ([7, Lemma 6.1.19]), where c Γ V is the completion ofthe value group of V with respect to some uniform structure. Now we can define λ ∞ ( M ) := | F ( M ) a | ∈ R ≥ Remark that λ ∞ ( N ) ≤ λ ∞ ( M ) for N ⊆ M , where N is a finitely generated V -module. Moreover,we define for arbitrary modules M ∈ Mod { m V } as follows: λ ∞ ( M ) := sup (cid:8) λ ∞ ( N ) (cid:12)(cid:12) N ⊆ M is a finitely generated V -module (cid:9) ∈ R ≥ ∪ {∞} . The normalized length has the following properties.
Proposition 3.7.
The followings assertions hold.(1) Let → L → M → N → be a short exact sequence of V -modules in Mod { m V } . Then λ ∞ ( M ) = λ ∞ ( L ) + λ ∞ ( N ) . (2) For the above short exact sequence and a, b ∈ m V , λ ∞ ( abM ) ≤ λ ∞ ( aL ) + λ ∞ ( bN ) . (3) If λ ∞ ( M ) = 0 , then M is m V -almost zero, and if M is contained in a finitely presented V -module in Mod { m V } , then M = 0 . NOTHER PROOF OF THE ALMOST PURITY THEOREM FOR PERFECTOID VALUATION RINGS 5 (4) If a V -module M is an m V -almost finitely generated module in Mod { m V } , then λ ∞ ( bM ) < ∞ for any b ∈ m V .Proof. (1) is [9, Proposition 14.5.12], (2) is [9, Lemma 14.5.80], (3) is [9, Proposition 14.5.12] and[9, Theorem 14.5.75], and (4) is [9, Lemma 14.5.83]. (cid:3) Remark 3.8.
Even if λ ∞ ( M ) = 0, it may not be true that M = 0. Such an example is providedby M = V / m V .The most important part of normalized length is contained in the following theorem. Theorem 3.9 (Frobenius pull-back formula) . Suppose that M is a V /pV -module. Then λ ∞ ( M [ F ] ) = 1 p · λ ∞ ( M ) . Here, M [ F ] is regarded as a V -module through the restriction of scalars via the Frobenius endomor-phism on V /pV .Proof.
This is [8, Proposition 4.3.15]. (cid:3)
Proposition 3.10.
Assume that V is a valuation ring of rank one with field of fractions K and abasic setup ( V, I ) . Let W be the integral closure of V in a finite separable extension of K . Then W is an I -almost finitely presented V -module.Proof. The proof uses only basic properties of trace mapping and Fitting ideals. For details, werefer the reader to [7, Proposition 6.3.8]. (cid:3) The almost purity theorem in perfectoid valuation ring
Perfectoid valuation rings.
We introduce perfectoid valuation rings . Definition 4.1.
Let p > V be a valuation ring such that pV = V . We saythat V is a perfectoid valuation ring if the following hold:(1) The valuation of V is not discrete of rank one(2) V is complete in the valuation topology.(3) The Frobenius endomorphism on V /pV is surjective.The definition includes the case F p ⊂ V , in which case V is a perfect valuation ring. Remarkthat ( V, m V ) is a basic setup. Definition 4.2.
Let V be a ring and let p > V ♭ := lim ←− F V V /pV is called the tilt of V , where F V is the Frobenius endomorphism on V /pV .The tilting operation is well-behaved for perfectoid valuation rings. See [14, Lemma 3.4] for thenext proposition.
Proposition 4.3.
Let V be a perfectoid valuation ring. The following assertions hold.(1) There exists an element ̟ ∈ V such that ̟ p = pu for some u ∈ V × . Moreover, one canchoose ̟ such that ̟ ♭ := ( . . . , ̟ p , ̟ p , ̟ ) ∈ V ♭ is well-defined.(2) V is ̟ -adically complete and separated. S.ISHIRO AND K.SHIMOMOTO (3) There is a short exact sequence → V ♭ ( ̟ p ) ♭ −−−→ V ♭ pr V −−→ V /pV → where pr V is the projection onto the first factor. Moreover, this short exact sequence givesan isomorphism V /̟V ∼ = V ♭ /̟ ♭ V ♭ . Proof of the main theorem.
We recall that a ring map f : A → B is weakly ´etale if both A → B and the diagonal map µ B : B ⊗ A B → B are flat. Lemma 4.4. ([7, Theorem 3.5.13.])
Let f : A → B be a ring map of F p -algebras. If the map f isweakly ´etale, then F B/A : A [ F ] ⊗ A B → B [ F ] is an isomorphism.Proof. This is [7, Theorem 3.5.13]. (cid:3)
Lemma 4.5.
Let f : A → B be a ring homomorphism. Then f is surjective if and only if thereexists a faithfully flat A -algebra C such that C → C ⊗ A B is surjective.Proof. If f is surjective, we just take C = A . If there exists a faithfully flat A -algebra C such that C → C ⊗ A B is surjective, C ⊗ A Coker f = 0. Since C is faithfully flat, we have Coker f = 0. (cid:3) Lemma 4.6.
Let ( V, ̟ ∞ ) be a basic setup and let M be an almost finitely generated V -module.If M is an almost projective module, then the following holds. Assume that φ : P → M is ahomomorphism of V -modules, where P is a V -module and Coker φ is killed by ̟ pn for a fixedinteger n > . Then for any k > there exists g : M → P such that φ ◦ g = ̟ pk + pn id M .Proof. Take the exact sequence P → M → Coker φ →
0. Let M ′ be the image of φ . Then weobtain the short exact sequence 0 → K → P → M ′ →
0, which induces the long exact sequence0 → Hom V ( M ′ , K ) → Hom V ( M ′ , P ) → Hom V ( M ′ , M ′ ) → Ext V ( M ′ , K ) → · · · . Next we take 0 → M ′ → M → M/M ′ → · · · → Ext V ( M, K ) → Ext V ( M ′ , K ) → Ext V ( M/M ′ , K ) → · · · . Then ̟ pk Ext V ( M, K ) = 0 for any k > ̟ pn Ext V ( M/M ′ , K ) = 0. Sowe obtain ̟ pn + pk Ext V ( M ′ , K ) = 0. Considering the first long exact sequence, we can find anelement f ∈ Hom V ( M ′ , P ) such that φ ◦ f = ̟ pk + pn id M ′ . Finally, since ̟ pn M ⊂ M ′ , we obtain g = f ◦ ̟ pn , as desired. (cid:3) Proposition 4.7.
Let ( V, ̟ ∞ ) be a basic setup and let M be an almost finitely generated V -modulewith M/̟ n M is an almost projective V /̟ n V -module for any n > . Suppose that M and V are ̟ -adically complete. Then M is an almost projective V -module.Proof. Since M is almost finitely generated by assumption, we can find a V -module homomorphism p n : P n → M for any n >
0, such that P n is a finite free module and ̟ pn Coker p n = 0. Remarkthat P n is ̟ -adically complete. Since M/̟M is almost projective, there is a
V /̟V -module map s n : M/̟M → P n /̟P n such that s n ◦ ( p n mod ̟ ) = ̟ pn + pk id M/̟M by Lemma 4.6. We willconstruct an almost splitting s n : M/̟ M → P n /̟ P n out of s n . Set N := π − (Im( s n )),where π : P n → P n /̟ P n . Then N is a submodule of P n and the induced map N → M induces t : N /̟N → M/̟M and t : N /̟ N → M/̟ M . By assumption, ̟ pn Coker t = 0. NOTHER PROOF OF THE ALMOST PURITY THEOREM FOR PERFECTOID VALUATION RINGS 7
Since Coker t /̟ Coker t ∼ = Coker t , we obtain ̟ pn Coker t = ̟ Coker t . By using this identityrepeatedly, we get ̟ pn Coker t = ̟ l − ( l − pn Coker t . Since l − ( l − p n ≥ l ≫
0, we have ̟ pn Coker t = 0. Since M/̟ M is an almost projective V /̟ V -module, there is a V /̟ V -module map g : M/̟ M → N /̟ N such that t ◦ g = ̟ pn + pk id M/̟ M by Lemma 4.6. Then the composition of g with the natural map N /̟ N → P n /̟ P n will yield s n . In a similar manner, we can construct s n l +1 from s n l inductively. Finally, P n and M are ̟ -adically complete, so we obtain s n = lim ←− l> s n l such that p n ◦ s n = ̟ pn + pk id M .As n, k are chosen arbitrarily, we conclude that M is almost projective. (cid:3) Theorem 4.8.
Let V be a perfectoid valuation ring with field of fractions K , and let W be theintegral closure of V in a finite ´etale extension K → L .(1) W is also a perfectoid valuation ring.(2) V → W is faithfully flat and almost finite ´etale.Proof. First we prove the both assertions in the prime characteristic p > W is a valuation ring of rank one and its valuation is induced by the valuation of V . In order toprove the completeness of W , it suffices to prove that W is complete and separated in the ̟ -adictopology, where ̟ ∈ V is an element that defines the adic topology of V . Since ̟ is topologicallynilpotent in W , W is equipped with the ̟ -adic topology. Applying [13, Proposition 4.1], we findthat W is also ̟ -adically complete and separated. Finally, since K is perfect and K ֒ → L is a finite´etale extension, it follows that L is perfect and W is thus perfect. The Frobenius endomorphismon W/pW is surjective. So (1) holds. Next, since V → W is flat and local, it is faithfully flat. So itsuffices to show that V → W is almost unramified. Since V and W are microbial valuation rings,we have K = V [ ̟ ] and L = W [ ̟ ]. There are several ways for proving the almost unramifiednessof V → W , which we briefly mention below and indicate the sources. • The almost purity theorem in prime characteristic. The proof of this fact uses only basicfacts on the trace maps. The details are found in [7, Theorme 3.5.28]. • Ramification theory and differentials. This proof seems to work for only valuation rings.The proof is found in [7, Proposition 6.6.2]. • Galois theory of commutative rings. This idea was suggested in Andr´e’s paper (see [1,Proposition 3.4.2]).So the theorem is proved in the equal characteristic p > K has characteristic 0.(1): By the same reasoning of the case in characteristic p > W is a non-discrete valuationring of rank one and complete in the valuation topology. So it suffices to prove that the Frobeniusendomorphism on W/pW is surjective. First, note that the Frobenius surjectivity on
V /pV impliesthat the residue field of V is perfect. Noting that the strict henselization V → V sh is an ind-´etalelocal extension, V sh is a valuation ring of rank one and F V sh is surjective. Put W ′ := V sh ⊗ V W .Then W ′ is a reduced and normal ring such that W → W ′ is ind-´etale. In particular, the relativeFrobenius map(4.1) ( W/pW ) [ F ] ⊗ W/pW W ′ /pW ′ → ( W ′ /pW ′ ) [ F ] is an isomorphism. So Q ( W ′ ) ∼ = L × · · · × L n for some n > W ′ ∼ = W × · · · × W n , where each W i is the integral closure of V sh in L i . Noticing the Frobenius surjectivity on V sh /pV sh , supposethat we have proved that the Frobenius endomorphism F W i is surjective. Then F W ′ is surjective S.ISHIRO AND K.SHIMOMOTO and in view of Lemma 4.5 and (4 . F W is surjective. So we may assume that the residue fieldof V is separably closed, that is, algebraically closed. Finally after taking the completion d V sh , wemay assume that V has the algebraically closed residue field. Let ̟ be the element of V providedby Proposition 4.3. Set A := V /̟V, B := W/̟W and C := W/pW.
Then the Frobenius endomorphism on C induces an injection B ֒ → C [ F ] , and this injection inducesthe exact sequence of A -modules:(4.2) 0 → B F W −−→ C [ F ] → N → , where N is the cokernel of F W . Our goal is to show that N is almost zero. To this aim, let usfix an arbitrary element b ∈ m V . Then F W restricts to an injection bB → ( b p C ) [ F ] , which givesanother exact sequence:(4.3) 0 → bB F W −−→ ( b p C ) [ F ] → N b → N b its cokernel. Combining (4 .
2) and (4 . −−−−→ bB F W −−−−→ ( b p C ) [ F ] −−−−→ N b −−−−→ y y y −−−−→ B F W −−−−→ C [ F ] −−−−→ N −−−−→ A -module map N b → N are annihilated by b . Remark that wehave B ∈ Mod { m V } and λ ∞ ( bB ) < ∞ for any b ∈ m V by Proposition 3.7 and Proposition 3.10.Taking the length, we get(4.4) λ ∞ (( b p C ) [ F ] ) = λ ∞ ( bB ) + λ ∞ ( N b ) . It follow from Theorem 3.9 that(4.5) λ ∞ (( b p C ) [ F ] ) = p − · λ ∞ ( b p C ) . Since C admits a filtration given by Fil • ( W ) := ( ̟ k C | ≤ k ≤ p ), each of whose graded componentof the associated graded module gr • (C) is isomorphic to B , we deduce from Proposition 3.7 that(4.6) λ ∞ ( b p C ) ≤ p · λ ∞ ( bB ) . Combining (4 .
5) and (4 . λ ∞ (( b p C ) [ F ] ) ≤ λ ∞ ( bB ) . Then this together with (4 .
4) and Proposition 3 . λ ∞ ( N b ) = 0. That is to say, m V N b = 0. On the other hand, both kernel and cokernel of N b → N are annihilated by b and b ∈ m V is arbitrary, we find that m V N = 0.Since V / m V is algebraically closed and the unique maximal ideal of W is m V W (this follows fromthe fact that the valuation of V is non-discrete of rank one), it follows that V / m V → W/ m V W isa trivial extension. In other words,(4.7) W = m V W + V. Pick an arbitrary element c ∈ W . Then (4 .
7) gives us an element d ∈ V such that c − d ∈ m V W .Let c − d be the image of c − d under W ։ W/pW . Then since m V N = 0, it follows that c − d is NOTHER PROOF OF THE ALMOST PURITY THEOREM FOR PERFECTOID VALUATION RINGS 9 contained in the subring Im( F W ) ⊂ W/pW . However, as the Frobenius endomorphism on
V /pV issurjective, we have d ∈ V /pV ⊂ Im( F W ). Hence c ∈ Im( F W ) and Im( F W ) = W/pW , as desired.(2): By the same reason in positive characteristic, it is true that V → W is faithfully flat. Andby Proposition 3.10, W is an almost finitely presented V -module. Hence it suffices to show that V → W is almost unramified. By the assertion (1), we have(4.8) V /̟V ∼ = V ♭ /̟ ♭ V ♭ and W/̟W ∼ = W ♭ /̟ ♭ W ♭ . Then using (4 . V ♭ /̟ ♭ V ♭ → W ♭ /̟ ♭ W ♭ is almost finitely generated as a V ♭ -modulemap. So V ♭ → W ♭ is almost finitely generated by Proposition 3.3 and an almost integral extensionby Lemma 3.5. Thus, it is an integral extension by Lemma 3.6. As V ♭ is a perfect F p -algebra, Q ( V ♭ ) → Q ( W ♭ ) is finite separable. Then the discussion in the beginning of the proof gives that V ♭ → W ♭ is almost finite ´etale. This fact, combined with (4 . W/̟ ⊗ V/̟
W/̟ → W/̟ is almost projective. On the other hand, the almost finite generatedness of V → W impliesthat W/̟ n ⊗ V/̟ n W/̟ n → W/̟ n is almost finitely presented for n >
0. By [7, Proposition2.4.18 and Theorem 5.2.12(i)],
W/̟ n ⊗ V/̟ n W/̟ n → W/̟ n is almost projective for any n > Applying [13, Proposition 4.1], we find that W ⊗ V W is ̟ -adically complete and separated, so W is an almost projective W ⊗ V W -module in view of Proposition 4.7. It implies that W is almostunramified over V , as desired. (cid:3) Frobenius map on valuation rings
One of the main results of the paper [4] states that if V is a valuation domain such that F p ⊂ V ,then the Frobenius endomorphism on V is faithfully flat. We remark that any perfect F p -algebrahas this property. In the mixed characteristic case, we have the following result. Proposition 5.1.
Let V be a valuation domain of mixed characteristic p > . Assume that thereis an element ̟ ∈ V together with an element u ∈ V × such that ̟ p = pu . Then the Frobeniusendomorphism on V /pV induces a faithfully flat map
V /̟V → ( V /pV ) [ F ] .Proof. This proof is inspired by [8, Lemma 4.3.9]. By [11, Theorem 7.8], it suffices to show thatTor
V/̟V (cid:0) V /I, ( V /pV ) [ F ] (cid:1) = 0 for an arbitrary nonzero finitely generated ideal I ⊂ V . By [7,Lemma 6.1.14], we may assume that I = aV , where ̟V ⊂ aV . Then the short exact sequence0 → a ( V /̟V ) → V /̟V → V /aV → → Tor
V/̟V (cid:0) V /aV, ( V /pV ) [ F ] (cid:1) → a ( V /̟V ) ⊗ V/̟V ( V /pV ) [ F ] φ −→ ( V /pV ) [ F ] → V /aV ⊗ V/̟V ( V /pV ) [ F ] → . Therefore, it suffices to show that the map φ is injective. In other words, we show that the naturalmap(5.1) a ( V /̟V ) ⊗ V/̟V ( V /pV ) [ F ] → a ( V /pV ) [ F ] ∼ = ( a p V /pV ) [ F ] is an isomorphism. Since ̟V ⊂ aV , there is an element b ∈ V for which ̟ = ab . Then the naturalsurjection V a −→ a ( V /̟V ) yields an isomorphism
V /bV ∼ = a ( V /̟V ). In view of the identities ̟ p = a p b p and ̟ p = pu with u ∈ V × , we get a ( V /̟V ) ⊗ V/̟V ( V /pV ) [ F ] ∼ = V /bV ⊗ V/̟V ( V /pV ) [ F ] ∼ = ( V /pV ) [ F ] /b ( V /pV ) [ F ] ∼ = ( V /b p V ) [ F ] ∼ = ( a p V /pV ) [ F ] , One could alternatively apply [7, Theorem 5.2.12(ii)]. However, almost finite presentedness of W ⊗ V W → W allows us to use [7, Theorem 5.2.12(i)] whose proof is relatively simple. which is (5 . (cid:3) This proposition is partly inspired by [10, Theorem 3.2] which discusses the regularity of Noe-therian rings via the Frobenius map. Indeed, Proposition 5.1 would follow if one knows that V canbe written as a colimit of regular subrings. This is expected to be true in general, as it is known asa consequence of Zariski’s uniformization theorem. In [17], Zariski proved that any valuation ringcontaining Q is described as a colimit of smooth Q -subalgebras. See also [2].Suppose that V = lim −→ A i , where A i are regular. Without loss of generality, we may even assumethat ̟ ∈ A i . But then the Frobenius induces a faithfully flat map A i /̟A i → A i /pA i in view of[10, Theorem 3.2]. The same holds for lim −→ A i /̟A i → lim −→ A i /pA i . Therefore, V /̟V → V /pV isfaithfully flat.Another motivation of Proposition 5.1 comes from a slight modification of the proof of Theorem4.8. Keeping notation as in Theorem 4.8, we know that A → B → C [ F ] is flat, so that we have bA ⊗ A C [ F ] = bC [ F ] = ( b p C ) [ F ] . By applying base change to (4 . → bB → ( b p C ) [ F ] → bA ⊗ A N →
0, and the length computation yields that bA ⊗ A N and itshomomorphic image N are almost zero. We end this article with the following question. Question 1.
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Local uniformization on algebraic varieties , Ann. of Math. (1940), 852–896. Department of Mathematics, College of Humanities and Sciences, Nihon University, Setagaya-ku,Tokyo 156-8550, Japan
Email address : [email protected] Department of Mathematics, College of Humanities and Sciences, Nihon University, Setagaya-ku,Tokyo 156-8550, Japan
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